parameter estimation for 3-parameter pareto

8/9/2019 Parameter Estimation for 3-Parameter Pareto

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Hydrological Sciences-Journal-des Sciences Hydrologiques,40,2,April 1995 1 6 5

Parameter estimation for 3-parameter

generalized pareto distribution by the

principle of maximum entropy (POME)

V. P. SINGH & H. GUO

DepartmentofCivilEngineering,Louisiana State University, BatonRouge,

Louisiana 70803-6405, USA

Abstract

The principle of maximum entropy (POME) is employed to

derive a new method of parameter estimation for the 3-parameter

generalized Pareto (GP) distribution. Monte Carlo simulated data are

used to evaluate this method and compare it with the methods of

moments (MOM), probability weighted moments (PW M), and maximum

likelihood estimation (MLE). The parameter estimates yielded by the

POME are either superior or comparable for high skewness.

Estimation des paramtres d'une loi de Pareto gnralise

trois paramtres par la mthode du maximum d'entropie

Rsum

Nous avons utilis le principe du maximum d'entropie en vue

d'tablir une nouvelle mthode d'estimation des paramtres de la

distribution de Pareto gnralise trois paramtres. Des donnes

synthtiques gnres selon une procdure de Monte Carlo ont t

utilises pour valuer cette mthode et pour la comparer aux mthodes

des moments, des moments pondrs et du maximum de vraisemblance.

L'estimation des paramtres s'appuyant sur le principe du maximum

d'entropie est prfrable ou comparable celle des autres mthodes en

particulier lorsque l'asymtrie est forte.

G EN ERA LIZED P A RETO D IS TRIBU TIO N

Consider a random variable Y with the standard exponential distribution. Let

a random variable Xbe defined as X = b{\

exp(-aY))/a, whereaandb are

parameters. Then the distribution of X is the 2-parameter generalized Pareto

distribution. If c is the threshold or lower bound ofX, then the distribution of

X

is the 3-parameter generalized Pareto (GP) distribution which can be

expressed as:

F(x)

=

1

-

1 -

=

1

- exp

a(x

b

x

c)

c

a jt

0

a = 0

(la)

( lb)

where cis a location parameter, b is a scale parameter, a is a shape parameter,

Openfordiscussion until 1 October 1995


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166

V. P. Singh N. Guo

andF{x)is the distribution function. The probability density function (PDF) of

the GP distribution is given by:

m

1

b

1 -

exp

aixc)

b

xc

b

a * 0

0

(2a)

(2b)

The Pareto distributions are obtained for a < 0. Figure1shows the PD F

for c = 0, b = 1.0, and various values of a. P ickands (1975) has shown that

the GP distribution given by equation (1) occurs as a limiting distribution for

excesses over thresholds if and only if the parent distribution is in the domain

of attraction of one of the extreme value distributions. The GP distribution

reduces to the 2-parameter GP distribution for c = 0, the exponential distri

bution for a 0 and c = 0, and the uniform distribution on [0,b]for c = 0

and(3= 1.

b)

z

o

rj

>

0.5-

< 7

-

4

o

0.3

0.0 0 .2 0.4 0.6 08 1.0 1.2 1.4 1.6

Line: a = 0.5; plus: a = 0.7 5;

star: a = 1.0; and dash: a = 1.25

0.0 0.2 0.4 0.6 0.8

1.2 1.4 1.6 1.8 2.0

Line: a = - 0 . 1 ; d a s h : a = - 0 . 5 ;

p lus : a = - 1 . 0

Fig. 1 Probability density function of generalized Pareto distribution with

(a)c = 0, b = 1.0, a = 0.5, 0.75, 1.0 and 1.25; and (b) with c = 0, b =

1.0,

a

= - 0 . 1 , - 0 . 5 and - 1 . 0 .

Some important properties of the GP distribution are worth mentioning:

(1) By com parison with the expon ential distribution, the GP distribution has

a heavier tail for a 0 (short-tailed distribution). When a

0; and

c

c,corresponding to a higher threshold Q

0

+ calso has a GP distri

bution. This is one of the properties that justifies the use of GP

distribution to model excesses.

Let Z = max(c, X

{

, X

2

, ..., X

N

), whereN > 0 is a number. IfX

h

i =

1,

2, .. . , N, are independent and identically distributed as a GP

distribution, and

N

has a Poisson distribution, then Z has a generalized

extreme value distribution (GEV) (Smith, 1984; Jin & Stedinger, 1989;

Wang, 1990), as defined by Jenkinson (1955). Thus, a Poisson process

of exceedance times with generalized Pareto excesses implies the

classical extreme value distributions. As a special case, the maximum of

a Poisson number of exponential varites lias a Gumbel distribution. So

exponential peaks lead to Gumbel maxima, and GP distribution peaks

lead to GEV maxima. The GEV can be expressed as:

F Z)

exp

exp

\l-

Z

~

y

13

-

exp

z-y

(3

l

I

-

-, -

0,

z

> 0

0

(3a)

(3b)

where the parameters


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168

V.

P.

Singh &N. Guo

reliability studies and the analysis of environmental extremes. Davison Smith

(1990) pointed out that the GP distribution might form the basis of a broad

modelling approach to high-level exceedances. DuMouchel (1983) applied it to

estimate the stable indexato measure tail thickness, whereas Davison (1984a,

1984b) modelled contamination due to long-range atmospheric transport of

radionuclides, van Montfort & Witter (1985, 1986) and van Montfort & Otten

(1991) applied the GP distribution to model the peaks over a threshold (POT)

streamflows and rainfall series, and Smith (1984, 1987, 1991) applied it to

analyse flood frequencies and wave heights. Similarly, Joe (1987) employed it

to estimate quantiles of the maximum of

iVobservations.

Wang (1991) applied

it to develop a POT model for flood peaks with Poisson arrival time, whereas

Rosbjerget

al.

(1992) compared the use ofthe2-parameter GP and exponential

distributions as distributionmodelsfor exceedances with the parent distribution

being a generalized GP distribution. In an extreme value analysis of the flow

of Burbage Brook, Barrett (1992) used the GP distribution to model the POT

flood series with Poisson inter-arrival times. Davison Smith (1990) presented

a comprehensive analysis of the extremes of data by use of the GP distribution

for modelling the sizes and occurrences of exceedances over high thresholds.

Methods for estimatingtheparameters of the 2-parameter GP distribution

were reviewed by Hosking & Wallis (1987). Quandt (1966) used the method

of moments (MOM ), while Baxter (1980) and Cook Mumme (1981) used the

method of maximum likelihood estimation (MLE) for the Pareto distribution.

The MOM, MLE and probability weighted moments (PWM) were included in

the review, van Montfort & Witter (1986) used the MLE to fit the GP distri

bution to represent the Dutch POT rainfall series and used an empirical

correction formula to reduce bias of the scale and shape parameter estimates.

Davison & Smith (1990) used the MLE, PWM, a graphical method and least

squares to estimate the GP distribution parameters. Wang (1991) derived the

PWM for both known and unknown thresholds.

OBJECTIVE OF

STUDY

The objective of this paper is to develop a new competitive method of

parameter estimation based on the principle of maximum entropy (POME), and

to compare it with the MOM, MLE and PWM using Monte Carlo simulated

data. The review of the literature shows that the POME does not appear to

have been employed for estimating parameters of the GP distribution.

DERIVATION OF PARAMETER ESTIMATION METHOD BY

POME

Shannon (1948) defined entropy as a numerical measure of uncertainty, or

conversely the information content associated with a probability distribution,


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Parameter estimation

for

generalized Pareto distribution 169

f(x;8), with a parameter vector0and usedtodescribearandom variableX.The

Shannon entropy function

H(f)

for continuousX canbeexpressedas:

H(f)

= -

fl.x;6) \nf(x;0)x

with

[/(x;0)dx

= l

4)

whereH f)is theentropy

off(x;0),

and can bethoughtof as themean valueof

-\nf(x;d).

AccordingtoJaynes (1961),theminimally biased distributionofXisthe

one which maximizes entropy subjectto given information, or which satisfies

the principleof maximum entropy (POME). Therefore, theparameters of the

distribution canbeobtainedbyachievingthem aximumof

H(f).

The useofthis

principlefor generatingthe least-biased probability distributionson thebasis

of limited

and

incomplete data

has

been discussed

by

several authors

and has

been applied to many diverse problems (e.g. a recent review by Singh

Fiorentino (1992)). Jaynes (1968)has reasoned thatthePOMEis thelogical

and rational criterionfor choosing some specific

f(x;d)

that maximizes Hand

satisfies the given information expressed as constraints. In other words, for

given information (e.g. mean, variance, skewness, lower limit, upper limit,

etc.),

thedistribution derivedby thePOM E w ould best represent

X;

implicitly,

this distribution would best represent the sample from whichthe information

was derived. Inversely,if it isdesiredto fit aparticular probab ility distribution

to

a

sample

of

data, then the POME can uniquely specify

the

constraints

(or the

information) neededtoderive that distribution. T he distribution parametersare

then related to these constraints. An excellent discussion of the underlying

mathematical rationaleisgiveninLevine Tribus (1979).

Givenmlinearly independent constraintsC

h

i = 1,2, ...,m,inthe form

C. =

\wfx)f{x;6)x,

i = 1,2,...,m (5)

where

w

t

(x )

are some functions whose averages

over f(x;6)

arespecified, then

the maximumofH subjecttoequation(5) isgivenby thedistribution:

f(x;6) = exp

-a

0

~ a,-w,-(x)

(=i

(6a)

where a

h

i = 0, 1, 2, ..., m, are the Lagrange multipliers, and can be

determined from equations(5) and (6a). Inserting equation (6a)inequation(4)

yieldstheentropy

of (x;6)

intermsofthe constraintsandLagrange mu ltipliers:

m

H(f) =

+

Y

j

a,C

i

(6b)

MaximizationofHthen establishestherelationships betw een constraints

and Lagrange multipliers. Thus,toderive a method usingthePOMEfor the

estimation of the parameters a, b and c of equation (2), three steps are


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170

V.

P. Singh N. Guo

involved: (i) specification of the appropriate constraints; (ii) derivation of the

entropy of the distribution; and (iii) derivation of the relationships between the

Lagrange multipliers and constraints. A complete mathematical discussion of

this method can be found in Tribus (1969), Jaynes (1968), Levine & Tribus

(1979) and Singh & Rajagopal (1986).

Specification of constraints

The entropy of the GP distribution can be derived by inserting equation (1) in

equation (4):

H(f) = lnof/fr ;0)dc-

1-1

a

In

. __a(x

c)

_

f(x;d)dx (

6 c

)

Comparing equation (6c) with equation (6b), the constraints appropriate for

equation (3) can be written (Singh & Rajagopal, 1986) as:

\f{x;d)

x = 1

7)

In

, _

a(xc)

f(x;6)dx = E In

1

_ a(xc)

b

(8)

in which E[*] denotes expectation of the bracketed quantity. These constraints

are unique and specify the information that is sufficient for the GP distribution.

The first constraint specifies the total probability. The second constraint

specifies the mean of the logarithm of the inverse ratio of the scale parameter

to the failure rate. Conceptually, this defines the expected value of the negative

logarithm of the scaled failure rate. The distribution parameters are related to

these constraints.

Construction

of

th e

entrop y function

The PDF of the GP distribution corresponding to the POME and consistent

with equations (7) and (8) takes the form:

f{x;d) =exp

a

Q

fljln

1

a(x - c)

(9)

where a

Q

and a

x

are Lagrange multipliers. The mathematical rationale for

equation (9) has been presented by Tribus (1969).

By applying equation (3) to the total probability condition in equation

(7),

one obtains:


7/18

exp(a

0

)

Parameter estimationforgeneralized Pareto distribution

a(x - c)

exp

-jln

1 - . dx

which yields the partition function:

exp(a

0

) = -

1

a

1

- a ,

The zeroth Lagrange multiplier is given by:

a

0

= In

b 1

a Ia,

Inserting equation (11) in equation (9) yields:

Ax-B)

a(\ a,)

1 -

a(xc)

A comparison of equation (13) with equation (3) yields:

1

Ia, =

a

Taking logarithms of equation (13) gives:

lnf(x;d) = l na+ l n ( l - a

x

) -Inb-a^n

aix - c)

b

Therefore, the entropyH(J) of the GP distribution follows:

H(f) = lna ln(l a{) +lnb+a

l

E\ In

1 -

a(xc)

111

(10)

(11)

(12)

(13)

(14)

(15)

(16)

Relationships between distr ibution parameters and constraints

According to Singh & Rajagopal (1986), the relationships between the

distribution parameters and constraints are obtained by taking partial derivatives

of the entropy

H(f)

with respect to the Lagrange multipliers as well as the

distribution parameters, and then equating these derivatives to ze ro, and m aking

use of the constraints. To that end, taking partial derivatives of equation (16)

with respect to a

x

, a, b and c separately and equating each derivative to zero

yields:

dH

da,

1

I

a,

E

In 1 a(x - c)

_

0

dH

da

=

-~a

t

E

(x - c)lb

1 - a(x - c)lb

0

(17)

(18)


8/18

172

V. P. Singh N. Guo

=

2

~db 1

dH

dc

= a j E

j j - E

(x - c)/6

1a(x

-

1

1a(xc)lb

-c)lb

= 0

(19)

(20)

Simplification of equations (17) to (20) yields, respectively:

1

In

1

a(xc)

b

(x - c)lb

1 a(xc)/b

(x

c)lb

1

a(x

c)/ft

1

I-a,

aa,

aa,

1 - a(xc)lb

(21)

22)

23)

24)

Clearly, equation (24) does not hold. Equation (22) is the same as equation

(23).

In order to get a unique solution, additional equations are needed which

can be obtained by differentiating the zeroth Lagrange multiplier with respect

to the Lagrange multipliers and equating the derivatives to zero. To that end,

equation (10) is written as:

a

Q

=

In exp fljln

1 -

a(xc)

dx

(25)

Differentiating equation (25) with respect to a

x

:

00

exp{fljln[l

a(xc)/b]}ln[l a(xc)/b]dx

da,

exp[a

0

ln{l a(xc)/b}]dx

^{-o.-aMl-^-cVbmi-aix-Omx

-E{[1-a(x-c)/b]}

(26)


9/18

Parameter estimation

for

generalized Pareto distribution

Following Tribus (1969):

var{ln[l

- a(x- c)/b]}

da,

173

27)

where var[] is the variance of the bracketed quantity. From equation (11):

a

0

=\n b/a)-\n l-a

l

) (28)

Differentiating equation (28) with respect to

a

{

:

(29)

30)

da

0

_

da

x

d \

1

1 flj

1

da,

a-^r

Equating equation (29) to equation (26) leads to:

In 1 -

a(x- c)

b

1

I

a,

(31)

which is the same as equation (21). When equation (30) is equated to equation

(27),

the following is obtained:

var

In

1

a(x

c)

1

( l - ^ )

2

32)

Therefore, theparam eter estimation equations for the POM E consistof

equations (21), (22) and (32). Inserting a

x

= 1 - lia from equation (14) into

these three equations, one gets:

1 -

a(x

c)

1

1

a(xc)lb

var

In

a(x

c)

_

=

a

I a

= a

33)

34)

35)

T H R E E O T H E R M E T H O D S O F P A R A M E T E R E S T I M A T I O N

Three of the most popular methods of parameter estimation are the method of

moments (MOM), the methodofprobability-weighted m oments (PW M ), and

the method ofmaximum likelihood estimation (M LE ). Th e POM E does not


10/18

174

V, P. Singh N. Guo

appear to have been used for estimating parameters of the GP distribution.

Therefore, virtually no literature exists on the comparison of parameter

estimates by the POME with those by the MLE, PWM and MOM. For the

sake of completeness, these methods are briefly summarized.

Method of moments (MOM)

Moment estimators of the GP distribution were derived by Hosking & Wallis

(1987). No te that E (l -

a(x

-

c)lb)

r

=

1/(1 +

ar)

if 1 + ra > 0. The rth

moment of X exists if a > - 1 / r . Provided that they exist, then the moment

estimators are:

x = c+Ji-

(36)

l+a

9

b

2

S

2

=

(37)

( l + f l )

2

( l +2a )

G = 2 ( l - Q ) ( l + 2 f l )

0

-

5

( 3 8 )

1

+3a

where x, S

2

and G are the mean, variance and skewness, respectively. First,

the moment estimate of a is obtained by solving equation (38). The relation

between G and

a

is illustrated in Fig. 2. With

a

calculated,

b

and

c

follow

from equation (36) and (37) as:

b =

S l+a) l+2af

5

(39)

c = x-- (40)

b+a

Probability-weighted moments

(PWM)

The PWM estimators for the GP distribution (Hosking & Wallis, 1987) are

given as:

a -

W

o~

SW

i-

9W

2

(41)

- W

0

+ 4W

1

3W

2

b = (

W

o-

2W

J(

W

o-

3W

2)(-4W

l+

6W

2

)

( - W

0

+ 4 W j - 3 F

2

)

2

2 W

Q

F

1

-6 W

0

W

2 +

6W

1

W

2

~W

0

+4W

l

-3W

2


11/18

Para meter estimation for generalized Pareto distribution

175

i.o-

0.8

1

xi 0.6

CC

0.4-

t n o . 2 i

< 0-0

a:

< -0.2

-0.4-1

-0.6

-0.8

-1.0

o i :

SKEWNESS G

Fig. 2 Parametera vsskewness Gfor GPD3.

where the rth probability-weighted moment

W

r

is:

l

W

r

= E[x(F)(l~F(x)Y] =

{c

+

- [ l - ( l - F )

a

] } ( l - f ) ' ' d F

1

r + 1

ft

1

a a+r+1

r = 0 ,1 ,2 , . .

44)

Method of maximum likelihood estimation

The MLE estimators can be expressed as:

j , Xj-cVb

=

^ _

frf

1

a(x

(

. -

c)lb Ia

45)

J2 ln[l - a(x

(

- c)/b] = na

(46)

A maximum likelihood estimator cannot be obtained for c, because the

likelihood function is unbounded with respect to c, as shown in Fig. 3. Since

c is the lower bound of the random variable X, we may use the constraint

c


12/18

V. P. Singh N. Guo

g

o

ZD

CL

8

X

_ l

UJ

o-

- i -

-2-

-3 -

-4 -

-5-

-6 -

-7-

- 8 -

- 9 -

-10-

- I I -

-12-

-13-

-14-

-15-^

OJO

0.1 0.2 0 3 0 .4 0.5

PARAMETER c

0.6

0.7

L in e: a = - 0 .1 1 6 , b 0.387 , c = 0 .562;

dash:

a 0 .544 , b = 1.116 c = 0.277

Fig . 3 Likelihood function of GPD 3 vs parameter c for sample size 10.

APPLICATION TO MONTE CARLO-SIMULATED DATA

Monte Carlo samples

To assess the performance ofthe POME estimation method by comparison with

the MOM, PWM and MLE, Monte Carlo sampling experiments were con

ducted. Two distribution population cases, listed in Table 1, were considered.

For each population case, 1000 random samples of size 20, 50 and 100 were

generated, and then parameters and quantiles were estimated.

Table 1 GP distribution population cases considered in the sampling

exper iment

GP distribution

population

Case 1

Case 2

c

v

0.5

0.5

G

0.5

2.5

Parameters

a b

0.554 1.116

- 0 .069 0 .433

c

0.277

0.536

C = coefficient of variation.


13/18

Parameter estimationforgeneralized Pareto distribution 111

Performance indices

The performance of the POM E was evaluated using the following performance

indices:

Standard bias BIAS =

(*>~* (47)

x

Root mean square error RMSE =

E

K

X - X

)

1

' (48)

A

, 2 i0 .5

X

wherex is an estimate ofx (parameter or quantile) and:

N

W) = jt*i

N

i=

i

whereTVis the number of Monte Carlo samples(N = 1000 in this study). 1000

may arguably not be a large enough number of samples to produce the true

values of BIAS and RMSE, but will suffice to compare the performances of the

estimation methods.

BIAS in parameter estimation

The bias of parameters estimated by the four methods is summarized in

Table 2. For G = 0.5, in absolute terms the MOM produced the least bias of

the four methods for all sample sizes. The MLE had the second least bias in

the parameter estimates. With increasing sample size, there was significant

reduction in bias for all four methods. The POME produced less bias than the

PWM in estimates ofb and c for all sample sizes, but that was not uniformly

true in the case of the estimate of parameter

a.

When G = 2.5, these methods

performed quite differently. For all samples sizes, the MLE and the POME

Table 2 BIAS of parameter estimates

Sample size

20

50

100

Method

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

M O M

PWM

MLE

POME

G =

0.5

a

0.156

0.488

0.217

- 0 . 3 9 7

0.063

0.230

0.132

- 0 . 4 0 7

0.040

0.132

0.086

- 0 . 2 8 8

b

0.094

0.632

0.037

- 0 . 1 2 2

0.042

0.258

0.060

- 0 . 0 9 6

0.028

0.138

0.048

- 0 . 0 6 0

c

- 0 . 0 5 3

- 0 . 9 4 8

0.215

- 0 . 0 9 4

- 0 . 0 2 5

- 0 . 3 9 6

0.067

- 0 . 1 5 6

- 0 . 0 1 9

- 0 . 2 0 8

0.039

- 0 . 1 2 6

G = 2.5

a

- 4 . 1 4 4

- 9 . 1 4 1

0.474

0.013

- 1 . 9 8 1

- 3 . 8 2 1

0.244

0.009

- 1 . 1 9 6

- 1 . 9 6 4

0.185

0.012

b

0.509

1.799

- 0 . 0 7 7

0.147

0.260

0.626

- 0 . 0 2 4

0.115

0.165

0.304

- 0 . 0 1 7

0.099

c

- 0 . 1 4 3

- 0 . 5 8 4

0.034

- 0 . 0 9 4

- 0 . 0 8 5

- 0 . 2 3 1

0.009

- 0 . 0 7 9

- 0 . 0 5 7

- 0 . 1 1 6

0.008

- 0 . 0 6 8


14/18

178

V. P. Singh N. Guo

were comparable, producing the least bias. For the a and c parameter

estimates, the POME had the least bias, but the MLE had the least bias for the

b

parameter estimate. The PWM had the highest bias in all three parameter

estimates for all sample sizes. Thus, if the value of G is high, the POME or

MLE may be the preferred method. For lower values of G, the MOM or MLE

may be preferable, especially when the sample size is small.

RMSE in parameter estimation

The values of RMSE of parameters estimated by the four methods are given in

Table 3. For G = 0.5, of the four methods the MOM produced the least

RMSE in thea parameter estimate. However, as the sample size increased, the

MOM, PWM and MLE became comparable. In the cases of the b and cpara

meter estimates, the MLE had the least RMSE, but all four methods were

comparable. For G = 2.5, the comparative behaviour of the four methods was

markedly different. In absolute terms, the MOM and the PWM produced the

highest RMSE in parameter estimates for all sample sizes, with the POME

having the least bias in the

a

parameter estimate but the MLE in the

b

and

c

parameter estimates. Thus, it may be concluded that for lower values of G, the

MOM or PWM may be the preferred method, but for higher values of G, the

MLE or POME is the preferred method.

BIAS in quantile estimation

The results of bias in quantile estimates by the GP distribution are summarized

in Table 4. The performance of the four estimation methods varied with the

value of G, and probability of non-exceedance P. For G = 0.5, all four

methods had comparable bias for P

0.99, the MOM and the PWM produced the smallest bias and the POME the

Tab le 3 RMSE of parameter estimates

Method

M OM

PWM

MLE

POME

M O M

PWM

MLE

POME

MOM

PWM

MLE

POME

a

0.448

0.780

0.502

0.785

0.301

0.419

0.329

0.696

0.203

0.268

0.224

0.590

b

0.310

0.820

0.284

0.371

0.213

0.365

0.234

0.271

0.144

0.211

0.176

0.233

c

0.336

0.984

0.357

0.348

0.201

0.427

0.146

0.262

0.139

0.237

0.056

0.185

a

- 5 . 1 7 8

--10.990

- 1 . 9 2 6

- 0 . 0 6 7

- 2 . 7 8 5

- 4 . 8 3 0

- 1 . 4 7 5

- 0 . 0 6 1

- 1 . 9 2 5

- 2 . 7 1 0

- 1 . 2 0 5

- 0 . 0 6 1

b

0.688

2.005

2.580

0.394

0.376

0.710

0.177

0.250

0.249

0.360

0.127

0.181

c

0.205

0.593

0.053

0.182

0.120

0.236

0.019

0.125

0.083

0.121

0.011

0.097


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Param eter estimation for generalized Pa reto distribution 179

Tab le 4 BIAS and RMSE of quantile estimates

p

0.8

0.9

0.99

0.999

Sample size

20

50

100

20

50

100

20

50

100

20

50

100

Method

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

MOM

PWM

MLE

POME

G

= 0.5

BIAS

0.000

0.091

- 0 . 0 1 1

- 0 . 0 3 0

0.001

0.041

0.010

0.000

- 0 . 0 1 2

0.076

- 0 . 0 3 7

0.031

- 0 . 0 2 1

0.153

- 0 . 0 8 2

- 0 . 0 6 2

- 0 . 0 0 4

0.032

- 0 . 0 0 5

0.066

0.000

0.018

0.004

0.048

- 0 . 0 2 6

0.036

- 0 . 0 6 7

0.287

- 0 . 0 0 9

0.007

- 0 . 0 2 9

0.323

- 0 . 0 0 5

0.002

0.014

0.230

- 0 . 0 2 2

0.028

- 0 . 0 6 3

0.582

- 0 . 0 0 5

0.000

- 0 . 0 3 4

0.612

- 0 . 0 0 4

- 0 . 0 0 4

- 0 . 0 1 9

0.439

RMSE

0.112

0.152

0.118

0.128

0.078

0.090

0.093

0.076

0.098

0.131

0.153

0.151

0.149

0.231

0.157

0.158

0.065

0.074

0.072

0.126

0.043

0.048

0.055

0.092

0.113

0.153

0.128

0.484

0.070

0.087

0.063

0.491

0.048

0.060

0.039

0.399

0.141

0.192

0.174

0.888

0.090

0.113

0.079

0.906

0.063

0.078

0.047

0.474

G =

2.5

BIAS

0.058

0.169

- 0 . 0 1 8

0.046

0.037

0.083

- 0 . 0 0 4

0.033

0.024

0.115

- 0 . 0 6 8

0.068

0.015

0.186

- 0 . 0 4 7

0.064

0.024

0.063

- 0 . 0 0 1

0.051

0.019

0.038

0.001

0.044

- 0 . 1 3 1

- 0 . 1 2 9

0.031

0.104

- 0 . 0 5 9

- 0 . 0 7 4

0.031

0.080

- 0 . 0 3 1

- 0 . 0 6 5

0.023

0.070

- 0 . 2 6 6

- 0 . 2 9 6

0.152

0.121

- 0 . 1 4 1

- 0 . 1 9 8

0.100

0.093

- 0 . 0 8 3

- 0 . 1 2 0

0.069

0.081

RMSE

0.172

0.224

0.134

0.176

0.107

0.125

0.090

0.109

0.197

0.225

0.221

0.221

0.273

0.348

0.224

0.271

0.123

0.131

0.106

0.137

0.084

0.085

0.073

0.098

0.309

0.372

0.286

0.297

0.205

0.235

0.203

0.186

0.154

0.165

0.150

0.135

0.427

0.600

0.572

0.332

0.310

0.393

0.406

0.207

0.252

0.289

0.295

0.151

highest, with the MLE in the intermediate range. However, for G = 2.5, the

POME produced the least bias, especially when P was greater than 0.99. For

all sample sizes, all four methods were somewhat comparable. In conclusion,


16/18

180

V. P. Singh N. Guo

for lower values of G, anyone of the four methods may be used for P 0.99 , the performance of the

POME deteriorated. When

G =

2.5, all methods produced comparable values

of RMSE for all sample sizes for P < 0.9; for P > 0.99 the POME had the

least RMSE. Thus, it is inferred that the MOM, PWM or MLE may be used

for smaller values of G, but for higher values of G, the POME may be the

preferred method.

C O N C L U S I O N S

The following conclusions can be drawn from this study: (1) the POME offers

an alternative method for estimating the parameters of the 3-parameter

generalized Pareto distribution; (2) when the skewness was high (G = 2.5), the

PO M E yielded superior param eter estimates; (3) for low skewness (G = 0.5 ),

the POME was better in parameter estimates than the MLE and PWM but

worse than the MOM; however, for large sample size, its performance

improved significantly; (4) the PO M E produced either better or comparable

quantile estimates as compared with the MOM, MLE and PWM for high

skewness (G = 2.5); (5) for low skewness (G = 0.5), the POME was

comparable to the MOM, the MLE and the PWM for lower probabilities of

nonexceedance which for higher values, the MOM or PWM was better than the

POME.

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Received 8 February 1993; 22 September 1994


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parameter estimation for 3-parameter pareto

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