package ‘vares’ · computes value at risk and expected shortfall, two most popular measures of...
TRANSCRIPT
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Package ‘VaRES’February 19, 2015
Type Package
Title Computes value at risk and expected shortfall for over 100parametric distributions
Version 1.0
Date 2013-8-25
Author Saralees Nadarajah, Stephen Chan and Emmanuel Afuecheta
Maintainer Saralees Nadarajah <[email protected]>
Depends R (>= 2.15.0)
Description Computes Value at risk and expected shortfall, two most popular measures of finan-cial risk, for over one hundred parametric distributions, including all commonly known distribu-tions. Also computed are the corresponding probability density function and cumulative distri-bution function.
License GPL (>= 2)
NeedsCompilation no
Repository CRAN
Date/Publication 2013-08-27 08:07:57
R topics documented:VaRES-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4aep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6ast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8asylaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9asypower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11beard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13betaburr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15betaburr7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16betadist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17betaexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19betafrechet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20betagompertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1
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2 R topics documented:
betagumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23betagumbel2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24betalognorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25betalomax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26betanorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28betapareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29betaweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30BS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32burr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33burr7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37clg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38compbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39dagum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41dweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42expexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44expext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45expgeo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46explog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47explogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50exppois . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51exppower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52expweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55FR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56frechet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59genbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60genbeta2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61geninvbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63genlogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64genlogis3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65genlogis4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67genpareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68genpowerweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69genunif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71gev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72gompertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73gumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75gumbel2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76halfcauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77halflogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78halfnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80halfT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81HBlaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82HL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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R topics documented: 3
Hlogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84invbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86invexpexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87invgamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88kum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89kumburr7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91kumexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92kumgamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93kumgumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95kumhalfnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96kumloglogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97kumnormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99kumpareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100kumweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103lfr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104LNbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106logbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107logcauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108loggamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109logisexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111logisrayleigh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113loglaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114loglog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116loglogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117lognorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119lomax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Mlaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121moexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123moweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124MRbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125nakagami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129paretostable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130PCTAlaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132perks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133power1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134power2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136quad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137rgamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138RS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140schabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142stacygamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144TL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
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4 aep
TL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148tsp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152xie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Index 156
VaRES-package Computes value at risk and expected shortfall for over 100 parametricdistributions
Description
Computes Value at risk and expected shortfall, two most popular measures of financial risk, for overone hundred parametric distributions, including all commonly known distributions. Also computedare the corresponding probability density function and cumulative distribution function.
Details
Package: VaRESType: PackageVersion: 1.0Date: 2013-8-25License: GPL(>=2)
Author(s)
Saralees Nadarajah, Stephen Chan and Emmanuel Afuecheta
Maintainer: Saralees Nadarajah <[email protected]>
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
aep Asymmetric exponential power distribution
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aep 5
Description
Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric exponential powerdistribution due to Zhu and Zinde-Walsh (2009) given by
f(x) =
α
α∗K (q1) exp
[− 1
q1
∣∣∣ x2α∗
∣∣∣q1] , if x ≤ 0,
1− α1− α∗
K (q2) exp
[− 1
q2
∣∣∣∣ x
2− 2α∗
∣∣∣∣q2] , if x > 0
F (x) =
αQ
(1
q1
(| x |2α∗
)q1,
1
q1
), if x ≤ 0,
1− (1− α)Q
(1
q2
(| x |
2− 2α∗
)q2,
1
q2
), if x > 0
VaRp(X) =
−2α∗
[q1Q
−1
(p
α,
1
q1
)] 1q1
, if p ≤ α,
2 (1− α∗)[q2Q
−1
(1− p1− α
,1
q2
)] 1q2
, if p > α,
ESp(X) =
−2α∗
p
∫ p
0
[q1Q
−1
(v
α,
1
q1
)] 1q1
dv, if p ≤ α,
−2α∗
p
∫ α
0
[q1Q
−1
(v
α,
1
q1
)] 1q1
dv
+2 (1− α∗)
p
∫ p
α
[q2Q
−1
(1− v1− α
,1
q2
)] 1q2
dv, if p > α
for −∞ < x < ∞, 0 < p < 1, 0 < α < 1, the scale parameter, q1 > 0, the first shape parame-ter, and q2 > 0, the second shape parameter, where α∗ = αK (q1) / {αK (q1) + (1− α)K (q2)},K(q) = 1
2q1/qΓ(1+1/q), Q(a, x) =
∫∞xta−1 exp (−t) dt/Γ(a) denotes the regularized complemen-
tary incomplete gamma function, Γ(a) =∫∞
0ta−1 exp (−t) dt denotes the gamma function, and
Q−1(a, x) denotes the inverse of Q(a, x).
Usage
daep(x, q1=1, q2=1, alpha=0.5, log=FALSE)paep(x, q1=1, q2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)varaep(p, q1=1, q2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)esaep(p, q1=1, q2=1, alpha=0.5)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
alpha the value of the scale parameter, must be in the unit interval, the default is 0.5
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6 arcsine
q1 the value of the first shape parameter, must be positive, the default is 1
q2 the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)daep(x)paep(x)varaep(x)esaep(x)
arcsine Arcsine distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the arcsine distribution given by
f(x) =1
π√
(x− a)(b− x),
F (x) =2
πarcsin
(√x− ab− a
),
VaRp(X) = a+ (b− a) sin2(πp
2
),
ESp(X) = a+b− ap
∫ p
0
sin2(πv
2
)dv
for a ≤ x ≤ b, 0 < p < 1, −∞ < a <∞, the first location parameter, and −∞ < a < b <∞, thesecond location parameter.
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arcsine 7
Usage
darcsine(x, a=0, b=1, log=FALSE)parcsine(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)vararcsine(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)esarcsine(p, a=0, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first location parameter, can take any real value, the default iszero
b the value of the second location parameter, can take any real value but must begreater than a, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)darcsine(x)parcsine(x)vararcsine(x)esarcsine(x)
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8 ast
ast Generalized asymmetric Student’s t distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized asymmetric Student’st distribution due to Zhu and Galbraith (2010) given by
f(x) =
α
α∗K (ν1)
[1 +
1
ν1
( x
2α∗
)2]− ν1+1
2
, if x ≤ 0,
1− α1− α∗
K (ν2)
[1 +
1
ν2
(x
2 (1− α∗)
)2]− ν2+1
2
, if x > 0
F (x) = 2αFν1
(min(x, 0)
2α∗
)− 1 + α+ 2(1− α)Fν2
(max(x, 0)
2− 2α∗
),
VaRp(X) = 2α∗F−1ν1
(min(p, α)
2α
)+ 2 (1− α∗)F−1
ν2
(max(p, α) + 1− 2α
2− 2α
),
ESp(X) =2α∗
p
∫ p
0
F−1ν1
(min(v, α)
2α
)dv +
2 (1− α∗)p
∫ p
0
F−1ν2
(max(v, α) + 1− 2α
2− 2α
)dv
for −∞ < x < ∞, 0 < p < 1, 0 < α < 1, the scale parameter, ν1 > 0, the first de-gree of freedom parameter, and ν2 > 0, the second degree of freedom parameter, where α∗ =αK (ν1) / {αK (ν1) + (1− α)K (ν2)}, K(ν) = Γ ((ν + 1)/2) / [
√πνΓ(ν/2)], Fν(·) denotes the
cdf of a Student’s t random variable with ν degrees of freedom, and F−1ν (·) denotes the inverse of
Fν(·).
Usage
dast(x, nu1=1, nu2=1, alpha=0.5, log=FALSE)past(x, nu1=1, nu2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)varast(p, nu1=1, nu2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)esast(p, nu1=1, nu2=1, alpha=0.5)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedalpha the value of the scale parameter, must be in the unit interval, the default is 0.5nu1 the value of the first degree of freedom parameter, must be positive, the default
is 1nu2 the value of the second degree of freedom parameter, must be positive, the de-
fault is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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asylaplace 9
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dast(x)past(x)varast(x)esast(x)
asylaplace Asymmetric Laplace distribution
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10 asylaplace
Description
Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric Laplace distributiondue to Kotz et al. (2001) given by
f(x) =
κ√
2
τ (1 + κ2)exp
(−κ√
2
τ|x− θ|
), if x ≥ θ,
κ√
2
τ (1 + κ2)exp
(−√
2
κτ|x− θ|
), if x < θ,
F (x) =
1− 1
1 + κ2exp
(κ√
2(θ − x)
τ
), if x ≥ θ,
κ2
1 + κ2exp
(√2(x− θ)κτ
), if x < θ,
VaRp(X) =
θ − τ√
2κlog[(1− p)
(1 + κ2
)], if p ≥ κ2
1+κ2 ,
θ +κτ√
2log[p(1 + κ−2
)], if p < κ2
1+κ2 ,
ESp(X) =
θ
p+ θ − τ√
2κlog(1 + κ2
)+
√2τ(1 + 2κ2
)2κ (1 + κ2) p
log(1 + κ2
)−√
2τκ log κ
(1 + κ2) p− θκ2
(1 + κ2) p+
τ(1− κ4
)√
2κ (1 + κ2) p
−τ(1− p)√2κp
+τ(1− p)√
2κplog(1− p), if p ≥ κ2
1+κ2 ,
θ +κτ√
2log(1 + κ−2
)+κτ√
2(log p− 1), if p < κ2
1+κ2
for −∞ < x < ∞, 0 < p < 1, −∞ < θ < ∞, the location parameter, κ > 0, the first scaleparameter, and τ > 0, the second scale parameter.
Usage
dasylaplace(x, tau=1, kappa=1, theta=0, log=FALSE)pasylaplace(x, tau=1, kappa=1, theta=0, log.p=FALSE, lower.tail=TRUE)varasylaplace(p, tau=1, kappa=1, theta=0, log.p=FALSE, lower.tail=TRUE)esasylaplace(p, tau=1, kappa=1, theta=0)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
theta the value of the location parameter, can take any real value, the default is zero
kappa the value of the first scale parameter, must be positive, the default is 1
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asypower 11
tau the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dasylaplace(x)pasylaplace(x)varasylaplace(x)esasylaplace(x)
asypower Asymmetric power distribution
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12 asypower
Description
Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric power distributiondue to Komunjer (2007) given by
f(x) =
δ1/λ
Γ(1 + 1/λ)exp
[− δ
aλ|x|λ
], if x ≤ 0,
δ1/λ
Γ(1 + 1/λ)exp
[− δ
(1− a)λ|x|λ
], if x > 0
F (x) =
a− aI
(δ
aλ
√λ|x|λ, 1/λ
), if x ≤ 0,
a− (1− a)I(
δ
(1− a)λ
√λ|x|λ, 1/λ
), if x > 0
VaRp(X) =
−[aλ
δ√λ
]1/λ [I−1
(1− p
a,
1
λ
)]1/λ
, if p ≤ a,
−[
(1− a)λ
δ√λ
]1/λ [I−1
(1− 1− p
1− a,
1
λ
)]1/λ
, if p > a,
ESp(X) =
−1
p
[aλ
δ√λ
]1/λ ∫ p
0
[I−1
(1− v
a,
1
λ
)]1/λ
dv, if p ≤ a,
−1
p
[aλ
δ√λ
]1/λ ∫ a
0
[I−1
(1− v
a,
1
λ
)]1/λ
dv
−1
p
[(1− a)λ
δ√λ
]1/λ ∫ p
a
[I−1
(1− 1− v
1− a,
1
λ
)]1/λ
dv, if p > a
for −∞ < x < ∞, 0 < p < 1, 0 < a < 1, the first scale parameter, δ > 0, the second scaleparameter, and λ > 0, the shape parameter, where I(x, γ) = 1
Γ(γ)
∫ x√γ0
tγ−1 exp(−t)dt.
Usage
dasypower(x, a=0.5, lambda=1, delta=1, log=FALSE)pasypower(x, a=0.5, lambda=1, delta=1, log.p=FALSE, lower.tail=TRUE)varasypower(p, a=0.5, lambda=1, delta=1, log.p=FALSE, lower.tail=TRUE)esasypower(p, a=0.5, lambda=1, delta=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be in the unit interval, the default is0.5
delta the value of the second scale parameter, must be positive, the default is 1
lambda the value of the shape parameter, must be positive, the default is 1
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beard 13
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dasypower(x)pasypower(x)varasypower(x)esasypower(x)
beard Beard distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Beard distribution due to Beard(1959) given by
f(x) =a exp(bx) [1 + aρ]
ρ−1/b
[1 + aρ exp(bx)]1+ρ−1/b
,
F (x) = 1− [1 + aρ]ρ−1/b
[1 + aρ exp(bx)]ρ−1/b
,
VaRp(X) =1
blog
[1 + aρ
aρ(1− p)ρ1/b− 1
aρ
],
ESp(X) =1
pb
∫ p
0
log
[− 1
aρ+
1 + aρ
aρ(1− v)ρ1/b
]dv
for x > 0, 0 < p < 1, a > 0, the first scale parameter, b > 0, the second scale parameter, andρ > 0, the shape parameter.
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14 beard
Usage
dbeard(x, a=1, b=1, rho=1, log=FALSE)pbeard(x, a=1, b=1, rho=1, log.p=FALSE, lower.tail=TRUE)varbeard(p, a=1, b=1, rho=1, log.p=FALSE, lower.tail=TRUE)esbeard(p, a=1, b=1, rho=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be positive, the default is 1
b the value of the second scale parameter, must be positive, the default is 1
rho the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbeard(x)pbeard(x)varbeard(x)esbeard(x)
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betaburr 15
betaburr Beta Burr distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Burr distribution due toParana\’iba et al. (2011) given by
f(x) =babd
B(c, d)xbd+1
[1 + (x/a)
−b]−c−d
,
F (x) = I 1
1+(x/a)−b(c, d),
VaRp(X) = a[I−1p (c, d)
]1/b [1− I−1
p (c, d)]−1/b
,
ESp(X) =a
p
∫ p
0
[I−1v (c, d)
]1/b [1− I−1
v (c, d)]−1/b
dv
for x > 0, 0 < p < 1, a > 0, the scale parameter, b > 0, the first shape parameter, c > 0,the second shape parameter, and d > 0, the third shape parameter, where Ix(a, b) =
∫ x0ta−1(1 −
t)b−1dt/B(a, b) denotes the incomplete beta function ratio,B(a, b) =∫ 1
0ta−1(1−t)b−1dt denotes
the beta function, and I−1x (a, b) denotes the inverse function of Ix(a, b).
Usage
dbetaburr(x, a=1, b=1, c=1, d=1, log=FALSE)pbetaburr(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)varbetaburr(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esbetaburr(p, a=1, b=1, c=1, d=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the scale parameter, must be positive, the default is 1
b the value of the first shape parameter, must be positive, the default is 1
c the value of the second shape parameter, must be positive, the default is 1
d the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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16 betaburr7
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetaburr(x)pbetaburr(x)varbetaburr(x)esbetaburr(x)
betaburr7 Beta Burr XII distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Burr XII distribution givenby
f(x) =kcxc−1
B(a, b)
[1− (1 + xc)
−k]a−1
(1 + xc)−bk−1
,
F (x) = I1−(1+xc)−k(a, b),
VaRp(X) ={[
1− I−1p (a, b)
]−1/k − 1}1/c
,
ESp(X) =1
p
∫ p
0
{[1− I−1
v (a, b)]−1/k − 1
}1/c
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, c > 0,the third shape parameter, and k > 0, the fourth shape parameter.
Usage
dbetaburr7(x, a=1, b=1, c=1, k=1, log=FALSE)pbetaburr7(x, a=1, b=1, c=1, k=1, log.p=FALSE, lower.tail=TRUE)varbetaburr7(p, a=1, b=1, c=1, k=1, log.p=FALSE, lower.tail=TRUE)esbetaburr7(p, a=1, b=1, c=1, k=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first shape parameter, must be positive, the default is 1
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betadist 17
b the value of the second shape parameter, must be positive, the default is 1
c the value of the third shape parameter, must be positive, the default is 1
k the value of the fourth shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetaburr7(x)pbetaburr7(x)varbetaburr7(x)esbetaburr7(x)
betadist Beta distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta distribution given by
f(x) =xa−1(1− x)b−1
B(a, b),
F (x) = Ix(a, b),VaRp(X) = I−1
p (a, b),
ESp(X) =1
p
∫ p
0
I−1v (a, b)dv
for 0 < x < 1, 0 < p < 1, a > 0, the first parameter, and b > 0, the second shape parameter.
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18 betadist
Usage
dbetadist(x, a=1, b=1, log=FALSE)pbetadist(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varbetadist(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esbetadist(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be positive, the default is 1
b the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetadist(x)pbetadist(x)varbetadist(x)esbetadist(x)
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betaexp 19
betaexp Beta exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta exponential distribution dueto Nadarajah and Kotz (2006) given by
f(x) =λ exp(−bλx)
B(a, b)[1− exp(−λx)]
a−1,
F (x) = I1−exp(−λx)(a, b),
VaRp(X) = − 1
λlog[1− I−1
p (a, b)],
ESp(X) = − 1
pλ
∫ p
0
log[1− I−1
v (a, b)]dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, andλ > 0, the scale parameter, where Ix(a, b) =
∫ x0ta−1(1− t)b−1dt/B(a, b) denotes the incomplete
beta function ratio, B(a, b) =∫ 1
0ta−1(1− t)b−1dt denotes the beta function, and I−1
x (a, b) denotesthe inverse function of Ix(a, b).
Usage
dbetaexp(x, lambda=1, a=1, b=1, log=FALSE)pbetaexp(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varbetaexp(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esbetaexp(p, lambda=1, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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20 betafrechet
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetaexp(x)pbetaexp(x)varbetaexp(x)esbetaexp(x)
betafrechet Beta Frechet distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Fr\’echet distribution due toBarreto-Souza et al. (2011) given by
f(x) =ασα
xα+1B(a, b)exp
{−a(σx
)α}[1− exp
{−(σx
)α}]b−1
,
F (x) = Iexp{−(σx )α}(a, b),
VaRp(X) = σ[− log I−1
p (a, b)]−1/α
,
ESp(X) =σ
p
∫ p
0
[− log I−1
v (a, b)]−1/α
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, σ > 0, the scale parameter, b > 0, thesecond shape parameter, and α > 0, the third shape parameter.
Usage
dbetafrechet(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)pbetafrechet(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varbetafrechet(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esbetafrechet(p, a=1, b=1, alpha=1, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
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betagompertz 21
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
alpha the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetafrechet(x)pbetafrechet(x)varbetafrechet(x)esbetafrechet(x)
betagompertz Beta Gompertz distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Gompertz distribution dueto Cordeiro et al. (2012b) given by
f(x) =bη exp(bx)
B(c, d)exp (dη) exp [−dη exp(bx)] {1− exp [η − η exp(bx)]}c−1
,
F (x) = I1−exp[η−η exp(bx)](c, d),
VaRp(X) =1
blog
{1− 1
ηlog[1− I−1
p (c, d)]}
,
ESp(X) =1
pb
∫ p
0
log
{1− 1
ηlog[1− I−1
v (c, d)]}
dv
for x > 0, 0 < p < 1, b > 0, the first scale parameter, η > 0, the second scale parameter, c > 0,the first shape parameter, and d > 0, the second shape parameter.
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22 betagompertz
Usage
dbetagompertz(x, b=1, c=1, d=1, eta=1, log=FALSE)pbetagompertz(x, b=1, c=1, d=1, eta=1, log.p=FALSE, lower.tail=TRUE)varbetagompertz(p, b=1, c=1, d=1, eta=1, log.p=FALSE, lower.tail=TRUE)esbetagompertz(p, b=1, c=1, d=1, eta=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
b the value of the first scale parameter, must be positive, the default is 1
eta the value of the second scale parameter, must be positive, the default is 1
c the value of the first shape parameter, must be positive, the default is 1
d the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetagompertz(x)pbetagompertz(x)varbetagompertz(x)esbetagompertz(x)
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betagumbel 23
betagumbel Beta Gumbel distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Gumbel distribution due toNadarajah and Kotz (2004) given by
f(x) =1
σB(a, b)exp
(µ− xσ
)exp
[−a exp
µ− xσ
]{1− exp
[− exp
µ− xσ
]}b−1
,
F (x) = Iexp[− exp µ−xσ ](a, b),
VaRp(X) = µ− σ log[− log I−1
p (a, b)],
ESp(X) = µ− σ
p
∫ p
0
log[− log I−1
v (a, b)]dv
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,a > 0, the first shape parameter, and b > 0, the second shape parameter.
Usage
dbetagumbel(x, a=1, b=1, mu=0, sigma=1, log=FALSE)pbetagumbel(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varbetagumbel(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esbetagumbel(p, a=1, b=1, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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24 betagumbel2
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetagumbel(x)pbetagumbel(x)varbetagumbel(x)esbetagumbel(x)
betagumbel2 Beta Gumbel 2 distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Gumbel II distribution givenby
f(x) =abx−a−1
B(c, d)exp
(−bdx−a
) [1− exp
(−bx−a
)]c−1,
F (x) = I1−exp(−bx−a)(c, d),
VaRp(X) = b1/a{− log
[1− I−1
p (c, d)]}−1/a
,
ESp(X) =b1/a
p
∫ p
0
{− log
[1− I−1
v (c, d)]}−1/a
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the scale parameter, c > 0, thesecond shape parameter, and d > 0, the third shape parameter.
Usage
dbetagumbel2(x, a=1, b=1, c=1, d=1, log=FALSE)pbetagumbel2(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)varbetagumbel2(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esbetagumbel2(p, a=1, b=1, c=1, d=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedb the value of the scale parameter, must be positive, the default is 1a the value of the first shape parameter, must be positive, the default is 1c the value of the second shape parameter, must be positive, the default is 1d the value of the third shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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betalognorm 25
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetagumbel2(x)pbetagumbel2(x)varbetagumbel2(x)#esbetagumbel2(x)
betalognorm Beta lognormal distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta lognormal distribution dueto Castellares et al. (2013) given by
f(x) =1
σxB(a, b)φ
(log x− µ
σ
)Φa−1
(log x− µ
σ
)Φb−1
(µ− log x
σ
),
F (x) = IΦ( log x−µσ )(a, b),
VaRp(X) = exp[µ+ σΦ−1
(I−1p (a, b)
)],
ESp(X) =exp(µ)
p
∫ p
0
exp[σΦ−1
(I−1v (a, b)
)]dv
for x > 0, 0 < p < 1, −∞ < µ < ∞, the location parameter, σ > 0, the scale parameter, a > 0,the first shape parameter, and b > 0, the second shape parameter, where φ(·) denotes the pdf of astandard normal random variable, and Φ(·) denotes the cdf of a standard normal random variable.
Usage
dbetalognorm(x, a=1, b=1, mu=0, sigma=1, log=FALSE)pbetalognorm(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varbetalognorm(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esbetalognorm(p, a=1, b=1, mu=0, sigma=1)
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26 betalomax
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetalognorm(x)pbetalognorm(x)varbetalognorm(x)esbetalognorm(x)
betalomax Beta Lomax distribution
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betalomax 27
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Lomax distribution due toLemonte and Cordeiro (2013) given by
f(x) =α
λB(a, b)
(1 +
x
λ
)−bα−1[1−
(1 +
x
λ
)−α]a−1
,
F (x) = I1−(1+ x
λ )−α(a, b),
VaRp(X) = λ[1− I−1
p (a, b)]−1/α − λ,
ESp(X) =λ
p
∫ p
0
[1− I−1
v (a, b)]−1/α
dv − λ
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, α > 0,the third shape parameter, and λ > 0, the scale parameter.
Usage
dbetalomax(x, a=1, b=1, alpha=1, lambda=1, log=FALSE)pbetalomax(x, a=1, b=1, alpha=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varbetalomax(p, a=1, b=1, alpha=1, lambda=1, log.p=FALSE, lower.tail=TRUE)esbetalomax(p, a=1, b=1, alpha=1, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
a the value of the first scale parameter, must be positive, the default is 1
b the value of the second scale parameter, must be positive, the default is 1
alpha the value of the third scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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28 betanorm
Examples
x=runif(10,min=0,max=1)dbetalomax(x)pbetalomax(x)varbetalomax(x)esbetalomax(x)
betanorm Beta normal distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta normal distribution due toEugene et al. (2002) given by
f(x) =1
σB(a, b)φ
(x− µσ
)Φa−1
(x− µσ
)Φb−1
(µ− xσ
),
F (x) = IΦ( x−µσ )(a, b),
VaRp(X) = µ+ σΦ−1(I−1p (a, b)
),
ESp(X) = µ+σ
p
∫ p
0
Φ−1(I−1v (a, b)
)dv
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,a > 0, the first shape parameter, and b > 0, the second shape parameter.
Usage
dbetanorm(x, mu=0, sigma=1, a=1, b=1, log=FALSE)pbetanorm(x, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varbetanorm(p, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esbetanorm(p, mu=0, sigma=1, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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betapareto 29
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetanorm(x)pbetanorm(x)varbetanorm(x)esbetanorm(x)
betapareto Beta Pareto distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Pareto distribution due toAkinsete et al. (2008) given by
f(x) =aKadx−ad−1
B(c, d)
[1−
(K
x
)a]c−1
,
F (x) = I1−(Kx )a(c, d),
VaRp(X) = K[1− I−1
p (c, d)]−1/a
,
ESp(X) =K
p
∫ p
0
[1− I−1
v (c, d)]−1/a
dv
for x ≥ K, 0 < p < 1, K > 0, the scale parameter, a > 0, the first shape parameter, c > 0, thesecond shape parameter, and d > 0, the third shape parameter.
Usage
dbetapareto(x, K=1, a=1, c=1, d=1, log=FALSE)pbetapareto(x, K=1, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)varbetapareto(p, K=1, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esbetapareto(p, K=1, a=1, c=1, d=1)
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30 betaweibull
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
K the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
c the value of the second shape parameter, must be positive, the default is 1
d the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dbetapareto(x)pbetapareto(x)varbetapareto(x)esbetapareto(x)
betaweibull Beta Weibull distribution
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betaweibull 31
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta Weibull distribution due toCordeiro et al. (2012b) given by
f(x) =αxα−1
σαB(a, b)exp
{−b(xσ
)α}[1− exp
{−(xσ
)α}]a−1
,
F (x) = I1−exp{−( xσ )α}(a, b),
VaRp(X) = σ{− log
[1− I−1
p (a, b)]}1/α
,
ESp(X) =σ
p
∫ p
0
{− log
[1− I−1
v (a, b)]}1/α
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, α > 0,the third shape parameter, and σ > 0, the scale parameter.
Usage
dbetaweibull(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)pbetaweibull(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varbetaweibull(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esbetaweibull(p, a=1, b=1, alpha=1, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
alpha the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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32 BS
Examples
x=runif(10,min=0,max=1)dbetaweibull(x)pbetaweibull(x)varbetaweibull(x)esbetaweibull(x)
BS Birnbaum-Saunders distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Birnbaum-Saunders distributiondue to Birnbaum and Saunders (1969a, 1969b) given by
f(x) =x1/2 + x−1/2
2γxφ
(x1/2 − x−1/2
γ
),
F (x) = Φ
(x1/2 − x−1/2
γ
),
VaRp(X) =1
4
{γΦ−1(p) +
√4 + γ2 [Φ−1(p)]
2
}2
,
ESp(X) =1
4p
∫ p
0
{γΦ−1(v) +
√4 + γ2 [Φ−1(v)]
2
}2
dv
for x > 0, 0 < p < 1, and γ > 0, the scale parameter.
Usage
dBS(x, gamma=1, log=FALSE)pBS(x, gamma=1, log.p=FALSE, lower.tail=TRUE)varBS(p, gamma=1, log.p=FALSE, lower.tail=TRUE)esBS(p, gamma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
gamma the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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burr 33
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dBS(x)pBS(x)varBS(x)esBS(x)
burr Burr distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Burr distribution due to Burr(1942) given by
f(x) =bab
xb+1
[1 + (x/a)
−b]−2
,
F (x) =1
1 + (x/a)−b ,
VaRp(X) = ap1/b(1− p)−1/b,
ESp(X) =a
pBp (1/b+ 1, 1− 1/b)
for x > 0, 0 < p < 1, a > 0, the scale parameter, and b > 0, the shape parameter, whereBx(a, b) =
∫ x0ta−1(1− t)b−1dt denotes the incomplete beta function.
Usage
dburr(x, a=1, b=1, log=FALSE)pburr(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varburr(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esburr(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the scale parameter, must be positive, the default is 1
b the value of the shape parameter, must be positive, the default is 1
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34 burr7
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dburr(x)pburr(x)varburr(x)esburr(x)
burr7 Burr XII distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Burr XII distribution due to Burr(1942) given by
f(x) =kcxc−1
(1 + xc)k+1
,
F (x) = 1− (1 + xc)−k,
VaRp(X) =[(1− p)−1/k − 1
]1/c,
ESp(X) =1
p
∫ p
0
[(1− v)−1/k − 1
]1/cdv
for x > 0, 0 < p < 1, c > 0, the first shape parameter, and k > 0, the second shape parameter.
Usage
dburr7(x, k=1, c=1, log=FALSE)pburr7(x, k=1, c=1, log.p=FALSE, lower.tail=TRUE)varburr7(p, k=1, c=1, log.p=FALSE, lower.tail=TRUE)esburr7(p, k=1, c=1)
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Cauchy 35
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
k the value of the first shape parameter, must be positive, the default is 1
c the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dburr7(x)pburr7(x)varburr7(x)esburr7(x)
Cauchy Cauchy distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Cauchy distribution given by
f(x) =1
π
σ
(x− µ)2 + σ2,
F (x) =1
2+
1
πarctan
(x− µσ
),
VaRp(X) = µ+ σ tan
(π
(p− 1
2
)),
ESp(X) = µ+σ
p
∫ p
0
tan
(π
(v − 1
2
))dv
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36 Cauchy
for −∞ < x < ∞, 0 < p < 1, −∞ < µ < ∞, the location parameter, and σ > 0, the scaleparameter.
Usage
dCauchy(x, mu=0, sigma=1, log=FALSE)pCauchy(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varCauchy(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esCauchy(p, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dCauchy(x)pCauchy(x)varCauchy(x)esCauchy(x)
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chen 37
chen Chen distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Chen distribution due to Chen(2000) given by
f(x) = λbxb−1 exp(xb)
exp[λ− λ exp
(xb)],
F (x) = 1− exp[λ− λ exp
(xb)],
VaRp(X) =
{log
[1− log(1− p)
λ
]}1/b
,
ESp(X) =1
p
∫ p
0
{log
[1− log(1− v)
λ
]}1/b
dv
for x > 0, 0 < p < 1, b > 0, the shape parameter, and λ > 0, the scale parameter.
Usage
dchen(x, b=1, lambda=1, log=FALSE)pchen(x, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varchen(p, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)eschen(p, b=1, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
b the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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38 clg
Examples
x=runif(10,min=0,max=1)dchen(x)pchen(x)varchen(x)eschen(x)
clg Compound Laplace gamma distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the compound Laplace gamma dis-tribution given by
f(x) =ab
2{1 + b |x− θ|}−(a+1)
,
F (x) =
1
2{1 + b |x− θ|}−a , if x ≤ θ,
1− 1
2{1 + b |x− θ|}−a , if x > θ,
VaRp(X) =
θ − 1
b− (2p)−1/a
b, if p ≤ 1/2,
θ − 1
b+
(2(1− p))−1/a
b, if p > 1/2,
ESp(X) =
θ − 1
b− (2p)−1/a
b(1− 1/a), if p ≤ 1/2,
θ − 1
b− [2(1− p)]1−1/a
2pb(1− 1/a), if p > 1/2
for −∞ < x <∞, 0 < p < 1, −∞ < θ <∞, the location parameter, b > 0, the scale parameter,and a > 0, the shape parameter.
Usage
dclg(x, a=1, b=1, theta=0, log=FALSE)pclg(x, a=1, b=1, theta=0, log.p=FALSE, lower.tail=TRUE)varclg(p, a=1, b=1, theta=0, log.p=FALSE, lower.tail=TRUE)esclg(p, a=1, b=1, theta=0)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
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compbeta 39
theta the value of the location parameter, can take any real value, the default is zero
b the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dclg(x)pclg(x)varclg(x)esclg(x)
compbeta Complementary beta distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the complementary beta distributiondue to Jones (2002) given by
f(x) = B(a, b){I−1x (a, b)
}1−a {1− I−1
x (a, b)}1−b
,F (x) = I−1
x (a, b),VaRp(X) = Ip(a, b),
ESp(X) =1
p
∫ p
0
Iv(a, b)dv
for 0 < x < 1, 0 < p < 1, a > 0, the first shape parameter, and b > 0, the second shape parameter.
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40 compbeta
Usage
dcompbeta(x, a=1, b=1, log=FALSE)pcompbeta(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varcompbeta(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)escompbeta(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dcompbeta(x)pcompbeta(x)varcompbeta(x)escompbeta(x)
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dagum 41
dagum Dagum distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Dagum distribution due to Dagum(1975, 1977, 1980) given by
f(x) =acbaxac−1
[xa + ba]c+1 ,
F (x) =
[1 +
(b
x
)a]−c,
VaRp(X) = b(
1− p−1/c)−1/a
,
ESp(X) =b
p
∫ p
0
(1− v−1/c
)−1/a
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the scale parameter, and c > 0, thesecond shape parameter.
Usage
ddagum(x, a=1, b=1, c=1, log=FALSE)pdagum(x, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)vardagum(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)esdagum(p, a=1, b=1, c=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
b the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
c the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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42 dweibull
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)ddagum(x)pdagum(x)vardagum(x)esdagum(x)
dweibull Double Weibull distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the double Weibull distribution dueto Balakrishnan and Kocherlakota (1985) given by
f(x) =c
2σ
∣∣∣∣x− µσ∣∣∣∣c−1
exp
{−∣∣∣∣x− µσ
∣∣∣∣c} ,F (x) =
1
2exp
{−(µ− xσ
)c}, if x ≤ µ,
1− 1
2exp
{−(x− µσ
)c}, if x > µ,
VaRp(X) =
µ− σ [− log (2p)]1/c
, if p ≤ 1/2,
µ+ σ [− log (2(1− p))]1/c , if p > 1/2,
ESp(X) =
µ− σ
p
∫ p
0
[− log 2− log v]1/c
dv, if p ≤ 1/2,
µ− σ
p
∫ 1/2
0
[− log 2− log v]1/c
dv
+σ
p
∫ p
1/2
[− log 2− log(1− v)]1/c
dv, if p > 1/2
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,and c > 0, the shape parameter.
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dweibull 43
Usage
ddweibull(x, c=1, mu=0, sigma=1, log=FALSE)pdweibull(x, c=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vardweibull(p, c=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esdweibull(p, c=1, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
c the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)ddweibull(x)pdweibull(x)vardweibull(x)esdweibull(x)
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44 expexp
expexp Exponentiated exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated exponential distri-bution due to Gupta and Kundu (1999, 2001) given by
f(x) = aλ exp(−λx) [1− exp(−λx)]a−1
,F (x) = [1− exp(−λx)]
a,
VaRp(X) = − 1
λlog(
1− p1/a),
ESp(X) = − 1
pλ
∫ p
0
log(
1− v1/a)dv
for x > 0, 0 < p < 1, a > 0, the shape parameter and λ > 0, the scale parameter.
Usage
dexpexp(x, lambda=1, a=1, log=FALSE)pexpexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varexpexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)esexpexp(p, lambda=1, a=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedlambda the value of the scale parameter, must be positive, the default is 1a the value of the shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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expext 45
Examples
x=runif(10,min=0,max=1)dexpexp(x)pexpexp(x)varexpexp(x)esexpexp(x)
expext Exponential extension distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponential extension distributiondue to Nadarajah and Haghighi (2011) given by
f(x) = aλ(1 + λx)a−1 exp [1− (1 + λx)a] ,F (x) = 1− exp [1− (1 + λx)a] ,
VaRp(X) =[1− log(1− p)]1/a − 1
λ,
ESp(X) = − 1
λ+
1
λp
∫ p
0
[1− log(1− v)]1/a
dv
for x > 0, 0 < p < 1, a > 0, the shape parameter and λ > 0, the scale parameter.
Usage
dexpext(x, lambda=1, a=1, log=FALSE)pexpext(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varexpext(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)esexpext(p, lambda=1, a=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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46 expgeo
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dexpext(x)pexpext(x)varexpext(x)esexpext(x)
expgeo Exponential geometric distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponential geometric distributiondue to Adamidis and Loukas (1998) given by
f(x) =λθ exp(−λx)
[1− (1− θ) exp(−λx)]2 ,
F (x) =θ exp(−λx)
1− (1− θ) exp(−λx),
VaRp(X) = − 1
λlog
p
θ + (1− θ)p,
ESp(X) = − log p
λ− θ log θ
λp(1− θ)+θ + (1− θ)pλp(1− θ)
log [θ + (1− θ)p]
for x > 0, 0 < p < 1, 0 < θ < 1, the first scale parameter, and λ > 0, the second scale parameter.
Usage
dexpgeo(x, theta=0.5, lambda=1, log=FALSE)pexpgeo(x, theta=0.5, lambda=1, log.p=FALSE, lower.tail=TRUE)varexpgeo(p, theta=0.5, lambda=1, log.p=FALSE, lower.tail=TRUE)esexpgeo(p, theta=0.5, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
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explog 47
theta the value of the first scale parameter, must be in the unit interval, the default is0.5
lambda the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dexpgeo(x)pexpgeo(x)varexpgeo(x)esexpgeo(x)
explog Exponential logarithmic distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponential logarithmic distribu-tion due to Tahmasbi and Rezaei (2008) given by
f(x) = − b(1− a) exp(−bx)
log a [1− (1− a) exp(−bx)],
F (x) = 1− log [1− (1− a) exp(−bx)]
log a,
VaRp(X) = −1
blog
[1− a1−p
1− a
],
ESp(X) = − 1
bp
∫ p
0
log
[1− a1−v
1− a
]dv
for x > 0, 0 < p < 1, 0 < a < 1, the first scale parameter, and b > 0, the second scale parameter.
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48 explog
Usage
dexplog(x, a=0.5, b=1, log=FALSE)pexplog(x, a=0.5, b=1, log.p=FALSE, lower.tail=TRUE)varexplog(p, a=0.5, b=1, log.p=FALSE, lower.tail=TRUE)esexplog(p, a=0.5, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be in the unit interval, the default is0.5
b the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dexplog(x)pexplog(x)varexplog(x)esexplog(x)
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explogis 49
explogis Exponentiated logistic distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated logistic distributiongiven by
f(x) = (a/b) exp(−x/b) [1 + exp(−x/b)]−a−1,
F (x) = [1 + exp(−x/b)]−a ,VaRp(X) = −b log
[p−1/a − 1
],
ESp(X) = − bp
∫ p
0
log[v−1/a − 1
]dv
for −∞ < x <∞, 0 < p < 1, a > 0, the shape parameter, and b > 0, the scale parameter.
Usage
dexplogis(x, a=1, b=1, log=FALSE)pexplogis(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varexplogis(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esexplogis(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
b the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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50 exponential
Examples
x=runif(10,min=0,max=1)dexplogis(x)pexplogis(x)varexplogis(x)esexplogis(x)
exponential Exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponential distribution given by
f(x) = λ exp(−λx),F (x) = 1− exp(−λx),
VaRp(X) = − 1
λlog(1− p),
ESp(X) = − 1
pλ{log(1− p)p− p− log(1− p)}
for x > 0, 0 < p < 1, and λ > 0, the scale parameter.
Usage
dexponential(x, lambda=1, log=FALSE)pexponential(x, lambda=1, log.p=FALSE, lower.tail=TRUE)varexponential(p, lambda=1, log.p=FALSE, lower.tail=TRUE)esexponential(p, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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exppois 51
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dexponential(x)pexponential(x)varexponential(x)esexponential(x)
exppois Exponential Poisson distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponential Poisson distributiondue to Kus (2007) given by
f(x) =bλ exp [−bx− λ+ λ exp(−bx)]
1− exp(−λ),
F (x) =1− exp [−λ+ λ exp(−bx)]
1− exp(−λ),
VaRp(X) = −1
blog
{1
λlog [1− p+ p exp(−λ)] + 1
},
ESp(X) = − 1
bp
∫ p
0
log
{1
λlog [1− v + v exp(−λ)] + 1
}dv
for x > 0, 0 < p < 1, b > 0, the first scale parameter, and λ > 0, the second scale parameter.
Usage
dexppois(x, b=1, lambda=1, log=FALSE)pexppois(x, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varexppois(p, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)esexppois(p, b=1, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedb the value of the first scale parameter, must be positive, the default is 1lambda the value of the second scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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52 exppower
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dexppois(x)pexppois(x)varexppois(x)esexppois(x)
exppower Exponential power distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponential power distributiondue to Subbotin (1923) given by
f(x) =1
2a1/aσΓ (1 + 1/a)exp
{−| x− µ |
a
aσa
},
F (x) =
1
2Q
(1
a,
(µ− x)a
aσa
), if x ≤ µ,
1− 1
2Q
(1
a,
(x− µ)a
aσa
), if x > µ,
VaRp(X) =
µ− a1/aσ
[Q−1
(1
a, 2p
)]1/a
, if p ≤ 1/2,
µ+ a1/aσ[Q−1
(1a , 2(1− p)
)]1/a, if p > 1/2,
ESp(X) =
µ− a1/aσ
p
∫ p
0
[Q−1
(1
a, 2v
)]1/a
dv, if p ≤ 1/2,
µ− a1/aσ
p
∫ 1/2
0
[Q−1
(1
a, 2v
)]1/a
dv
+a1/aσ
p
∫ p
1/2
[Q−1
(1
a, 2(1− v)
)]1/a
dv, if p > 1/2
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exppower 53
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,and a > 0, the shape parameter.
Usage
dexppower(x, mu=0, sigma=1, a=1, log=FALSE)pexppower(x, mu=0, sigma=1, a=1, log.p=FALSE, lower.tail=TRUE)varexppower(p, mu=0, sigma=1, a=1, log.p=FALSE, lower.tail=TRUE)esexppower(p, mu=0, sigma=1, a=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dexppower(x)pexppower(x)varexppower(x)esexppower(x)
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54 expweibull
expweibull Exponentiated Weibull distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated Weibull distributiondue to Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) given by
f(x) = aασ−αxα−1 exp [−(x/σ)α] {1− exp [−(x/σ)α]}a−1,
F (x) = {1− exp [−(x/σ)α]}a ,
VaRp(X) = σ[− log
(1− p1/a
)]1/α,
ESp(X) =σ
p
∫ p
0
[− log
(1− v1/a
)]1/αdv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, α > 0, the second shape parameter, andσ > 0, the scale parameter.
Usage
dexpweibull(x, a=1, alpha=1, sigma=1, log=FALSE)pexpweibull(x, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varexpweibull(p, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esexpweibull(p, a=1, alpha=1, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
alpha the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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F 55
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dexpweibull(x)pexpweibull(x)varexpweibull(x)esexpweibull(x)
F F distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the F distribution given by
f(x) =1
B(d12 ,
d22
) (d1
d2
) d12
xd12 −1
(1 +
d1
d2x
)− d1+d22
,
F (x) = I d1x
d1x+d2
(d1
2,d2
2
),
VaRp(X) =d2
d1
I−1p
(d12 ,
d22
)1− I−1
p
(d12 ,
d22
) ,ESp(X) =
d2
d1p
∫ p
0
I−1v
(d12 ,
d22
)1− I−1
v
(d12 ,
d22
)dvfor x ≥ K, 0 < p < 1, d1 > 0, the first degree of freedom parameter, and d2 > 0, the seconddegree of freedom parameter.
Usage
dF(x, d1=1, d2=1, log=FALSE)pF(x, d1=1, d2=1, log.p=FALSE, lower.tail=TRUE)varF(p, d1=1, d2=1, log.p=FALSE, lower.tail=TRUE)esF(p, d1=1, d2=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
d1 the value of the first degree of freedom parameter, must be positive, the defaultis 1
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56 FR
d2 the value of the second degree of freedom parameter, must be positive, the de-fault is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dF(x)pF(x)varF(x)esF(x)
FR Freimer distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Freimer distribution due toFreimer et al. (1988) given by
VaRp(X) =1
a
[pb − 1
b− (1− p)c − 1
c
],
ESp(X) =1
a
(1
c− 1
b
)+
pb
ab(b+ 1)+
(1− p)c+1 − 1
pac(c+ 1)
for 0 < p < 1, a > 0, the scale parameter, b > 0, the first shape parameter, and c > 0, the secondshape parameter.
Usage
varFR(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)esFR(p, a=1, b=1, c=1)
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frechet 57
Arguments
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the scale parameter, must be positive, the default is 1
b the value of the first shape parameter, must be positive, the default is 1
c the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)varFR(x)esFR(x)
frechet Frechet distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Fr\’echet distribution due toFr\’echet (1927) given by
f(x) =ασα
xα+1exp
{−(σx
)α},
F (x) = exp{−(σx
)α},
VaRp(X) = σ [− log p]−1/α
,
ESp(X) =σ
pΓ (1− 1/α,− log p)
for x > 0, 0 < p < 1, α > 0, the shape parameter, and σ > 0, the scale parameter, whereΓ(a, x) =
∫∞xta−1 exp (−t) dt denotes the complementary incomplete gamma function.
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58 frechet
Usage
dfrechet(x, alpha=1, sigma=1, log=FALSE)pfrechet(x, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varfrechet(p, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esfrechet(p, alpha=1, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
alpha the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dfrechet(x)pfrechet(x)varfrechet(x)esfrechet(x)
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Gamma 59
Gamma Gamma distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the gamma distribution given by
f(x) =baxa−1 exp(−bx)
Γ(a),
F (x) =γ(a, bx)
Γ(a),
VaRp(X) =1
bQ−1(a, 1− p),
ESp(X) =1
bp
∫ p
0
Q−1(a, 1− v)dv
for x > 0, 0 < p < 1, b > 0, the scale parameter, and a > 0, the shape parameter, where γ(a, x) =∫ x0ta−1 exp (−t) dt denotes the incomplete gamma function,Q(a, x) =
∫∞xta−1 exp (−t) dt/Γ(a)
denotes the regularized complementary incomplete gamma function, Γ(a) =∫∞
0ta−1 exp (−t) dt
denotes the gamma function, and Q−1(a, x) denotes the inverse of Q(a, x).
Usage
dGamma(x, a=1, b=1, log=FALSE)pGamma(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varGamma(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esGamma(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
b the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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60 genbeta
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dGamma(x)pGamma(x)varGamma(x)esGamma(x)
genbeta Generalized beta distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized beta distributiongiven by
f(x) =(x− c)a−1(d− x)b−1
B(a, b)(d− c)a+b−1,
F (x) = I x−cd−c
(a, b),
VaRp(X) = c+ (d− c)I−1p (a, b),
ESp(X) = c+d− cp
∫ p
0
I−1v (a, b)dv
for c ≤ x ≤ d, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter,−∞ < c <∞, the first location parameter, and −∞ < c < d <∞, the second location parameter.
Usage
dgenbeta(x, a=1, b=1, c=0, d=1, log=FALSE)pgenbeta(x, a=1, b=1, c=0, d=1, log.p=FALSE, lower.tail=TRUE)vargenbeta(p, a=1, b=1, c=0, d=1, log.p=FALSE, lower.tail=TRUE)esgenbeta(p, a=1, b=1, c=0, d=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
c the value of the first location parameter, can take any real value, the default iszero
d the value of the second location parameter, can take any real value but must begreater than c, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
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genbeta2 61
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgenbeta(x)pgenbeta(x)vargenbeta(x)esgenbeta(x)
genbeta2 Generalized beta II distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized beta II distributiongiven by
f(x) =cxac−1 (1− xc)b−1
B(a, b),
F (x) = Ixc(a, b),
VaRp(X) =[I−1p (a, b)
]1/c,
ESp(X) =1
p
∫ p
0
[I−1v (a, b)
]1/cdv
for 0 < x < 1, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, andc > 0, the third shape parameter.
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62 genbeta2
Usage
dgenbeta2(x, a=1, b=1, c=1, log=FALSE)pgenbeta2(x, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)vargenbeta2(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)esgenbeta2(p, a=1, b=1, c=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
c the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgenbeta2(x)pgenbeta2(x)vargenbeta2(x)esgenbeta2(x)
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geninvbeta 63
geninvbeta Generalized inverse beta distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized inverse beta distribu-tion given by
f(x) =axac−1
B(c, d) (1 + xa)c+d
,
F (x) = I xa
1+xa(c, d),
VaRp(X) =
[I−1p (c, d)
1− I−1p (c, d)
]1/a
,
ESp(X) =1
p
∫ p
0
[I−1v (c, d)
1− I−1v (c, d)
]1/a
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, c > 0, the second shape parameter, andd > 0, the third shape parameter.
Usage
dgeninvbeta(x, a=1, c=1, d=1, log=FALSE)pgeninvbeta(x, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)vargeninvbeta(p, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esgeninvbeta(p, a=1, c=1, d=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first shape parameter, must be positive, the default is 1
c the value of the second shape parameter, must be positive, the default is 1
d the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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64 genlogis
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgeninvbeta(x)pgeninvbeta(x)vargeninvbeta(x)esgeninvbeta(x)
genlogis Generalized logistic distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic distributiongiven by
f(x) =a exp
(−x−µσ
)σ{
1 + exp(−x−µσ
)}1+a ,
F (x) =1{
1 + exp(−x−µσ
)}a ,VaRp(X) = µ− σ log
(p−1/a − 1
),
ESp(X) = µ− σ
p
∫ p
0
log(v−1/a − 1
)dv
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,and a > 0, the shape parameter.
Usage
dgenlogis(x, a=1, mu=0, sigma=1, log=FALSE)pgenlogis(x, a=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vargenlogis(p, a=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esgenlogis(p, a=1, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
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genlogis3 65
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgenlogis(x)pgenlogis(x)vargenlogis(x)esgenlogis(x)
genlogis3 Generalized logistic III distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic III distributiongiven by
f(x) =1
σB(α, α)exp
(αx− µσ
){1 + exp
(x− µσ
)}−2α
,
F (x) = I 1
1+exp(− x−µσ )
(α, α),
VaRp(X) = µ− σ log1− I−1
p (α, α)
I−1p (α, α)
,
ESp(X) = µ− σ
p
∫ p
0
log1− I−1
v (α, α)
I−1v (α, α)
dv
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,and α > 0, the shape parameter.
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66 genlogis3
Usage
dgenlogis3(x, alpha=1, mu=0, sigma=1, log=FALSE)pgenlogis3(x, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vargenlogis3(p, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esgenlogis3(p, alpha=1, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
alpha the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgenlogis3(x)pgenlogis3(x)vargenlogis3(x)esgenlogis3(x)
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genlogis4 67
genlogis4 Generalized logistic IV distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic IV distributiongiven by
f(x) =1
σB(α, a)exp
(−αx− µ
σ
){1 + exp
(−x− µ
σ
)}−α−a,
F (x) = I 1
1+exp(− x−µσ )
(α, a),
VaRp(X) = µ− σ log1− I−1
p (α, a)
I−1p (α, a)
,
ESp(X) = µ− σ
p
∫ p
0
log1− I−1
v (α, a)
I−1v (α, a)
dv
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,α > 0, the first shape parameter, and a > 0, the second shape parameter.
Usage
dgenlogis4(x, a=1, alpha=1, mu=0, sigma=1, log=FALSE)pgenlogis4(x, a=1, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vargenlogis4(p, a=1, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esgenlogis4(p, a=1, alpha=1, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
alpha the value of the first shape parameter, must be positive, the default is 1
a the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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68 genpareto
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgenlogis4(x)pgenlogis4(x)vargenlogis4(x)esgenlogis4(x)
genpareto Generalized Pareto distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized Pareto distributiondue to Pickands (1975) given by
f(x) =1
k
(1− cx
k
)1/c−1
,
F (x) = 1−(
1− cx
k
)1/c
,
VaRp(X) =k
c[1− (1− p)c] ,
ESp(X) =k
c+k(1− p)c+1
pc(c+ 1)− k
pc(c+ 1)
for x < k/c if c > 0, x > k/c if c < 0, x > 0 if c = 0, 0 < p < 1, k > 0, the scale parameter and−∞ < c <∞, the shape parameter.
Usage
dgenpareto(x, k=1, c=1, log=FALSE)pgenpareto(x, k=1, c=1, log.p=FALSE, lower.tail=TRUE)vargenpareto(p, k=1, c=1, log.p=FALSE, lower.tail=TRUE)esgenpareto(p, k=1, c=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
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genpowerweibull 69
k the value of the scale parameter, must be positive, the default is 1
c the value of the shape parameter, can take any real value, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgenpareto(x)pgenpareto(x)vargenpareto(x)esgenpareto(x)
genpowerweibull Generalized power Weibull distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized power Weibull dis-tribution due to Nikulin and Haghighi (2006) given by
f(x) = aθxa−1 [1 + xa]θ−1
exp{
1− [1 + xa]θ},
F (x) = 1− exp{
1− [1 + xa]θ},
VaRp(X) ={
[1− log(1− p)]1/θ − 1}1/a
,
ESp(X) =1
p
∫ p
0
{[1− log(1− v)]
1/θ − 1}1/a
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, and θ > 0, the second shape parameter.
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70 genpowerweibull
Usage
dgenpowerweibull(x, a=1, theta=1, log=FALSE)pgenpowerweibull(x, a=1, theta=1, log.p=FALSE, lower.tail=TRUE)vargenpowerweibull(p, a=1, theta=1, log.p=FALSE, lower.tail=TRUE)esgenpowerweibull(p, a=1, theta=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first shape parameter, must be positive, the default is 1
theta the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgenpowerweibull(x)pgenpowerweibull(x)vargenpowerweibull(x)esgenpowerweibull(x)
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genunif 71
genunif Generalized uniform distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized uniform distributiongiven by
f(x) = hkc(x− a)c−1 [1− k(x− a)c]h−1
,
F (x) = 1− [1− k(x− a)c]h,
VaRp(X) = a+ k−1/c[1− (1− p)1/h
]1/c,
ESp(X) = a+k−1/c
p
∫ p
0
[1− (1− v)1/h
]1/cdv
for a ≤ x ≤ a + k−1/c, 0 < p < 1, −∞ < a < ∞, the location parameter, c > 0, the first shapeparameter, k > 0, the scale parameter, and h > 0, the second shape parameter.
Usage
dgenunif(x, a=0, c=1, h=1, k=1, log=FALSE)pgenunif(x, a=0, c=1, h=1, k=1, log.p=FALSE, lower.tail=TRUE)vargenunif(p, a=0, c=1, h=1, k=1, log.p=FALSE, lower.tail=TRUE)esgenunif(p, a=0, c=1, h=1, k=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the location parameter, can take any real value, the default is zero
k the value of the scale parameter, must be positive, the default is 1
c the value of the first scale parameter, must be positive, the default is 1
h the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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72 gev
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgenunif(x)pgenunif(x)vargenunif(x)esgenunif(x)
gev Generalized extreme value distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized extreme value distri-bution due to Fisher and Tippett (1928) given by
f(x) =1
σ
[1 + ξ
(x− µσ
)]−1/ξ−1
exp
{−[1 + ξ
(x− µσ
)]−1/ξ},
F (x) = exp
{−[1 + ξ
(x− µσ
)]−1/ξ},
VaRp(X) = µ− σ
ξ+σ
ξ(− log p)−ξ,
ESp(X) = µ− σ
ξ+
σ
pξ
∫ p
0
(− log v)−ξdv
for x ≥ µ − σ/ξ if ξ > 0, x ≤ µ − σ/ξ if ξ < 0, −∞ < x < ∞ if ξ = 0, 0 < p < 1,−∞ < µ < ∞, the location parameter, σ > 0, the scale parameter, and −∞ < ξ < ∞, the shapeparameter.
Usage
dgev(x, mu=0, sigma=1, xi=1, log=FALSE)pgev(x, mu=0, sigma=1, xi=1, log.p=FALSE, lower.tail=TRUE)vargev(p, mu=0, sigma=1, xi=1, log.p=FALSE, lower.tail=TRUE)esgev(p, mu=0, sigma=1, xi=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
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gompertz 73
sigma the value of the scale parameter, must be positive, the default is 1
xi the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgev(x)pgev(x)vargev(x)esgev(x)
gompertz Gompertz distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Gompertz distribution due toGompertz (1825) given by
f(x) = bη exp(bx) exp [η − η exp(bx)] ,F (x) = 1− exp [η − η exp(bx)] ,
VaRp(X) =1
blog
[1− 1
ηlog(1− p)
],
ESp(X) =1
pb
∫ p
0
log
[1− 1
ηlog(1− v)
]dv
for x > 0, 0 < p < 1, b > 0, the first scale parameter and η > 0, the second scale parameter.
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74 gompertz
Usage
dgompertz(x, b=1, eta=1, log=FALSE)pgompertz(x, b=1, eta=1, log.p=FALSE, lower.tail=TRUE)vargompertz(p, b=1, eta=1, log.p=FALSE, lower.tail=TRUE)esgompertz(p, b=1, eta=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
b the value of the first scale parameter, must be positive, the default is 1
eta the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgompertz(x)pgompertz(x)vargompertz(x)esgompertz(x)
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gumbel 75
gumbel Gumbel distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Gumbel distribution given by dueto Gumbel (1954) given by
f(x) =1
σexp
(µ− xσ
)exp
[− exp
(µ− xσ
)],
F (x) = exp
[− exp
(µ− xσ
)],
VaRp(X) = µ− σ log(− log p),
ESp(X) = µ− σ
p
∫ p
0
log(− log v)dv
for −∞ < x < ∞, 0 < p < 1, −∞ < µ < ∞, the location parameter, and σ > 0, the scaleparameter.
Usage
dgumbel(x, mu=0, sigma=1, log=FALSE)pgumbel(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vargumbel(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esgumbel(p, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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76 gumbel2
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgumbel(x)pgumbel(x)vargumbel(x)esgumbel(x)
gumbel2 Gumbel II distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Gumbel II distribution
f(x) = abx−a−1 exp(−bx−a
),
F (x) = 1− exp(−bx−a
),
VaRp(X) = b1/a [− log(1− p)]−1/a,
ESp(X) =b1/a
p
∫ p
0
[− log(1− v)]−1/a
dv
for x > 0, 0 < p < 1, a > 0, the shape parameter, and b > 0, the scale parameter.
Usage
dgumbel2(x, a=1, b=1, log=FALSE)pgumbel2(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)vargumbel2(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esgumbel2(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the scale parameter, must be positive, the default is 1
b the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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halfcauchy 77
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dgumbel2(x)pgumbel2(x)vargumbel2(x)#esgumbel2(x)
halfcauchy Half Cauchy distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the half Cauchy distribution given by
f(x) =2
π
σ
x2 + σ2,
F (x) =2
πarctan
(xσ
),
VaRp(X) = σ tan(πp
2
),
ESp(X) =σ
p
∫ p
0
tan(πv
2
)dv
for x > 0, 0 < p < 1, and σ > 0, the scale parameter.
Usage
dhalfcauchy(x, sigma=1, log=FALSE)phalfcauchy(x, sigma=1, log.p=FALSE, lower.tail=TRUE)varhalfcauchy(p, sigma=1, log.p=FALSE, lower.tail=TRUE)eshalfcauchy(p, sigma=1)
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78 halflogis
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dhalfcauchy(x)phalfcauchy(x)varhalfcauchy(x)eshalfcauchy(x)
halflogis Half logistic distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the half logistic distribution given by
f(x) =2λ exp (−λx)
[1 + exp (−λx)]2 ,
F (x) =1− exp (−λx)
1 + exp (−λx),
VaRp(X) = − 1
λlog
1− p1 + p
,
ESp(X) = − 1
λlog
1− p1 + p
+1
λplog(1− p2
)for x > 0, 0 < p < 1, and λ > 0, the scale parameter.
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halflogis 79
Usage
dhalflogis(x, lambda=1, log=FALSE)phalflogis(x, lambda=1, log.p=FALSE, lower.tail=TRUE)varhalflogis(p, lambda=1, log.p=FALSE, lower.tail=TRUE)eshalflogis(p, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dhalflogis(x)phalflogis(x)varhalflogis(x)eshalflogis(x)
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80 halfnorm
halfnorm Half normal distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for Half normal distribution given by
f(x) =2
σφ(xσ
),
F (x) = 2Φ(xσ
)− 1,
VaRp(X) = σΦ−1
(1 + p
2
),
ESp(X) =σ
p
∫ p
0
Φ−1
(1 + v
2
)dv
for x > 0, 0 < p < 1, and σ > 0, the scale parameter.
Usage
dhalfnorm(x, sigma=1, log=FALSE)phalfnorm(x, sigma=1, log.p=FALSE, lower.tail=TRUE)varhalfnorm(p, sigma=1, log.p=FALSE, lower.tail=TRUE)eshalfnorm(p, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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halfT 81
Examples
x=runif(10,min=0,max=1)dhalfnorm(x)phalfnorm(x)varhalfnorm(x)eshalfnorm(x)
halfT Half Student’s t distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the half Student’s t distribution givenby
f(x) =2Γ(n+1
2
)√nπΓ
(n2
) (1 +x2
n
)−n+12
,
F (x) = I x2
x2+n
(1
2,n
2
),
VaRp(X) =
√nI−1p
(12 ,
n2
)1− I−1
p
(12 ,
n2
) ,ESp(X) =
√n
p
∫ p
0
√I−1v
(12 ,
n2
)1− I−1
v
(12 ,
n2
)dvfor −∞ < x <∞, 0 < p < 1, and n > 0, the degree of freedom parameter.
Usage
dhalfT(x, n=1, log=FALSE)phalfT(x, n=1, log.p=FALSE, lower.tail=TRUE)varhalfT(p, n=1, log.p=FALSE, lower.tail=TRUE)eshalfT(p, n=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedn the value of the degree of freedom parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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82 HBlaplace
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dhalfT(x)phalfT(x)varhalfT(x)eshalfT(x)
HBlaplace Holla-Bhattacharya Laplace distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Holla-Bhattacharya Laplacedistribution due to Holla and Bhattacharya (1968) given by
f(x) =
aφ exp {φ (x− θ)} , if x ≤ θ,
(1− a)φ exp {φ (θ − x)} , if x > θ,
F (x) =
a exp (φx− θφ) , if x ≤ θ,
1− (1− a) exp (θφ− φx) , if x > θ,
VaRp(X) =
θ +
1
φlog(pa
), if p ≤ a,
θ − 1
φlog
(1− p1− a
), if p > a,
ESp(X) =
θ − 1
φ+
1
φlog
p
a, if p ≤ a,
1
p
[θ(1 + p− a) +
p− 2a− (1− a) log a
φ+
1− pφ
log1− p1− a
], if p > a
for −∞ < x < ∞, 0 < p < 1, −∞ < θ < ∞, the location parameter, 0 < a < 1, the first scaleparameter, and φ > 0, the second scale parameter.
Usage
dHBlaplace(x, a=0.5, theta=0, phi=1, log=FALSE)pHBlaplace(x, a=0.5, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)varHBlaplace(p, a=0.5, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)esHBlaplace(p, a=0.5, theta=0, phi=1)
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HL 83
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedtheta the value of the location parameter, can take any real value, the default is zeroa the value of the first scale parameter, must be in the unit interval, the default is
0.5phi the value of the second scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dHBlaplace(x)pHBlaplace(x)varHBlaplace(x)esHBlaplace(x)
HL Hankin-Lee distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Hankin-Lee distribution due toHankin and Lee (2006) given by
VaRp(X) =cpa
(1− p)b,
ESp(X) =c
pBp(a+ 1, 1− b)
for 0 < p < 1, c > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the secondshape parameter.
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84 Hlogis
Usage
varHL(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)esHL(p, a=1, b=1, c=1)
Arguments
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
c the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)varHL(x)esHL(x)
Hlogis Hosking logistic distribution
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Hlogis 85
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Hosking logistic distribution dueto Hosking (1989, 1990) given by
f(x) =(1− kx)1/k−1[
1 + (1− kx)1/k]2 ,
F (x) =1
1 + (1− kx)1/k,
VaRp(X) =1
k
[1−
(1− pp
)k],
ESp(X) =1
k− 1
kpBp(1− k, 1 + k)
for x < 1/k if k > 0, x > 1/k if k < 0, −∞ < x < ∞ if k = 0, and −∞ < k < ∞, the shapeparameter.
Usage
dHlogis(x, k=1, log=FALSE)pHlogis(x, k=1, log.p=FALSE, lower.tail=TRUE)varHlogis(p, k=1, log.p=FALSE, lower.tail=TRUE)esHlogis(p, k=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
k the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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86 invbeta
Examples
x=runif(10,min=0,max=1)dHlogis(x)pHlogis(x)varHlogis(x)esHlogis(x)
invbeta Inverse beta distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the inverse beta distribution given by
f(x) =xa−1
B(a, b)(1 + x)a+b,
F (x) = I x1+x
(a, b),
VaRp(X) =I−1p (a, b)
1− I−1p (a, b)
,
ESp(X) =1
p
∫ p
0
I−1v (a, b)
1− I−1v (a, b)
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, and b > 0, the second shape parameter.
Usage
dinvbeta(x, a=1, b=1, log=FALSE)pinvbeta(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varinvbeta(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esinvbeta(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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invexpexp 87
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dinvbeta(x)pinvbeta(x)varinvbeta(x)esinvbeta(x)
invexpexp Inverse exponentiated exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the inverse exponentiated exponentialdistribution due to Ghitany et al. (2013) given by
f(x) = aλx−2 exp
(−λx
)[1− exp
(−λx
)]a−1
,
F (x) = 1−[1− exp
(−λx
)]a,
VaRp(X) = λ{− log
[1− (1− p)1/a
]}−1
,
ESp(X) =λ
p
∫ p
0
{− log
[1− (1− v)1/a
]}−1
dv
for x > 0, 0 < p < 1, a > 0, the shape parameter and λ > 0, the scale parameter.
Usage
dinvexpexp(x, lambda=1, a=1, log=FALSE)pinvexpexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varinvexpexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)esinvexpexp(p, lambda=1, a=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
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88 invgamma
a the value of the shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dinvexpexp(x)pinvexpexp(x)varinvexpexp(x)esinvexpexp(x)
invgamma Inverse gamma distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the inverse gamma distribution givenby
f(x) =ba exp(−b/x)
xa+1Γ(a),
F (x) = Q(a, b/x),
VaRp(X) = b[Q−1(a, p)
]−1,
ESp(X) =b
p
∫ p
0
[Q−1(a, v)
]−1dv
for x > 0, 0 < p < 1, a > 0, the shape parameter, and b > 0, the scale parameter.
Usage
dinvgamma(x, a=1, b=1, log=FALSE)pinvgamma(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varinvgamma(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esinvgamma(p, a=1, b=1)
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kum 89
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedb the value of the scale parameter, must be positive, the default is 1a the value of the shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dinvgamma(x)pinvgamma(x)varinvgamma(x)esinvgamma(x)
kum Kumaraswamy distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy distribution dueto Kumaraswamy (1980) given by
f(x) = abxa−1 (1− xa)b−1
,
F (x) = 1− (1− xa)b,
VaRp(X) =[1− (1− p)1/b
]1/a,
ESp(X) =1
p
∫ p
0
[1− (1− v)1/b
]1/adv
for 0 < x < 1, 0 < p < 1, a > 0, the first shape parameter, and b > 0, the second shape parameter.
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90 kum
Usage
dkum(x, a=1, b=1, log=FALSE)pkum(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varkum(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eskum(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkum(x)pkum(x)varkum(x)eskum(x)
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kumburr7 91
kumburr7 Kumaraswamy Burr XII distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Burr XII distri-bution due to Parana\’iba et al. (2013) given by
f(x) =abkcxc−1
(1 + xc)k+1
[1− (1 + xc)
−k]a−1 {
1−[1− (1 + xc)
−k]a}b−1
,
F (x) = 1−{
1−[1− (1 + xc)
−k]a}b
,
VaRp(X) =
[{1−
[1− (1− p)1/b
]1/a}−1/k
− 1
]1/c
,
ESp(X) =1
p
∫ p
0
[{1−
[1− (1− v)1/b
]1/a}−1/k
− 1
]1/c
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, c > 0,the third shape parameter, and k > 0, the fourth shape parameter.
Usage
dkumburr7(x, a=1, b=1, k=1, c=1, log=FALSE)pkumburr7(x, a=1, b=1, k=1, c=1, log.p=FALSE, lower.tail=TRUE)varkumburr7(p, a=1, b=1, k=1, c=1, log.p=FALSE, lower.tail=TRUE)eskumburr7(p, a=1, b=1, k=1, c=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
c the value of the third shape parameter, must be positive, the default is 1
k the value of the fourth shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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92 kumexp
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumburr7(x)pkumburr7(x)varkumburr7(x)eskumburr7(x)
kumexp Kumaraswamy exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy exponential dis-tribution due to Cordeiro and de Castro (2011) given by
f(x) = abλ exp(−λx) [1− exp(−λx)]a−1 {1− [1− exp(−λx)]
a}b−1,
F (x) = 1− {1− [1− exp(−λx)]a}b ,
VaRp(X) = − 1
λlog
{1−
[1− (1− p)1/b
]1/a},
ESp(X) = − 1
pλ
∫ p
0
log
{1−
[1− (1− v)1/b
]1/a}dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, andλ > 0, the scale parameter.
Usage
dkumexp(x, lambda=1, a=1, b=1, log=FALSE)pkumexp(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varkumexp(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eskumexp(p, lambda=1, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
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kumgamma 93
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumexp(x)pkumexp(x)varkumexp(x)eskumexp(x)
kumgamma Kumaraswamy gamma distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy gamma distribu-tion due to de Pascoa et al. (2011) given by
f(x) = cdbaxa−1 exp(−bx)γc−1(a, bx)
Γc(a)
[1− γc(a, bx)
Γc(a)
]d−1
,
F (x) = 1−[1− γc(a, bx)
Γc(a)
]d,
VaRp(X) =1
bQ−1
(a, 1−
[1− (1− p)1/d
]1/c),
ESp(X) =1
bp
∫ p
0
Q−1
(a, 1−
[1− (1− v)1/d
]1/c)dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the scale parameter, c > 0, thesecond shape parameter, and d > 0, the third shape parameter.
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94 kumgamma
Usage
dkumgamma(x, a=1, b=1, c=1, d=1, log=FALSE)pkumgamma(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)varkumgamma(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)eskumgamma(p, a=1, b=1, c=1, d=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
b the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
c the value of the second shape parameter, must be positive, the default is 1
d the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumgamma(x)pkumgamma(x)varkumgamma(x)eskumgamma(x)
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kumgumbel 95
kumgumbel Kumaraswamy Gumbel distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Gumbel distribu-tion due to Cordeiro et al. (2012a) given by
f(x) =ab
σexp
(µ− xσ
)exp
[−a exp
(µ− xσ
)]{1− exp
[−a exp
(µ− xσ
)]}b−1
,
F (x) = 1−{
1− exp
[−a exp
(µ− xσ
)]}b,
VaRp(X) = µ− σ log
{− log
[1− (1− p)1/b
]1/a},
ESp(X) = µ− σ
p
∫ p
0
log
{− log
[1− (1− v)1/b
]1/a}dv
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,a > 0, the first shape parameter, and b > 0, the second shape parameter.
Usage
dkumgumbel(x, a=1, b=1, mu=0, sigma=1, log=FALSE)pkumgumbel(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varkumgumbel(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eskumgumbel(p, a=1, b=1, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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96 kumhalfnorm
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumgumbel(x)pkumgumbel(x)varkumgumbel(x)eskumgumbel(x)
kumhalfnorm Kumaraswamy half normal distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy half normaldistribution due to Cordeiro et al. (2012c) given by
f(x) =2ab
σφ(xσ
) [2Φ(xσ
)− 1]a−1 {
1−[2Φ(xσ
)− 1]a}b−1
,
F (x) = 1−{
1−[2Φ(xσ
)− 1]a}b
,
VaRp(X) = σΦ−1
(1
2+
1
2
[1− (1− p)1/b
]1/a),
ESp(X) =σ
p
∫ p
0
Φ−1
(1
2+
1
2
[1− (1− v)1/b
]1/a)dv
for x > 0, 0 < p < 1, σ > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, thesecond shape parameter.
Usage
dkumhalfnorm(x, sigma=1, a=1, b=1, log=FALSE)pkumhalfnorm(x, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varkumhalfnorm(p, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eskumhalfnorm(p, sigma=1, a=1, b=1)
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kumloglogis 97
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumhalfnorm(x)pkumhalfnorm(x)varkumhalfnorm(x)eskumhalfnorm(x)
kumloglogis Kumaraswamy log-logistic distribution
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98 kumloglogis
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy log-logistic dis-tribution due to de Santana et al. (2012) given by
f(x) =abβαβxaβ−1
(αβ + xβ)a+1
[1− xaβ
(αβ + xβ)a
]b−1
,
F (x) =
[1− xaβ
(αβ + xβ)a
]b,
VaRp(X) = α
{[1− (1− p)1/b
]1/a− 1
}−1/β
,
ESp(X) =α
p
∫ p
0
{[1− (1− v)1/b
]1/a− 1
}−1/β
dv
for x > 0, 0 < p < 1, α > 0, the scale parameter, β > 0, the first shape parameter, a > 0, thesecond shape parameter, and b > 0, the third shape parameter.
Usage
dkumloglogis(x, a=1, b=1, alpha=1, beta=1, log=FALSE)pkumloglogis(x, a=1, b=1, alpha=1, beta=1, log.p=FALSE, lower.tail=TRUE)varkumloglogis(p, a=1, b=1, alpha=1, beta=1, log.p=FALSE, lower.tail=TRUE)eskumloglogis(p, a=1, b=1, alpha=1, beta=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
alpha the value of the scale parameter, must be positive, the default is 1
beta the value of the first shape parameter, must be positive, the default is 1
a the value of the second shape parameter, must be positive, the default is 1
b the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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kumnormal 99
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumloglogis(x)pkumloglogis(x)varkumloglogis(x)eskumloglogis(x)
kumnormal Kumaraswamy normal distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for Kumaraswamy normal distributiondue to Cordeiro and de Castro (2011) given by
f(x) =ab
σφ
(x− µσ
)Φa−1
(x− µσ
)[1− Φa
(x− µσ
)]b−1
,
F (x) = 1−[1− Φa
(x− µσ
)]b,
VaRp(X) = µ+ σΦ−1
([1− (1− p)1/b
]1/a),
ESp(X) = µ+σ
p
∫ p
0
Φ−1
([1− (1− v)1/b
]1/a)dv
for −∞ < x <∞, 0 < p < 1, −∞ < µ <∞, the location parameter, σ > 0, the scale parameter,a > 0, the first shape parameter, and b > 0, the second shape parameter.
Usage
dkumnormal(x, mu=0, sigma=1, a=1, b=1, log=FALSE)pkumnormal(x, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varkumnormal(p, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eskumnormal(p, mu=0, sigma=1, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
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100 kumpareto
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumnormal(x)pkumnormal(x)varkumnormal(x)eskumnormal(x)
kumpareto Kumaraswamy Pareto distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Pareto distributiondue to Pereira et al. (2013) given by
f(x) = abcKcx−c−1
[1−
(K
x
)c]a−1{1−
[1−
(K
x
)c]a}b−1
,
F (x) = 1−{
1−[1−
(K
x
)c]a}b,
VaRp(X) = K
{1−
[1− (1− p)1/b
]1/a}−1/c
,
ESp(X) =K
p
∫ p
0
{1−
[1− (1− v)1/b
]1/a}−1/c
dv
for x ≥ K, 0 < p < 1, K > 0, the scale parameter, c > 0, the first shape parameter, a > 0, thesecond shape parameter, and b > 0, the third shape parameter.
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kumpareto 101
Usage
dkumpareto(x, K=1, a=1, b=1, c=1, log=FALSE)pkumpareto(x, K=1, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)varkumpareto(p, K=1, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)eskumpareto(p, K=1, a=1, b=1, c=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
K the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
c the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumpareto(x)pkumpareto(x)varkumpareto(x)eskumpareto(x)
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102 kumweibull
kumweibull Kumaraswamy Weibull distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Weibull distribu-tion due to Cordeiro et al. (2010) given by
f(x) =abαxα−1
σαexp
[−(xσ
)α]{1− exp
[−(xσ
)α]}a−1[1−
{1− exp
[−(xσ
)α]}a]b−1
,
F (x) = 1−[1−
{1− exp
[−(xσ
)α]}a]b,
VaRp(X) = σ
[− log
{1−
[1− (1− p)1/b
]1/a}]1/α
,
ESp(X) =σ
p
∫ p
0
[− log
{1−
[1− (1− v)1/b
]1/a}]1/α
dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, α > 0,the third shape parameter, and σ > 0, the scale parameter.
Usage
dkumweibull(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)pkumweibull(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varkumweibull(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)eskumweibull(p, a=1, b=1, alpha=1, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
alpha the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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laplace 103
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dkumweibull(x)pkumweibull(x)varkumweibull(x)eskumweibull(x)
laplace Laplace distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Laplace distribution due to dueto Laplace (1774) given by
f(x) =1
2σexp
(−| x− µ |
σ
),
F (x) =
1
2exp
(x− µσ
), if x < µ,
1− 1
2exp
(−x− µ
σ
), if x ≥ µ,
VaRp(X) =
µ+ σ log(2p), if p < 1/2,
µ− σ log [2(1− p)] , if p ≥ 1/2,
ESp(X) =
µ+ σ [log(2p)− 1] , if p < 1/2,
µ+ σ − σ
p+ σ
1− pp
log(1− p) + σ1− pp
log 2, if p ≥ 1/2
for −∞ < x < ∞, 0 < p < 1, −∞ < µ < ∞, the location parameter, and σ > 0, the scaleparameter.
Usage
dlaplace(x, mu=0, sigma=1, log=FALSE)plaplace(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varlaplace(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eslaplace(p, mu=0, sigma=1)
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104 lfr
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedmu the value of the location parameter, can take any real value, the default is zerosigma the value of the scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlaplace(x)plaplace(x)varlaplace(x)eslaplace(x)
lfr Linear failure rate distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the linear failure rate distribution dueto Bain (1974) given by
f(x) = (a+ bx) exp(−ax− bx2/2
),
F (x) = 1− exp(−ax− bx2/2
),
VaRp(X) =−a+
√a2 − 2b log(1− p)
b,
ESp(X) = −ab
+1
bp
∫ p
0
√a2 − 2b log(1− v)dv
for x > 0, 0 < p < 1, a > 0, the first scale parameter, and b > 0, the second scale parameter.
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lfr 105
Usage
dlfr(x, a=1, b=1, log=FALSE)plfr(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varlfr(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eslfr(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be positive, the default is 1
b the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlfr(x)plfr(x)varlfr(x)eslfr(x)
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106 LNbeta
LNbeta Libby-Novick beta distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Libby-Novick beta distributiondue to Libby and Novick (1982) given by
f(x) =λaxa−1(1− x)b−1
B(a, b) [1− (1− λ)x]a+b
,
F (x) = I λx1+(λ−1)x
(a, b),
VaRp(X) =I−1p (a, b)
λ− (λ− 1)I−1p (a, b)
,
ESp(X) =1
p
∫ p
0
I−1v (a, b)
λ− (λ− 1)I−1v (a, b)
dv
for 0 < x < 1, 0 < p < 1, λ > 0, the scale parameter, a > 0, the first shape parameter, and b > 0,the second shape parameter.
Usage
dLNbeta(x, lambda=1, a=1, b=1, log=FALSE)pLNbeta(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varLNbeta(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esLNbeta(p, lambda=1, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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logbeta 107
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dLNbeta(x)pLNbeta(x)varLNbeta(x)esLNbeta(x)
logbeta Log beta distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the log beta distribution given by
f(x) =(log d− log c)1−a−b
xB(a, b)(log x− log c)a−1(log d− log x)b−1,
F (x) = I log x−log clog d−log c
(a, b),
VaRp(X) = c
(d
c
)I−1p (a,b)
,
ESp(X) =c
p
∫ p
0
(d
c
)I−1v (a,b)
dv
for 0 < c ≤ x ≤ d, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter,c > 0, the first location parameter, and d > 0, the second location parameter.
Usage
dlogbeta(x, a=1, b=1, c=1, d=2, log=FALSE)plogbeta(x, a=1, b=1, c=1, d=2, log.p=FALSE, lower.tail=TRUE)varlogbeta(p, a=1, b=1, c=1, d=2, log.p=FALSE, lower.tail=TRUE)eslogbeta(p, a=1, b=1, c=1, d=2)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
c the value of the first location parameter, must be positive, the default is 1
d the value of the second location parameter, must be positive and greater than c,the default is 2
a the value of the first scale parameter, must be positive, the default is 1
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108 logcauchy
b the value of the second scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlogbeta(x)plogbeta(x)varlogbeta(x)eslogbeta(x)
logcauchy Log Cauchy distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the log Cauchy distribution given by
f(x) =1
xπ
σ
(log x− µ)2 + σ2,
F (x) =1
πarctan
(log x− µ
σ
),
VaRp(X) = exp [µ+ σ tan (πp)] ,
ESp(X) =exp(µ)
p
∫ p
0
exp [σ tan (πv)] dv
for x > 0, 0 < p < 1, −∞ < µ <∞, the location parameter, and σ > 0, the scale parameter.
Usage
dlogcauchy(x, mu=0, sigma=1, log=FALSE)plogcauchy(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varlogcauchy(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eslogcauchy(p, mu=0, sigma=1)
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loggamma 109
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedmu the value of the location parameter, can take any real value, the default is zerosigma the value of the scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlogcauchy(x)plogcauchy(x)varlogcauchy(x)#eslogcauchy(x)
loggamma Log gamma distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the log gamma distribution due toConsul and Jain (1971) given by
f(x) =arxa−1(− log x)r−1
Γ(r),
F (x) = Q(r,−a log x),
VaRp(X) = exp
[−1
aQ−1(r, p)
],
ESp(X) =1
p
∫ p
0
exp
[−1
aQ−1(r, v)
]dv
for x > 0, 0 < p < 1, a > 0, the first shape parameter, and r > 0, the second shape parameter.
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110 loggamma
Usage
dloggamma(x, a=1, r=1, log=FALSE)ploggamma(x, a=1, r=1, log.p=FALSE, lower.tail=TRUE)varloggamma(p, a=1, r=1, log.p=FALSE, lower.tail=TRUE)esloggamma(p, a=1, r=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be positive, the default is 1
r the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dloggamma(x)ploggamma(x)varloggamma(x)esloggamma(x)
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logisexp 111
logisexp Logistic exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the logistic exponential distributiondue to Lan and Leemis (2008) given by
f(x) =aλ exp(λx) [exp(λx)− 1]
a−1
{1 + [exp(λx)− 1]a}2
,
F (x) =[exp(λx)− 1]
a
1 + [exp(λx)− 1]a ,
VaRp(X) =1
λlog
[1 +
(p
1− p
)1/a],
ESp(X) =1
pλ
∫ p
0
log
[1 +
(v
1− v
)1/a]dv
for x > 0, 0 < p < 1, a > 0, the shape parameter and λ > 0, the scale parameter.
Usage
dlogisexp(x, lambda=1, a=1, log=FALSE)plogisexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varlogisexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)eslogisexp(p, lambda=1, a=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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112 logisrayleigh
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlogisexp(x)plogisexp(x)varlogisexp(x)eslogisexp(x)
logisrayleigh Logistic Rayleigh distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the logistic Rayleigh distribution dueto Lan and Leemis (2008) given by
f(x) = aλx exp(λx2/2
) [exp
(λx2/2
)− 1]a−1
{1 +
[exp
(λx2/2
)− 1]a}−2
,
F (x) =
[exp
(λx2/2
)− 1]a
1 + [exp (λx2/2)− 1]a ,
VaRp(X) =
√2
λ
√√√√log
[1 +
(p
1− p
)1/a],
ESp(X) =
√2
p√λ
∫ p
0
{log
[1 +
(v
1− v
)1/a]}1/2
dv
for x > 0, 0 < p < 1, a > 0, the shape parameter, and λ > 0, the scale parameter.
Usage
dlogisrayleigh(x, a=1, lambda=1, log=FALSE)plogisrayleigh(x, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varlogisrayleigh(p, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)eslogisrayleigh(p, a=1, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
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logistic 113
log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlogisrayleigh(x)plogisrayleigh(x)varlogisrayleigh(x)eslogisrayleigh(x)
logistic Logistic distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the logistic distribution given by
f(x) =1
σexp
(−x− µ
σ
)[1 + exp
(−x− µ
σ
)]−2
,
F (x) =1
1 + exp(−x−µσ
) ,VaRp(X) = µ+ σ log [p(1− p)] ,ESp(X) = µ− 2σ + σ log p− σ 1− p
plog(1− p)
for −∞ < x < ∞, 0 < p < 1, −∞ < µ < ∞, the location parameter, and σ > 0, the scaleparameter.
Usage
dlogistic(x, mu=0, sigma=1, log=FALSE)plogistic(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varlogistic(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eslogistic(p, mu=0, sigma=1)
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114 loglaplace
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlogistic(x)plogistic(x)varlogistic(x)eslogistic(x)
loglaplace Log Laplace distribution
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loglaplace 115
Description
Computes the pdf, cdf, value at risk and expected shortfall for the log Laplace distribution given by
f(x) =
abxb−1
δb(a+ b), if x ≤ δ,
abδa
xa+1(a+ b), if x > δ,
F (x) =
axb
δb(a+ b), if x ≤ δ,
1− bδa
xa(a+ b), if x > δ,
VaRp(X) =
δ
[pa+ b
a
]1/b
, if p ≤ aa+b ,
δ
[(1− p)a+ b
a
]−1/a
, if p > aa+b ,
ESp(X) =
δb
b+ 1
[pa+ b
a
]1/b
, if p ≤ aa+b ,
aδ
p(1 + 1/b)(a+ b)+
a1/ab1−1/aδ
p(a+ b)(1− 1/a)
− δ(1− p)p(1− 1/a)
[a
(a+ b)(1− p)
]1/a
, if p > aa+b
for −∞ < x < ∞, 0 < p < 1, δ > 0, the scale parameter, a > 0, the first shape parameter, andb > 0, the second shape parameter.
Usage
dloglaplace(x, a=1, b=1, delta=0, log=FALSE)ploglaplace(x, a=1, b=1, delta=0, log.p=FALSE, lower.tail=TRUE)varloglaplace(p, a=1, b=1, delta=0, log.p=FALSE, lower.tail=TRUE)esloglaplace(p, a=1, b=1, delta=0)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computeddelta the value of the scale parameter, must be positive, the default is 1a the value of the first shape parameter, must be positive, the default is 1b the value of the second shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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116 loglog
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dloglaplace(x)ploglaplace(x)varloglaplace(x)esloglaplace(x)
loglog Loglog distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Loglog distribution due to Pham(2002) given by
f(x) = a log(λ)xa−1λxa
exp[1− λx
a],
F (x) = 1− exp[1− λx
a],
VaRp(X) =
{log [1− log(1− p)]
log λ
}1/a
,
ESp(X) =1
p(log λ)1/a
∫ p
0
{log [1− log(1− v)]}1/a dv
for x > 0, 0 < p < 1, a > 0, the shape parameter, and λ > 1, the scale parameter.
Usage
dloglog(x, a=1, lambda=2, log=FALSE)ploglog(x, a=1, lambda=2, log.p=FALSE, lower.tail=TRUE)varloglog(p, a=1, lambda=2, log.p=FALSE, lower.tail=TRUE)esloglog(p, a=1, lambda=2)
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loglogis 117
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be greater than 1, the default is 2
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dloglog(x)ploglog(x)varloglog(x)esloglog(x)
loglogis Log-logistic distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the log-logistic distribution given by
f(x) =babxb−1
(ab + xb)2 ,
F (x) =xb
ab + xb,
VaRp(X) = a
(p
1− p
)1/b
,
ESp(X) =a
pBp
(1 +
1
b, 1− 1
b
)
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118 loglogis
for x > 0, 0 < p < 1, a > 0, the scale parameter, and b > 0, the shape parameter, whereBx(a, b) =
∫ x0ta−1(1− t)b−1dt denotes the incomplete beta function.
Usage
dloglogis(x, a=1, b=1, log=FALSE)ploglogis(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varloglogis(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esloglogis(p, a=1, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the scale parameter, must be positive, the default is 1
b the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dloglogis(x)ploglogis(x)varloglogis(x)esloglogis(x)
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lognorm 119
lognorm Lognormal distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the lognormal distribution given by
f(x) =1
σxφ
(log x− µ
σ
),
F (x) = Φ
(log x− µ
σ
),
VaRp(X) = exp[µ+ σΦ−1(p)
],
ESp(X) =exp(µ)
p
∫ p
0
exp[σΦ−1(v)
]dv
for x > 0, 0 < p < 1, −∞ < µ <∞, the location parameter, and σ > 0, the scale parameter.
Usage
dlognorm(x, mu=0, sigma=1, log=FALSE)plognorm(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varlognorm(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eslognorm(p, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
mu the value of the location parameter, can take any real value, the default is zero
sigma the value of the scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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120 lomax
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlognorm(x)plognorm(x)varlognorm(x)eslognorm(x)
lomax Lomax distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Lomax distribution due to Lomax(1954) given by
f(x) =a
λ
(1 +
x
λ
)−a−1
,
F (x) = 1−(
1 +x
λ
)−a,
VaRp(X) = λ[(1− p)−1/a − 1
],
ESp(X) = −λ+λ− λ(1− p)1−1/a
p− p/afor x > 0, 0 < p < 1, a > 0, the shape parameter, and λ > 0, the scale parameter.
Usage
dlomax(x, a=1, lambda=1, log=FALSE)plomax(x, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varlomax(p, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)eslomax(p, a=1, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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Mlaplace 121
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dlomax(x)plomax(x)varlomax(x)eslomax(x)
Mlaplace McGill Laplace distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the McGill Laplace distribution dueto McGill (1962) given by
f(x) =
1
2ψexp
(x− θψ
), if x ≤ θ,
1
2φexp
(θ − xφ
), if x > θ,
F (x) =
1
2exp
(x− θψ
), if x ≤ θ,
1− 1
2exp
(θ − xφ
), if x > θ,
VaRp(X) =
θ + ψ log(2p), if p ≤ 1/2,
θ − φ log (2(1− p)) , if p > 1/2,
ESp(X) =
ψ + θ log(2p)− θp, if p ≤ 1/2,
θ + φ+ψ − φ− 2θ
2p+φ
plog 2− φ log 2
+φ
plog(1− p)− φ log(1− p), if p > 1/2
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122 Mlaplace
for −∞ < x < ∞, 0 < p < 1, −∞ < θ < ∞, the location parameter, φ > 0, the first scaleparameter, and ψ > 0, the second scale parameter.
Usage
dMlaplace(x, theta=0, phi=1, psi=1, log=FALSE)pMlaplace(x, theta=0, phi=1, psi=1, log.p=FALSE, lower.tail=TRUE)varMlaplace(p, theta=0, phi=1, psi=1, log.p=FALSE, lower.tail=TRUE)esMlaplace(p, theta=0, phi=1, psi=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
theta the value of the location parameter, can take any real value, the default is zero
phi the value of the first scale parameter, must be positive, the default is 1
psi the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dMlaplace(x)pMlaplace(x)varMlaplace(x)esMlaplace(x)
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moexp 123
moexp Marshall-Olkin exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin exponentialdistribution due to Marshall and Olkin (1997) given by
f(x) =λ exp(λx)
[exp(λx)− 1 + a]2 ,
F (x) =exp(λx)− 2 + a
exp(λx)− 1 + a,
VaRp(X) =1
λlog
2− a− (1− a)p
1− p,
ESp(X) =1
λlog [2− a− (1− a)p]− 2− a
λ(1− a)plog
2− a− (1− a)p
2− a+
1− pλp
log(1− p)
for x > 0, 0 < p < 1, a > 0, the first scale parameter and λ > 0, the second scale parameter.
Usage
dmoexp(x, lambda=1, a=1, log=FALSE)pmoexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varmoexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)esmoexp(p, lambda=1, a=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be positive, the default is 1
lambda the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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124 moweibull
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dmoexp(x)pmoexp(x)varmoexp(x)esmoexp(x)
moweibull Marshall-Olkin Weibull distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin Weibull distribu-tion due to Marshall and Olkin (1997) given by
f(x) = bλbxb−1 exp[(λx)b
] {exp
[(λx)b
]− 1 + a
}−2,
F (x) =exp
[(λx)b
]− 2 + a
exp[(λx)b
]− 1 + a
,
VaRp(X) =1
λ
[log
(1
1− p+ 1− a
)]1/b
,
ESp(X) =1
λp
∫ p
0
[log
(1
1− v+ 1− a
)]1/b
dv
for x > 0, 0 < p < 1, a > 0, the first scale parameter, b > 0, the shape parameter, and λ > 0, thesecond scale parameter.
Usage
dmoweibull(x, a=1, b=1, lambda=1, log=FALSE)pmoweibull(x, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varmoweibull(p, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)esmoweibull(p, a=1, b=1, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be positive, the default is 1
lambda the value of the second scale parameter, must be positive, the default is 1
b the value of the shape parameter, must be positive, the default is 1
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MRbeta 125
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dmoweibull(x)pmoweibull(x)varmoweibull(x)esmoweibull(x)
MRbeta McDonald-Richards beta distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the McDonald-Richards beta distri-bution due to McDonald and Richards (1987a, 1987b) given by
f(x) =xar−1 (bqr − xr)b−1
(bqr)a+b−1
B(a, b),
F (x) = I xrbqr
(a, b),
VaRp(X) = b1/rq[I−1p (a, b)
]1/r,
ESp(X) =b1/rq
p
∫ p
0
[I−1v (a, b)
]1/rdv
for 0 ≤ x ≤ b1/rq, 0 < p < 1, q > 0, the scale parameter, a > 0, the first shape parameter, b > 0,the second shape parameter, and r > 0, the third shape parameter.
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126 MRbeta
Usage
dMRbeta(x, a=1, b=1, r=1, q=1, log=FALSE)pMRbeta(x, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)varMRbeta(p, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)esMRbeta(p, a=1, b=1, r=1, q=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
q the value of the scale parameter, must be positive, the default is 1
a the value of the first shape parameter, must be positive, the default is 1
b the value of the second shape parameter, must be positive, the default is 1
r the value of the third shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dMRbeta(x)pMRbeta(x)varMRbeta(x)esMRbeta(x)
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nakagami 127
nakagami Nakagami distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Nakagami distribution due toNakagami (1960) given by
f(x) =2mm
Γ(m)amx2m−1 exp
(−mx
2
a
),
F (x) = 1−Q(m,
mx2
a
),
VaRp(X) =
√a
m
√Q−1(m, 1− p),
ESp(X) =
√a
p√m
∫ p
0
√Q−1(m, 1− v)dv
for x > 0, 0 < p < 1, a > 0, the scale parameter, and m > 0, the shape parameter.
Usage
dnakagami(x, m=1, a=1, log=FALSE)pnakagami(x, m=1, a=1, log.p=FALSE, lower.tail=TRUE)varnakagami(p, m=1, a=1, log.p=FALSE, lower.tail=TRUE)esnakagami(p, m=1, a=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the scale parameter, must be positive, the default is 1
m the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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128 normal
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dnakagami(x)pnakagami(x)varnakagami(x)esnakagami(x)
normal Normal distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the normal distribution due to deMoivre (1738) and Gauss (1809) given by
f(x) =1
σφ
(x− µσ
),
F (x) = Φ
(x− µσ
),
VaRp(X) = µ+ σΦ−1(p),
ESp(X) = µ+σ
p
∫ p
0
Φ−1(v)dv
for −∞ < x < ∞, 0 < p < 1, −∞ < µ < ∞, the location parameter, and σ > 0, the scaleparameter, where φ(·) denotes the pdf of a standard normal random variable, and Φ(·) denotes thecdf of a standard normal random variable.
Usage
dnormal(x, mu=0, sigma=1, log=FALSE)pnormal(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varnormal(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esnormal(p, mu=0, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedmu the value of the location parameter, can take any real value, the default is zerosigma the value of the scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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pareto 129
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dnormal(x)pnormal(x)varnormal(x)esnormal(x)
pareto Pareto distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Pareto distribution due to Pareto(1964) given by
f(x) = cKcx−c−1,
F (x) = 1−(K
x
)c,
VaRp(X) = K(1− p)−1/c,
ESp(X) =Kc
p(1− c)(1− p)1−1/c − Kc
p(1− c)
for x ≥ K, 0 < p < 1, K > 0, the scale parameter, and c > 0, the shape parameter.
Usage
dpareto(x, K=1, c=1, log=FALSE)ppareto(x, K=1, c=1, log.p=FALSE, lower.tail=TRUE)varpareto(p, K=1, c=1, log.p=FALSE, lower.tail=TRUE)espareto(p, K=1, c=1)
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130 paretostable
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedK the value of the scale parameter, must be positive, the default is 1c the value of the shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dpareto(x)ppareto(x)varpareto(x)espareto(x)
paretostable Pareto positive stable distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Pareto positive stable distributiondue to Sarabia and Prieto (2009) and Guillen et al. (2011) given by
f(x) =νλ
x
[log(xσ
)]ν−1
exp{−λ[log(xσ
)]ν},
F (x) = 1− exp{−λ[log(xσ
)]ν},
VaRp(X) = σ exp
{[− 1
λlog(1− p)
]1/ν},
ESp(X) =σ
p
∫ p
0
exp
{[− 1
λlog(1− v)
]1/ν}dv
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paretostable 131
for x > 0, 0 < p < 1, λ > 0, the first scale parameter, σ > 0, the second scale parameter, andν > 0, the shape parameter.
Usage
dparetostable(x, lambda=1, nu=1, sigma=1, log=FALSE)pparetostable(x, lambda=1, nu=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varparetostable(p, lambda=1, nu=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esparetostable(p, lambda=1, nu=1, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the first scale parameter, must be positive, the default is 1
sigma the value of the second scale parameter, must be positive, the default is 1
nu the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dparetostable(x)pparetostable(x)varparetostable(x)esparetostable(x)
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132 PCTAlaplace
PCTAlaplace Poiraud-Casanova-Thomas-Agnan Laplace distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Poiraud-Casanova-Thomas-Agnan Laplace distribution due to Poiraud-Casanova and Thomas-Agnan (2000) given by
f(x) =
a (1− a) exp {(1− a) (x− θ)} , if x ≤ θ,
a (1− a) exp {a (θ − x)} , if x > θ,
F (x) =
a exp {(1− a) (x− θ)} , if x ≤ θ,
1− (1− a) exp {a (θ − x)} , if x > θ,
VaRp(X) =
θ +
1
1− alog(pa
), if p ≤ a,
θ − 1
alog
(1− p1− a
), if p > a,
ESp(X) =
θ − log a
1− a+
log p− 1
(1− a)p, if p ≤ a,
θ − 1
a+
1
p− a
(1− a)p+
1− pap
log
(1− p1− a
), if p > a
for −∞ < x < ∞, 0 < p < 1, −∞ < θ < ∞, the location parameter, and a > 0, the scaleparameter.
Usage
dPCTAlaplace(x, a=0.5, theta=0, log=FALSE)pPCTAlaplace(x, a=0.5, theta=0, log.p=FALSE, lower.tail=TRUE)varPCTAlaplace(p, a=0.5, theta=0, log.p=FALSE, lower.tail=TRUE)esPCTAlaplace(p, a=0.5, theta=0)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
theta the value of the location parameter, can take any real value, the default is zero
a the value of the scale parameter, must be in the unit interval, the default is 0.5
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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perks 133
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dPCTAlaplace(x)pPCTAlaplace(x)varPCTAlaplace(x)esPCTAlaplace(x)
perks Perks distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Perks distribution due to Perks(1932) given by
f(x) =a exp(bx) [1 + a]
[1 + a exp(bx)]2 ,
F (x) = 1− 1 + a
1 + a exp(bx),
VaRp(X) =1
blog
a+ p
a(1− p),
ESp(X) = −(
1 +a
p
)log a
b+
(a+ p) log(a+ p)
bp+
(1− p) log(1− p)bp
for x > 0, 0 < p < 1, a > 0, the first scale parameter and b > 0, the second scale parameter.
Usage
dperks(x, a=1, b=1, log=FALSE)pperks(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varperks(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esperks(p, a=1, b=1)
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134 power1
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be positive, the default is 1
b the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dperks(x)pperks(x)varperks(x)esperks(x)
power1 Power function I distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the power function I distributiongiven by
f(x) = axa−1,F (x) = xa,
VaRp(X) = p1/a,
ESp(X) =p1/a
1/a+ 1
for 0 < x < 1, 0 < p < 1, and a > 0, the shape parameter.
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power1 135
Usage
dpower1(x, a=1, log=FALSE)ppower1(x, a=1, log.p=FALSE, lower.tail=TRUE)varpower1(p, a=1, log.p=FALSE, lower.tail=TRUE)espower1(p, a=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dpower1(x)ppower1(x)varpower1(x)espower1(x)
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136 power2
power2 Power function II distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the power function II distributiongiven by
f(x) = b(1− x)b−1,
F (x) = 1− (1− x)b,
VaRp(X) = 1− (1− p)1/b,
ESp(X) = 1 +b[(1− p)1/b+1 − 1
]p(b+ 1)
for 0 < x < 1, 0 < p < 1, and b > 0, the shape parameter.
Usage
dpower2(x, b=1, log=FALSE)ppower2(x, b=1, log.p=FALSE, lower.tail=TRUE)varpower2(p, b=1, log.p=FALSE, lower.tail=TRUE)espower2(p, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
b the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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quad 137
Examples
x=runif(10,min=0,max=1)dpower2(x)ppower2(x)varpower2(x)espower2(x)
quad Quadratic distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the quadratic distribution given by
f(x) = α(x− β)2,
F (x) =α
3
[(x− β)3 + (β − a)3
],
VaRp(X) = β +
[3p
α− (β − a)3
]1/3
,
ESp(X) = β +α
4p
{[3p
α− (β − a)3
]4/3
− (β − a)4
}
for a ≤ x ≤ b, 0 < p < 1, −∞ < a < ∞ , the first location parameter, and −∞ < a < b < ∞,the second location parameter, where α = 12
(b−a)3 and β = a+b2 .
Usage
dquad(x, a=0, b=1, log=FALSE)pquad(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)varquad(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)esquad(p, a=0, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first location parameter, can take any real value, the default iszero
b the value of the second location parameter, can take any real value but must begreater than a, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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138 rgamma
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dquad(x)pquad(x)varquad(x)esquad(x)
rgamma Reflected gamma distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the reflected gamma distribution dueto Borgi (1965) given by
f(x) =1
2φΓ (a)
∣∣∣∣x− θφ∣∣∣∣a−1
exp
{−∣∣∣∣x− θφ
∣∣∣∣} ,F (x) =
1
2Q
(a,θ − xφ
), if x ≤ θ,
1− 1
2Q
(a,x− θφ
), if x > θ,
VaRp(X) =
θ − φQ−1 (a, 2p) , if p ≤ 1/2,
θ + φQ−1 (a, 2(1− p)) , if p > 1/2,
ESp(X) =
θ − φ
p
∫ p
0
Q−1 (a, 2v) dv, if p ≤ 1/2,
θ − φ
p
∫ 1/2
0
Q−1 (a, 2v) dv +φ
p
∫ p
1/2
Q−1 (a, 2(1− v)) dv, if p > 1/2
for −∞ < x <∞, 0 < p < 1, −∞ < θ <∞, the location parameter, φ > 0, the scale parameter,and a > 0, the shape parameter.
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rgamma 139
Usage
drgamma(x, a=1, theta=0, phi=1, log=FALSE)prgamma(x, a=1, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)varrgamma(p, a=1, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)esrgamma(p, a=1, theta=0, phi=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
theta the value of the location parameter, can take any real value, the default is zero
phi the value of the scale parameter, must be positive, the default is 1
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)drgamma(x)prgamma(x)varrgamma(x)esrgamma(x)
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140 RS
RS Ramberg-Schmeiser distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Ramber-Schmeiser distributiondue to Ramberg and Schmeiser (1974) given by
VaRp(X) =pb − (1− p)c
d,
ESp(X) =pb
d(b+ 1)+
(1− p)c+1 − 1
pd(c+ 1)
for 0 < p < 1, b > 0, the first shape parameter, c > 0, the second shape parameter, and d > 0, thescale parameter.
Usage
varRS(p, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esRS(p, b=1, c=1, d=1)
Arguments
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
d the value of the scale parameter, must be positive, the default is 1
b the value of the first shape parameter, must be positive, the default is 1
c the value of the second shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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schabe 141
Examples
x=runif(10,min=0,max=1)varRS(x)esRS(x)
schabe Schabe distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Schabe distribution due to Schabe(1994) given by
f(x) =2γ + (1− γ)x/θ
θ(γ + x/θ)2 ,
F (x) =(1 + γ)x
x+ γθ,
VaRp(X) =pγθ
1 + γ − p,
ESp(X) = −θγ − θγ(1 + γ)
plog
1 + γ − p1 + γ
for x > 0, 0 < p < 1, 0 < γ < 1, the first scale parameter, and θ > 0, the second scale parameter.
Usage
dschabe(x, gamma=0.5, theta=1, log=FALSE)pschabe(x, gamma=0.5, theta=1, log.p=FALSE, lower.tail=TRUE)varschabe(p, gamma=0.5, theta=1, log.p=FALSE, lower.tail=TRUE)esschabe(p, gamma=0.5, theta=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
gamma the value of the first scale parameter, must be in the unit interval, the default is0.5
theta the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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142 secant
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dschabe(x)pschabe(x)varschabe(x)esschabe(x)
secant Hyperbolic secant distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the hyperbolic secant distributiongiven by
f(x) =1
2sech
(πx2
),
F (x) =2
πarctan
[exp
(πx2
)],
VaRp(X) =2
πlog[tan
(πp2
)],
ESp(X) =2
πp
∫ p
0
log[tan
(πv2
)]dv
for −∞ < x <∞, and 0 < p < 1.
Usage
dsecant(x, log=FALSE)psecant(x, log.p=FALSE, lower.tail=TRUE)varsecant(p, log.p=FALSE, lower.tail=TRUE)essecant(p)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computedlog if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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stacygamma 143
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dsecant(x)psecant(x)varsecant(x)essecant(x)
stacygamma Stacy distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for Stacy distribution due to Stacy (1962)given by
f(x) =cxcγ−1 exp [−(x/θ)c]
θcγΓ(γ),
F (x) = 1−Q(γ,(xθ
)c),
VaRp(X) = θ[Q−1(γ, 1− p)
]1/c,
ESp(X) =θ
p
∫ p
0
[Q−1(γ, 1− v)
]1/cdv
for x > 0, 0 < p < 1, θ > 0, the scale parameter, c > 0, the first shape parameter, and γ > 0, thesecond shape parameter.
Usage
dstacygamma(x, gamma=1, c=1, theta=1, log=FALSE)pstacygamma(x, gamma=1, c=1, theta=1, log.p=FALSE, lower.tail=TRUE)varstacygamma(p, gamma=1, c=1, theta=1, log.p=FALSE, lower.tail=TRUE)esstacygamma(p, gamma=1, c=1, theta=1)
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144 T
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
theta the value of the scale parameter, must be positive, the default is 1
c the value of the first scale parameter, must be positive, the default is 1
gamma the value of the second scale parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dstacygamma(x)pstacygamma(x)varstacygamma(x)esstacygamma(x)
T Student’s t distribution
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T 145
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Student’s t distribution due toGosset (1908) given by
f(x) =Γ(n+1
2
)√nπΓ
(n2
) (1 +x2
n
)−n+12
,
F (x) =1 + sign(x)
2− sign(x)
2I nx2+n
(n
2,
1
2
),
VaRp(X) =√nsign
(p− 1
2
)√1
I−1a
(n2 ,
12
) − 1,
where a = 2p if p < 1/2, a = 2(1− p) if p ≥ 1/2,
ESp(X) =
√n
p
∫ p
0
sign
(v − 1
2
)√1
I−1a
(n2 ,
12
) − 1dv,
where a = 2v if v < 1/2, a = 2(1− v) if v ≥ 1/2
for −∞ < x <∞, 0 < p < 1, and n > 0, the degree of freedom parameter.
Usage
dT(x, n=1, log=FALSE)pT(x, n=1, log.p=FALSE, lower.tail=TRUE)varT(p, n=1, log.p=FALSE, lower.tail=TRUE)esT(p, n=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
n the value of the degree of freedom parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
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146 TL
Examples
x=runif(10,min=0,max=1)dT(x)pT(x)varT(x)esT(x)
TL Tukey-Lambda distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Tukey-Lambda distribution dueto Tukey (1962) given by
VaRp(X) =pλ − (1− p)λ
λ,
ESp(X) =pλ+1 + (1− p)λ+1 − 1
pλ(λ+ 1)
for 0 < p < 1, and λ > 0, the shape parameter.
Usage
varTL(p, lambda=1, log.p=FALSE, lower.tail=TRUE)esTL(p, lambda=1)
Arguments
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
lambda the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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TL2 147
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)varTL(x)esTL(x)
TL2 Topp-Leone distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Topp-Leone distribution due toTopp and Leone (1955) given by
f(x) = 2b(x(2− x))b−1(1− x),
F (x) = (x(2− x))b,
VaRp(X) = 1−√
1− p1/b,
ESp(X) = 1− b
pBp1/b
(b,
3
2
)for x > 0, 0 < p < 1, and b > 0, the shape parameter.
Usage
dTL2(x, b=1, log=FALSE)pTL2(x, b=1, log.p=FALSE, lower.tail=TRUE)varTL2(p, b=1, log.p=FALSE, lower.tail=TRUE)esTL2(p, b=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
b the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
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148 triangular
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dTL2(x)pTL2(x)varTL2(x)esTL2(x)
triangular Triangular distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the triangular distribution given by
f(x) =
0, if x < a,
2(x− a)
(b− a)(c− a), if a ≤ x ≤ c,
2(b− x)
(b− a)(b− c), if c < x ≤ b,
0, if b < x,
F (x) =
0, if x < a,
(x− a)2
(b− a)(c− a), if a ≤ x ≤ c,
1− (b− x)2
(b− a)(b− c), if c < x ≤ b,
1, if b < x,
VaRp(X) =
a+
√p(b− a)(c− a), if 0 < p < c−a
b−a ,
b−√
(1− p)(b− a)(b− c), if c−ab−a ≤ p < 1,
ESp(X) =
a+
2
3
√p(b− a)(c− a), if 0 < p < c−a
b−a ,
b+a− cp
+2(2c− a− b)
3p+ 2√
(b− a)(b− c) (1− p)3/2
3p, if c−ab−a ≤ p < 1
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triangular 149
for a ≤ x ≤ b, 0 < p < 1, −∞ < a < ∞, the first location parameter, −∞ < a < c < ∞, thesecond location parameter, and −∞ < c < b <∞, the third location parameter.
Usage
dtriangular(x, a=0, b=2, c=1, log=FALSE)ptriangular(x, a=0, b=2, c=1, log.p=FALSE, lower.tail=TRUE)vartriangular(p, a=0, b=2, c=1, log.p=FALSE, lower.tail=TRUE)estriangular(p, a=0, b=2, c=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first location parameter, can take any real value, the default iszero
c the value of the second location parameter, can take any real value but must begreater than a, the default is 1
b the value of the third location parameter, can take any real value but must begreater than c, the default is 2
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dtriangular(x)ptriangular(x)vartriangular(x)estriangular(x)
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150 tsp
tsp Two sided power distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the two sided power distribution dueto van Dorp and Kotz (2002) given by
f(x) =
a(xθ
)a−1
, if 0 < x ≤ θ,
a
(1− x1− θ
)a−1
, if θ < x < 1,
F (x) =
θ(xθ
)a, if 0 < x ≤ θ,
1− (1− θ)(
1− x1− θ
)a, if θ < x < 1,
VaRp(X) =
θ(pθ
)1/a
, if 0 < p ≤ θ,
1− (1− θ)(
1− p1− θ
)1/a
, if θ < p < 1,
ESp(X) =
aθ
a+ 1
(pθ
)1/a
, if 0 < p ≤ θ,
1− θ
p+a(2θ − 1)
(a+ 1)p+a(1− θ)2
(a+ 1)p
(1− p1− θ
)1+1/a
, if θ < p < 1
for 0 < x < 1, 0 < p < 1, a > 0, the shape parameter, and −∞ < θ <∞, the location parameter.
Usage
dtsp(x, a=1, theta=0.5, log=FALSE)ptsp(x, a=1, theta=0.5, log.p=FALSE, lower.tail=TRUE)vartsp(p, a=1, theta=0.5, log.p=FALSE, lower.tail=TRUE)estsp(p, a=1, theta=0.5)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
theta the value of the location parameter, must take a value in the unit interval, thedefault is 0.5
a the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
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uniform 151
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dtsp(x)ptsp(x)vartsp(x)estsp(x)
uniform Uniform distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the uniform distribution given by
f(x) =1
b− a,
F (x) =x− ab− a
,
VaRp(X) = a+ p(b− a),
ESp(X) = a+p
2(b− a)
for a < x < b, 0 < p < 1, −∞ < a < ∞ , the first location parameter, and −∞ < a < b < ∞ ,the second location parameter.
Usage
duniform(x, a=0, b=1, log=FALSE)puniform(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)varuniform(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)esuniform(p, a=0, b=1)
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152 weibull
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs
to be computeda the value of the first location parameter, can take any real value, the default is
zerob the value of the second location parameter, can take any real value but must be
greater than a, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)duniform(x)puniform(x)varuniform(x)esuniform(x)
weibull Weibull distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Weibull distribution due toWeibull (1951) given by
f(x) =αxα−1
σαexp
{−(xσ
)α},
F (x) = 1− exp{−(xσ
)α},
VaRp(X) = σ [− log(1− p)]1/α ,ESp(X) =
σ
pγ (1 + 1/α,− log(1− p))
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weibull 153
for x > 0, 0 < p < 1, α > 0, the shape parameter, and σ > 0, the scale parameter.
Usage
dWeibull(x, alpha=1, sigma=1, log=FALSE)pWeibull(x, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varWeibull(p, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esWeibull(p, alpha=1, sigma=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
sigma the value of the scale parameter, must be positive, the default is 1
alpha the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dWeibull(x)pWeibull(x)varWeibull(x)esWeibull(x)
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154 xie
xie Xie distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Xie distribution due to Xie et al.(2002) given by
f(x) = λb(xa
)b−1
exp[(x/a)b
]exp (λa) exp
{−λa exp
[(x/a)b
]},
F (x) = 1− exp (λa) exp{−λa exp
[(x/a)b
]},
VaRp(X) = a
{log
[1− log(1− p)
λa
]}1/b
,
ESp(X) =a
p
∫ p
0
{log
[1− log(1− v)
λa
]}1/b
dv
for x > 0, 0 < p < 1, a > 0, the first scale parameter, b > 0, the shape parameter, and λ > 0, thesecond scale parameter.
Usage
dxie(x, a=1, b=1, lambda=1, log=FALSE)pxie(x, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varxie(p, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)esxie(p, a=1, b=1, lambda=1)
Arguments
x scaler or vector of values at which the pdf or cdf needs to be computed
p scaler or vector of values at which the value at risk or expected shortfall needsto be computed
a the value of the first scale parameter, must be positive, the default is 1
lambda the value of the second scale parameter, must be positive, the default is 1
b the value of the shape parameter, must be positive, the default is 1
log if TRUE then log(pdf) are returned
log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Value
An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.
Author(s)
Saralees Nadarajah
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xie 155
References
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted
Examples
x=runif(10,min=0,max=1)dxie(x)pxie(x)varxie(x)esxie(x)
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Index
∗Topic Value at risk, expectedshortfall
aep, 4arcsine, 6ast, 8asylaplace, 9asypower, 11beard, 13betaburr, 15betaburr7, 16betadist, 17betaexp, 19betafrechet, 20betagompertz, 21betagumbel, 23betagumbel2, 24betalognorm, 25betalomax, 26betanorm, 28betapareto, 29betaweibull, 30BS, 32burr, 33burr7, 34Cauchy, 35chen, 37clg, 38compbeta, 39dagum, 41dweibull, 42expexp, 44expext, 45expgeo, 46explog, 47explogis, 49exponential, 50exppois, 51exppower, 52expweibull, 54
F, 55FR, 56frechet, 57Gamma, 59genbeta, 60genbeta2, 61geninvbeta, 63genlogis, 64genlogis3, 65genlogis4, 67genpareto, 68genpowerweibull, 69genunif, 71gev, 72gompertz, 73gumbel, 75gumbel2, 76halfcauchy, 77halflogis, 78halfnorm, 80halfT, 81HBlaplace, 82HL, 83Hlogis, 84invbeta, 86invexpexp, 87invgamma, 88kum, 89kumburr7, 91kumexp, 92kumgamma, 93kumgumbel, 95kumhalfnorm, 96kumloglogis, 97kumnormal, 99kumpareto, 100kumweibull, 102laplace, 103lfr, 104
156
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INDEX 157
LNbeta, 106logbeta, 107logcauchy, 108loggamma, 109logisexp, 111logisrayleigh, 112logistic, 113loglaplace, 114loglog, 116loglogis, 117lognorm, 119lomax, 120Mlaplace, 121moexp, 123moweibull, 124MRbeta, 125nakagami, 127normal, 128pareto, 129paretostable, 130PCTAlaplace, 132perks, 133power1, 134power2, 136quad, 137rgamma, 138RS, 140schabe, 141secant, 142stacygamma, 143T, 144TL, 146TL2, 147triangular, 148tsp, 150uniform, 151weibull, 152xie, 154
∗Topic packageVaRES-package, 4
aep, 4arcsine, 6ast, 8asylaplace, 9asypower, 11
beard, 13betaburr, 15
betaburr7, 16betadist, 17betaexp, 19betafrechet, 20betagompertz, 21betagumbel, 23betagumbel2, 24betalognorm, 25betalomax, 26betanorm, 28betapareto, 29betaweibull, 30BS, 32burr, 33burr7, 34
Cauchy, 35chen, 37clg, 38compbeta, 39
daep (aep), 4dagum, 41darcsine (arcsine), 6dast (ast), 8dasylaplace (asylaplace), 9dasypower (asypower), 11dbeard (beard), 13dbetaburr (betaburr), 15dbetaburr7 (betaburr7), 16dbetadist (betadist), 17dbetaexp (betaexp), 19dbetafrechet (betafrechet), 20dbetagompertz (betagompertz), 21dbetagumbel (betagumbel), 23dbetagumbel2 (betagumbel2), 24dbetalognorm (betalognorm), 25dbetalomax (betalomax), 26dbetanorm (betanorm), 28dbetapareto (betapareto), 29dbetaweibull (betaweibull), 30dBS (BS), 32dburr (burr), 33dburr7 (burr7), 34dCauchy (Cauchy), 35dchen (chen), 37dclg (clg), 38dcompbeta (compbeta), 39ddagum (dagum), 41
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158 INDEX
ddweibull (dweibull), 42dexpexp (expexp), 44dexpext (expext), 45dexpgeo (expgeo), 46dexplog (explog), 47dexplogis (explogis), 49dexponential (exponential), 50dexppois (exppois), 51dexppower (exppower), 52dexpweibull (expweibull), 54dF (F), 55dfrechet (frechet), 57dGamma (Gamma), 59dgenbeta (genbeta), 60dgenbeta2 (genbeta2), 61dgeninvbeta (geninvbeta), 63dgenlogis (genlogis), 64dgenlogis3 (genlogis3), 65dgenlogis4 (genlogis4), 67dgenpareto (genpareto), 68dgenpowerweibull (genpowerweibull), 69dgenunif (genunif), 71dgev (gev), 72dgompertz (gompertz), 73dgumbel (gumbel), 75dgumbel2 (gumbel2), 76dhalfcauchy (halfcauchy), 77dhalflogis (halflogis), 78dhalfnorm (halfnorm), 80dhalfT (halfT), 81dHBlaplace (HBlaplace), 82dHlogis (Hlogis), 84dinvbeta (invbeta), 86dinvexpexp (invexpexp), 87dinvgamma (invgamma), 88dkum (kum), 89dkumburr7 (kumburr7), 91dkumexp (kumexp), 92dkumgamma (kumgamma), 93dkumgumbel (kumgumbel), 95dkumhalfnorm (kumhalfnorm), 96dkumloglogis (kumloglogis), 97dkumnormal (kumnormal), 99dkumpareto (kumpareto), 100dkumweibull (kumweibull), 102dlaplace (laplace), 103dlfr (lfr), 104dLNbeta (LNbeta), 106
dlogbeta (logbeta), 107dlogcauchy (logcauchy), 108dloggamma (loggamma), 109dlogisexp (logisexp), 111dlogisrayleigh (logisrayleigh), 112dlogistic (logistic), 113dloglaplace (loglaplace), 114dloglog (loglog), 116dloglogis (loglogis), 117dlognorm (lognorm), 119dlomax (lomax), 120dMlaplace (Mlaplace), 121dmoexp (moexp), 123dmoweibull (moweibull), 124dMRbeta (MRbeta), 125dnakagami (nakagami), 127dnormal (normal), 128dpareto (pareto), 129dparetostable (paretostable), 130dPCTAlaplace (PCTAlaplace), 132dperks (perks), 133dpower1 (power1), 134dpower2 (power2), 136dquad (quad), 137drgamma (rgamma), 138dschabe (schabe), 141dsecant (secant), 142dstacygamma (stacygamma), 143dT (T), 144dTL2 (TL2), 147dtriangular (triangular), 148dtsp (tsp), 150duniform (uniform), 151dWeibull (weibull), 152dweibull, 42dxie (xie), 154
esaep (aep), 4esarcsine (arcsine), 6esast (ast), 8esasylaplace (asylaplace), 9esasypower (asypower), 11esbeard (beard), 13esbetaburr (betaburr), 15esbetaburr7 (betaburr7), 16esbetadist (betadist), 17esbetaexp (betaexp), 19esbetafrechet (betafrechet), 20esbetagompertz (betagompertz), 21
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INDEX 159
esbetagumbel (betagumbel), 23esbetagumbel2 (betagumbel2), 24esbetalognorm (betalognorm), 25esbetalomax (betalomax), 26esbetanorm (betanorm), 28esbetapareto (betapareto), 29esbetaweibull (betaweibull), 30esBS (BS), 32esburr (burr), 33esburr7 (burr7), 34esCauchy (Cauchy), 35eschen (chen), 37esclg (clg), 38escompbeta (compbeta), 39esdagum (dagum), 41esdweibull (dweibull), 42esexpexp (expexp), 44esexpext (expext), 45esexpgeo (expgeo), 46esexplog (explog), 47esexplogis (explogis), 49esexponential (exponential), 50esexppois (exppois), 51esexppower (exppower), 52esexpweibull (expweibull), 54esF (F), 55esFR (FR), 56esfrechet (frechet), 57esGamma (Gamma), 59esgenbeta (genbeta), 60esgenbeta2 (genbeta2), 61esgeninvbeta (geninvbeta), 63esgenlogis (genlogis), 64esgenlogis3 (genlogis3), 65esgenlogis4 (genlogis4), 67esgenpareto (genpareto), 68esgenpowerweibull (genpowerweibull), 69esgenunif (genunif), 71esgev (gev), 72esgompertz (gompertz), 73esgumbel (gumbel), 75esgumbel2 (gumbel2), 76eshalfcauchy (halfcauchy), 77eshalflogis (halflogis), 78eshalfnorm (halfnorm), 80eshalfT (halfT), 81esHBlaplace (HBlaplace), 82esHL (HL), 83
esHlogis (Hlogis), 84esinvbeta (invbeta), 86esinvexpexp (invexpexp), 87esinvgamma (invgamma), 88eskum (kum), 89eskumburr7 (kumburr7), 91eskumexp (kumexp), 92eskumgamma (kumgamma), 93eskumgumbel (kumgumbel), 95eskumhalfnorm (kumhalfnorm), 96eskumloglogis (kumloglogis), 97eskumnormal (kumnormal), 99eskumpareto (kumpareto), 100eskumweibull (kumweibull), 102eslaplace (laplace), 103eslfr (lfr), 104esLNbeta (LNbeta), 106eslogbeta (logbeta), 107eslogcauchy (logcauchy), 108esloggamma (loggamma), 109eslogisexp (logisexp), 111eslogisrayleigh (logisrayleigh), 112eslogistic (logistic), 113esloglaplace (loglaplace), 114esloglog (loglog), 116esloglogis (loglogis), 117eslognorm (lognorm), 119eslomax (lomax), 120esMlaplace (Mlaplace), 121esmoexp (moexp), 123esmoweibull (moweibull), 124esMRbeta (MRbeta), 125esnakagami (nakagami), 127esnormal (normal), 128espareto (pareto), 129esparetostable (paretostable), 130esPCTAlaplace (PCTAlaplace), 132esperks (perks), 133espower1 (power1), 134espower2 (power2), 136esquad (quad), 137esrgamma (rgamma), 138esRS (RS), 140esschabe (schabe), 141essecant (secant), 142esstacygamma (stacygamma), 143esT (T), 144esTL (TL), 146
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160 INDEX
esTL2 (TL2), 147estriangular (triangular), 148estsp (tsp), 150esuniform (uniform), 151esWeibull (weibull), 152esxie (xie), 154expexp, 44expext, 45expgeo, 46explog, 47explogis, 49exponential, 50exppois, 51exppower, 52expweibull, 54
F, 55FR, 56frechet, 57
Gamma, 59genbeta, 60genbeta2, 61geninvbeta, 63genlogis, 64genlogis3, 65genlogis4, 67genpareto, 68genpowerweibull, 69genunif, 71gev, 72gompertz, 73gumbel, 75gumbel2, 76
halfcauchy, 77halflogis, 78halfnorm, 80halfT, 81HBlaplace, 82HL, 83Hlogis, 84
invbeta, 86invexpexp, 87invgamma, 88
kum, 89kumburr7, 91
kumexp, 92kumgamma, 93kumgumbel, 95kumhalfnorm, 96kumloglogis, 97kumnormal, 99kumpareto, 100kumweibull, 102
laplace, 103lfr, 104LNbeta, 106logbeta, 107logcauchy, 108loggamma, 109logisexp, 111logisrayleigh, 112logistic, 113loglaplace, 114loglog, 116loglogis, 117lognorm, 119lomax, 120
Mlaplace, 121moexp, 123moweibull, 124MRbeta, 125
nakagami, 127normal, 128
paep (aep), 4parcsine (arcsine), 6pareto, 129paretostable, 130past (ast), 8pasylaplace (asylaplace), 9pasypower (asypower), 11pbeard (beard), 13pbetaburr (betaburr), 15pbetaburr7 (betaburr7), 16pbetadist (betadist), 17pbetaexp (betaexp), 19pbetafrechet (betafrechet), 20pbetagompertz (betagompertz), 21pbetagumbel (betagumbel), 23pbetagumbel2 (betagumbel2), 24pbetalognorm (betalognorm), 25
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INDEX 161
pbetalomax (betalomax), 26pbetanorm (betanorm), 28pbetapareto (betapareto), 29pbetaweibull (betaweibull), 30pBS (BS), 32pburr (burr), 33pburr7 (burr7), 34pCauchy (Cauchy), 35pchen (chen), 37pclg (clg), 38pcompbeta (compbeta), 39PCTAlaplace, 132pdagum (dagum), 41pdweibull (dweibull), 42perks, 133pexpexp (expexp), 44pexpext (expext), 45pexpgeo (expgeo), 46pexplog (explog), 47pexplogis (explogis), 49pexponential (exponential), 50pexppois (exppois), 51pexppower (exppower), 52pexpweibull (expweibull), 54pF (F), 55pfrechet (frechet), 57pGamma (Gamma), 59pgenbeta (genbeta), 60pgenbeta2 (genbeta2), 61pgeninvbeta (geninvbeta), 63pgenlogis (genlogis), 64pgenlogis3 (genlogis3), 65pgenlogis4 (genlogis4), 67pgenpareto (genpareto), 68pgenpowerweibull (genpowerweibull), 69pgenunif (genunif), 71pgev (gev), 72pgompertz (gompertz), 73pgumbel (gumbel), 75pgumbel2 (gumbel2), 76phalfcauchy (halfcauchy), 77phalflogis (halflogis), 78phalfnorm (halfnorm), 80phalfT (halfT), 81pHBlaplace (HBlaplace), 82pHlogis (Hlogis), 84pinvbeta (invbeta), 86pinvexpexp (invexpexp), 87
pinvgamma (invgamma), 88pkum (kum), 89pkumburr7 (kumburr7), 91pkumexp (kumexp), 92pkumgamma (kumgamma), 93pkumgumbel (kumgumbel), 95pkumhalfnorm (kumhalfnorm), 96pkumloglogis (kumloglogis), 97pkumnormal (kumnormal), 99pkumpareto (kumpareto), 100pkumweibull (kumweibull), 102plaplace (laplace), 103plfr (lfr), 104pLNbeta (LNbeta), 106plogbeta (logbeta), 107plogcauchy (logcauchy), 108ploggamma (loggamma), 109plogisexp (logisexp), 111plogisrayleigh (logisrayleigh), 112plogistic (logistic), 113ploglaplace (loglaplace), 114ploglog (loglog), 116ploglogis (loglogis), 117plognorm (lognorm), 119plomax (lomax), 120pMlaplace (Mlaplace), 121pmoexp (moexp), 123pmoweibull (moweibull), 124pMRbeta (MRbeta), 125pnakagami (nakagami), 127pnormal (normal), 128power1, 134power2, 136ppareto (pareto), 129pparetostable (paretostable), 130pPCTAlaplace (PCTAlaplace), 132pperks (perks), 133ppower1 (power1), 134ppower2 (power2), 136pquad (quad), 137prgamma (rgamma), 138pschabe (schabe), 141psecant (secant), 142pstacygamma (stacygamma), 143pT (T), 144pTL2 (TL2), 147ptriangular (triangular), 148ptsp (tsp), 150
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162 INDEX
puniform (uniform), 151pWeibull (weibull), 152pxie (xie), 154
quad, 137
rgamma, 138RS, 140
schabe, 141secant, 142stacygamma, 143
T, 144TL, 146TL2, 147triangular, 148tsp, 150
uniform, 151
varaep (aep), 4vararcsine (arcsine), 6varast (ast), 8varasylaplace (asylaplace), 9varasypower (asypower), 11varbeard (beard), 13varbetaburr (betaburr), 15varbetaburr7 (betaburr7), 16varbetadist (betadist), 17varbetaexp (betaexp), 19varbetafrechet (betafrechet), 20varbetagompertz (betagompertz), 21varbetagumbel (betagumbel), 23varbetagumbel2 (betagumbel2), 24varbetalognorm (betalognorm), 25varbetalomax (betalomax), 26varbetanorm (betanorm), 28varbetapareto (betapareto), 29varbetaweibull (betaweibull), 30varBS (BS), 32varburr (burr), 33varburr7 (burr7), 34varCauchy (Cauchy), 35varchen (chen), 37varclg (clg), 38varcompbeta (compbeta), 39vardagum (dagum), 41vardweibull (dweibull), 42VaRES (VaRES-package), 4
VaRES-package, 4varexpexp (expexp), 44varexpext (expext), 45varexpgeo (expgeo), 46varexplog (explog), 47varexplogis (explogis), 49varexponential (exponential), 50varexppois (exppois), 51varexppower (exppower), 52varexpweibull (expweibull), 54varF (F), 55varFR (FR), 56varfrechet (frechet), 57varGamma (Gamma), 59vargenbeta (genbeta), 60vargenbeta2 (genbeta2), 61vargeninvbeta (geninvbeta), 63vargenlogis (genlogis), 64vargenlogis3 (genlogis3), 65vargenlogis4 (genlogis4), 67vargenpareto (genpareto), 68vargenpowerweibull (genpowerweibull), 69vargenunif (genunif), 71vargev (gev), 72vargompertz (gompertz), 73vargumbel (gumbel), 75vargumbel2 (gumbel2), 76varhalfcauchy (halfcauchy), 77varhalflogis (halflogis), 78varhalfnorm (halfnorm), 80varhalfT (halfT), 81varHBlaplace (HBlaplace), 82varHL (HL), 83varHlogis (Hlogis), 84varinvbeta (invbeta), 86varinvexpexp (invexpexp), 87varinvgamma (invgamma), 88varkum (kum), 89varkumburr7 (kumburr7), 91varkumexp (kumexp), 92varkumgamma (kumgamma), 93varkumgumbel (kumgumbel), 95varkumhalfnorm (kumhalfnorm), 96varkumloglogis (kumloglogis), 97varkumnormal (kumnormal), 99varkumpareto (kumpareto), 100varkumweibull (kumweibull), 102varlaplace (laplace), 103
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INDEX 163
varlfr (lfr), 104varLNbeta (LNbeta), 106varlogbeta (logbeta), 107varlogcauchy (logcauchy), 108varloggamma (loggamma), 109varlogisexp (logisexp), 111varlogisrayleigh (logisrayleigh), 112varlogistic (logistic), 113varloglaplace (loglaplace), 114varloglog (loglog), 116varloglogis (loglogis), 117varlognorm (lognorm), 119varlomax (lomax), 120varMlaplace (Mlaplace), 121varmoexp (moexp), 123varmoweibull (moweibull), 124varMRbeta (MRbeta), 125varnakagami (nakagami), 127varnormal (normal), 128varpareto (pareto), 129varparetostable (paretostable), 130varPCTAlaplace (PCTAlaplace), 132varperks (perks), 133varpower1 (power1), 134varpower2 (power2), 136varquad (quad), 137varrgamma (rgamma), 138varRS (RS), 140varschabe (schabe), 141varsecant (secant), 142varstacygamma (stacygamma), 143varT (T), 144varTL (TL), 146varTL2 (TL2), 147vartriangular (triangular), 148vartsp (tsp), 150varuniform (uniform), 151varWeibull (weibull), 152varxie (xie), 154
weibull, 152
xie, 154