test of normality against generalized exponential power ... · comes from a normal distribution....

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Test of normality against generalized exponential power alternatives Alain Desgagn´ e epartement de Math´ ematiques, Universit´ e du Qu´ ebec ` a Montr´ eal Pierre Lafaye de Micheaux 1 epartement de Math´ ematiques et Statistique, Universit´ e de Montr´ eal and Alexandre Leblanc Department of Statistics, University of Manitoba CRM-3314 Abstract The family of symmetric generalized exponential power (GEP) densities offers a wide range of tail behaviours, which may be exponential, polynomial and/or logarith- mic. In this paper, a test of normality based on Rao’s score statistic and this family of GEP alternatives is proposed. This test is tailored to detect departures from normality in the tails of the distribution. The main interest of this approach is that it provides a test with a large family of symmetric alternatives having non-normal tails. In addition, the test’s statistic consists of a combination of three quantities that can be interpreted as new measures of tail thickness. In a Monte-Carlo simulation study, the proposed test is shown to perform well in terms of power when compared to its competitors. 1 Corresponding Author

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Page 1: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Test of normality against

generalized exponential power alternatives

Alain Desgagne

Departement de Mathematiques, Universite du Quebec a Montreal

Pierre Lafaye de Micheaux1

Departement de Mathematiques et Statistique, Universite de Montreal

and

Alexandre Leblanc

Department of Statistics, University of Manitoba

CRM-3314

Abstract

The family of symmetric generalized exponential power (GEP) densities offers a

wide range of tail behaviours, which may be exponential, polynomial and/or logarith-

mic. In this paper, a test of normality based on Rao’s score statistic and this family of

GEP alternatives is proposed. This test is tailored to detect departures from normality

in the tails of the distribution. The main interest of this approach is that it provides a

test with a large family of symmetric alternatives having non-normal tails. In addition,

the test’s statistic consists of a combination of three quantities that can be interpreted

as new measures of tail thickness. In a Monte-Carlo simulation study, the proposed

test is shown to perform well in terms of power when compared to its competitors.

1Corresponding Author

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Key words: Test of normality, Rao’s score test, generalized exponential power, sym-

metric distributions, tail behaviour.

1991 MS Classification: 62E20, 62E25, 62F03, 60F05.

2

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1 Introduction

An important issue, in applied statistics, is the validity of the normality assumptions that

are often required for the use of many popular methods of statistical analysis. Consequently,

the problem of testing that a sample has been drawn from some normal distribution with

unknown mean and variance is one of the most common problems of goodness-of-fit in

statistical practice. For this, many test procedures have been proposed in the literature.

These tests can be mainly divided into three groups.

In the first group, the observed sample distribution is compared to the normal distribution

using some measure of discrepancy. This can be done in different ways, one of which is

the computation of a distance between the empirical distribution function (EDF) and the

normal cumulative distribution function (CDF). To accomplish this task, different norms

have been used. The tests proposed by Cramer (1928), Kolmogorov (1933) and Anderson

& Darling (1954) are very famous examples of this. See also Zhang & Wu (2005). Another

approach consists of assessing the closeness between a nonparametric estimate of the density

(usually a histogram, or some kernel estimator) and the normal density function. Again,

different norms have been used for this. Pearson’s χ2 test could be put into this category.

See also Fan (1994) and Cao & Lugosi (2005). Other approaches have also been considered.

For instance, we would also include in this group the procedure due to Epps & Pulley (1983)

which is based on the empirical characteristic function.

Moment based tests form the second group. Tests based on skewness or Pearson’s kurtosis,

or a combination of both, naturally fall into this group. See, for example, the tests proposed

by D’Agostino & Pearson (1973), Bowman & Shenton (1975), Jarque & Bera (1987) and

D’Agostino et al. (1990). We would also include here the test based on the ratio of the mean

(absolute) deviation to the standard deviation due to Geary (1935) and the test based on

G-Kurtosis introduced by Bonett & Seier (2002).

3

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The third group is composed of regression and correlation tests. See, for example, Shapiro &

Wilk (1965), D’Agostino (1971), Filliben (1975), Gan & Koehler (1990) and Coin (2008). The

Shapiro-Wilk test is quite possibly the most well-known and popular test of normality. It has

also been recognized as the best available omnibus tests by different authors who compared

the power of different tests for many different alternatives. See Shapiro et al. (1968), Pearson

et al. (1977) and D’Agostino & Stephens (1986, Section 9.4). Many authors have worked

on extensions of the Shapiro-Wilk test, including Shapiro & Francia (1972), Royston (1989,

1992) and Rahman & Govindarajulu (1997). Most of the tests in this group use a test

statistic that can essentially be viewed as the squared correlation R2 obtained from a normal

probability plot, that is, from a regression of the inverse normal CDF of the EDF on the

observations. The key here, is that R2 is expected to be close to unity when the sample

comes from a normal distribution.

There are, however, many other tests of normality in the literature, all based on different

ideas and characterizations of the normal distribution. Among them, we mention the test

of the ratio of the range to the standard deviation due to David et al. (1954) and the tests

related to entropy and the Kullback-Leibler information due to Vasicek (1976), Arizono

& Ohta (1989) and Park (1999). See also Locke & Spurrier (1976), Spiegelhalter (1977),

Oja (1983), LaRiccia (1986), Mudholkar et al. (2002) and Gel et al. (2007). For a compre-

hensive review of goodness-of-fit tests in general and in the particular case of normality, we

refer the reader to D’Agostino & Stephens (1986) and Thode (2002). Finally, more recent

papers on testing for normality are Carota (2010), Drezner et al. (2010), Goia et al. (2010),

Han (2010), Noughabi (2010), Sadooghi-Alvandi & Rasekhi (2009), Zghoul (2010). See

also the comprehensive simulation studies by Thadewald & Buning (2007) and Romao et

al. (2010).

In this paper, a new test of normality is proposed. This test is designed to detect non-normal

tail behaviour by considering a large family of symmetric alternatives and hence, is directional

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in nature. The strategy we adopt here, similar to the one used by Neyman (1937) or later by

Mardia et al. (1991), consists of using Rao’s score test (also known as the Lagrange multiplier

test, cf. Rao, 1948) and the generalized exponential power (GEP) family of densities which

can exhibit a large range of tail behaviours (cf. Desgagne & Angers, 2005). Specifically,

we develop a test of normality with unknown location and scale based on Rao’s score by

taking this vast family of densities as an alternative. Note that in the simple case where

location and scale are considered known, our procedure is equivalent to a test of a simple

null hypothesis about the GEP parameters. The resulting test statistic can be seen as a

combination of three measures of tail behaviour coming together to form an original and

simple test statistic having an asymptotically χ2 distribution with three degrees of freedom

under the null hypothesis.

For the remainder of this paper, we proceed as follows. In Section 2, the family of GEP

densities is presented and adapted for the purpose of our test. In Section 3, Rao’s score test

on the family of modified GEP densities is used to obtain a test of normality with unkown

location and scale. In Section 4, a Monte-Carlo simulation study is conducted to compare

the power of the proposed test with the power of other known tests of normality over a wide

range of alternatives. Finally, we briefly conclude in Section 5.

2 Generalized exponential power densities

The family of generalized exponential power densities covers a large range of tail behaviours,

ranging from exponential, to polynomial and logarithmic. This family of distributions has

been considered in applications where tail behaviour is of special interest, such as Bayesian

robustness (see Desgagne & Angers, 2007), the characterization of tails of distributions using

p-credence (see Angers, 2000) and the selection of an importance function in Monte Carlo

simulations (see Desgagne & Angers, 2005). In Section 3, we will use this family of densities

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as an embedding family of alternatives to develop a test of normality. This strategy can be

adopted because the normal density is included in the family of GEP densities. This idea is

not new and dates back to Neyman (1937). The main interest in the present approach is to

provide a simple and powerful test against symmetric alternatives.

Desgagne & Angers (2005) have defined the GEP density as having the following general

form,

p(x; γ, δ, α, β, z0) ∝ e−δ|x|γ

|x|−α (log |x|)−β if |x| ≥ z0,

and equal to p(z0; γ, δ, α, β, z0) for |x| < z0. Notice that the constant p(z0; γ, δ, α, β, z0) is

selected to make the density continuous. The density is symmetric with respect to the origin

and defined on the real line. The parameter space is a subset of R5 with other specific

conditions on the parameters guaranteeing that the density is proper, positive, bounded and

unimodal. For more details, see Desgagne & Angers (2005).

The exponential term, which includes the parameters γ and δ, provides a large spectrum

of tail behaviours. The polynomial term, controlled by the parameter α, provides densities

with heavy tails. The logarithmic term, using the parameter β, provides densities with even

heavier tails, sometimes called super heavy tails. The parameter z0 has little impact on the

tails of the density.

Certain known distributions are special cases of the GEP density. If the parameters α, β and

z0 are set to 0, we get the exponential power density introduced by Box & Tiao (1962). In

addition, if the parameter γ is set to 2, the normal density is obtained, and if it is instead

set to 1, we get the Laplace density. Also, the right tail of the GEP density (for x ≥ z0)

corresponds to certain known distributions. For example, the Weibull density is obtained by

letting α = 1 − γ and β = z0 = 0. Setting γ = 1, β = z0 = 0 and imposing the constraint

α < 1 gives the gamma density. The generalized gamma, Rayleigh and Maxwell-Boltzmann

densities can be obtained similarly. If the exponential term is removed (when γ = δ = 0),

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the GEP density is considered as a heavy-tailed distribution, with polynomial and possibly

logarithmic terms. For example, if we set γ = δ = β = 0 and impose the constraint α > 1,

the Pareto density is obtained. If we instead set γ = δ = 0 and z0 = 1 and impose the

constraints α > 1 and β < 1, we get the log-gamma density, an exponential transformation

of the gamma distribution. Letting γ = δ = 0 and α = 1 and imposing the constraints β > 1

and z0 > 1 leads to the super heavy-tailed log-Pareto density, an exponential transformation

of the Pareto distribution. Finally, note that GEP densities can approximate many classical

densities quite well in the tails, suggesting that a test of normality against GEP alternatives

should fare well against many non-GEP alternatives.

We now introduce a modified version of the GEP density to be used as a family of alternatives

for our test. This is done by setting certain parameters to fixed values and by adding carefully

chosen constants. Our rationale for doing this is the following.

First, note that the normal density N(0, σ2) is a special case of the GEP density, as it is

defined above, when γ = 2, δ = 12σ2 , α = 0, β = 0, z0 = 0. This means that δ is a function

of the scale parameter when normality is assumed. This is not desirable as the alternative

family will be extended to a location-scale family in the next section, our test statistic

incorporating classical maximum likelihood estimates of the location and scale parameters

under normality. In this context, we fix δ to avoid an additional degenerate term when

considering the score statistic (see (3) in the next section) and its use for the composite case.

We chose to set δ = 12

to include the N(0, 1) as a special case.

Secondly, it can be seen that the impact of the parameter z0 is greatest around the origin since

it essentially determines the constant part of the density. Its impact on the tail behaviour,

and consequently, its usefulness for detecting light and/or heavy tails can be expected to be

limited. We thus here set z0 = 0.

Finally, to ensure that the density remains positive and bounded for any values of the

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parameters γ, α and β, the constants 1 and e are added respectively to the polynomial and

logarithmic terms. Note that these constants have limited impact on the tail behaviour of

the densities.

The modified family of densities that results can be written in the following way. We will

write X ∼ GEP(θ) when the density of X is given by

g(x;θ) = k(θ) e−12|x|θ1 (1 + |x|

)−θ2( log(e + |x|))−θ3 , (1)

for all x ∈ R, where θ = (θ1, θ2, θ3)T is the vector of parameters, k(θ) is the normalizing

constant and θ1 ∈ R+, θ2 ∈ R and θ3 ∈ R. Note that the normalizing constant is generally

not available in closed form, although it is in simple cases. Also, the conditions θ2 ≥ 1 if

θ1 = 0, and θ3 > 1 if both θ1 = 0 and θ2 = 1, must be satisfied for the density to be proper.

We denote the resulting parameter space by Θ. Finally, note that the standard normal

density corresponds to the special case where θ = (2, 0, 0)T , and that this point is located in

the interior of the parameter space.

3 The proposed test for normality

Let X1, . . . , Xn be independent and identically distributed random variables with density

f(x;θ, µ, σ) =1

σg((x− µ)/σ;θ

),

where µ ∈ R and σ > 0 are respectively the unknown location and scale parameters, and

g(x;θ) is defined as in (1). We will write X ∼ GEP(θ;µ, σ) when the density of X is given

by f(x;θ, µ, σ).

Our goal, here, is to carry out a test that the measurements of X come from some N(µ, σ2)

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distribution (with µ and σ unspecified), considering GEP distributions as a family of alter-

natives or, equivalently, assuming that these measurements come from some GEP(θ;µ, σ)

distribution. In other words, we want to test

H0 : L(X) ∈ {N(µ, σ2);µ ∈ R, σ > 0} = {GEP(2, 0, 0;µ, σ);µ ∈ R, σ > 0},

against (2)

H1 : L(X) ∈ {GEP(θ;µ, σ);θ ∈ Θ/{(2, 0, 0)T}, µ ∈ R, σ > 0},

with Θ denoting the parameter space introduced in Section 2.

Different approaches could be used to achieve this. Maximum likelihood techniques obviously

come to mind, but the fact that the normalizing constant k(θ) in (1) is in general not

analytically tractable makes the use of these techniques difficult. We here propose to take

a different approach and make use of Rao’s score test, considering µ and σ as nuisance

parameters. If we define

d0(y) =∂

∂θlog g(y;θ)

∣∣∣θ=(2,0,0)T

,

it can be shown that

d0(y) =

0.18240929− 1

2y2 log |y|

0.53482230− log(1 + |y|)

0.20981558− log(

log(e +|y|))

. (3)

Then the score vector to be used for our test is defined, for given values of µ and σ, as

rn(µ, σ) =1

n

n∑i=1

d0(Yi), (4)

where Yi = (Xi− µ)/σ. Note that the three included constants ensure that each component

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of the score vector has zero mean under normality. Note also that the first element of the

vector in (3) is not defined for Yi = 0. However, by convention, we set it to zero when Yi = 0

since limy→0(y2 log |y|) = 0. It is worth noting that rn(µ, σ) consists of three new measures

of tail thickness given by

r(1)n (µ, σ) = 0.18240929− 1

2n

n∑i=1

Y 2i log |Yi|,

r(2)n (µ, σ) = 0.53482230− 1

n

n∑i=1

log(1 + |Yi|),

r(3)n (µ, σ) = 0.20981558− 1

n

n∑i=1

log(

log(e +|Yi|)).

As pointed out by a referee, it would be interesting to study more specifically the behaviour

of the above quantities and compare them, for example, to Pearson’s sample kurtosis or to

G-Kurtosis (cf. Bonett & Seir, 2002) in how they assess thickness of the tails of specific

distributions.

Now, under the null hypothesis of normality, the results of Rao (1948) imply that

n1/2rn(µ, σ)D−→ N3(0,J0) and n rn(µ, σ)TJ−1

0 rn(µ, σ)D−→ χ2

3, (5)

where the information matrix J0 is defined as

J0 = E0

[d0(Y )d0(Y )T

],

where Y = (X − µ)/σ and the notation E0[·] indicates that the expectation is taken under

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the null hypothesis. It is not difficult to show that

J0 =

0.42423099 0.13285410 0.05401580

0.13285410 0.10080860 0.04056528

0.05401580 0.04056528 0.01632673

, (6)

doing the involved numerical calculations by using Monte-Carlo simulations or any appro-

priate numerical integration method.

However, note that the result in (5) has no practical utility since, µ and σ being unknown,

rn(µ, σ) cannot be computed from the observations at hand. To circumvent this, the strat-

egy we adopt is quite natural, and similar to the one adopted by Jarque & Bera (1987).

Essentially, we still rely on the score vector, but use the maximum likelihood estimators of

µ and σ under normality, respectively

Xn =1

n

n∑i=1

Xi and Sn =

√√√√ 1

n

n∑i=1

(Xi − Xn)2,

as surrogates for the unknown µ and σ. Hence, we simply propose to base the composite test

of normality on rn(Xn, Sn). However, the repercussions of this substitution have to be well

understood. The remainder of this section is devoted to obtaining the asymptotic behaviour

of rn(Xn, Sn) and of the resulting test statistic.

For this, we first establish the (asymptotic) link between rn(Xn, Sn) and rn(µ, σ) so as to

make use of the results given in (5) above. An advantage of using this approach is that it

explicitly describes the effect of using Xn and Sn in lieu of the unknown parameters when

computing the test statistic.

Proposition 1. When X ∼ N(µ, σ2), we have that

rn(Xn, Sn) = rn(µ, σ)− 1

2(1− Tn)v0 + oP

(n−

12

)13,

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where

Tn =1

n

n∑i=1

(Xi − µσ

)2

,

and where 13 = (1, 1, 1)T and

v0 = −E0

[Y 2d0(Y )

]= (0.86481850, 0.38512925, 0.15594892)T . (7)

A proof of this result is given in the Appendix with the vector v0 requiring the use of

numerical integration to be calculated.

Based on the previous result, it is now possible to determine the asymptotic distribution of

rn(Xn, Sn) and to present the statistic to be used for the composite test of normality, denoted

hereafter by Rn. The presentation of a proof of Theorem 1 is delayed to the Appendix. This

is our main result.

Theorem 1. The modified score vector for the composite test of normality of the hypothe-

ses given in (2) is rn(Xn, Sn) where rn is defined as in (4). Furthermore, under the null

hypothesis,

n1/2rn(Xn, Sn)D−→ N3

(0,J0 −

1

2v0v

T0

),

where J0 is given in (6) and the vector v0 is given in (7). The proposed statistic to test the

composite hypothesis of normality is given by

Rn = n rn(Xn, Sn)T(J0 −

1

2v0v

T0

)−1

rn(Xn, Sn),

and, under the null hypothesis, we have that RnD−→ χ2

3.

Note that the test statistic is location and scale invariant. Note also that the asymptotic

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covariance matrix of n1/2rn(Xn, Sn) is given by

J0 −1

2v0v

T0 =

0.0502754623 −0.0336793487 −0.0134179540

−0.0336793487 0.0266463308 0.0105350321

−0.0134179540 0.0105350321 0.00416669944

,

which is quite different from the asymptotic covariance matrix of n1/2rn(µ, σ) given in (6).

We next study the power of the resulting procedure through some simulations. Comparisons

are made with many other tests that are available in the literature.

4 Practical Considerations and Simulations

In this section, we first give a formula to compute quantiles and p-values for our test for finite

sample sizes. This is done since the convergence in distribution of the statistic Rn seems

very slow. Then, we compare the power of the proposed test for the composite hypothesis

of normality with selected tests for different symmetric alternatives.

In order to see if the distribution of Rn, for finite samples, can be approximated by the

Chi-square distribution with 3 degrees of freedom, we estimated the power of our test under

the null hypothesis of normality, using the Chi-square quantiles, and we observed that the

results were too far from the exact level, for different levels and sample sizes. We concluded

that the use of simulated quantiles, or of a formula that estimates these quantiles, would be

necessary in practice. To derive such a formula, we simulated quantiles of different orders for

different sample sizes and were able to find a simple formula to estimate all these quantiles.

Specifically, we proceeded as follows.

For each sample size n given in Table 1, we found the best regression of the quantiles of

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order α explained by a power of α, ranging from 0.01 to 0.50. That is, we found the values

of an, bn and cn that maximize the R2 of the regression

qα;n = an + bnαcn , (8)

where qα;n is the simulated quantile such that Pr[Rn > qα;n] ≈ α. Doing this, we found the fit

to be remarkably good, R2 values ranging from 0.9995 to 0.9999 for all the considered sample

sizes. The coefficients an and bn and the exponent cn are given in Table 1. For example, if

n = 50 and the chosen level of the test is α = 0.05, then the hypothesis of normality of the

observations should be rejected if

R50 > q0.05;50 = −8.805 + 9.650(0.05)−0.17 = 7.2534.

Furthermore, a formula for p-values between 0.01 and 0.50 can easily be obtained by invert-

ing (8). This leads to

p-value ≈(Rn − an

bn

)1/cn, (9)

where Rn is the observed value of the test statistic. For example, if we observed R50 = 6.02,

the approximate p-value of the test would be

p-value ≈(6.02− (−8.805)

9.650

)1/(−0.17)

= 0.08,

and the hypothesis of normality would not be rejected at a level of α = 0.05. In addition to

the excellent fit of the regressions to estimate the quantiles, we assessed the quality of the

fit by performing a study of the empirical power under the null of Rn based on 1,000,000

simulations and on quantiles calculated from (8) and from the coefficients given in Table 1.

The results given in Table 2 suggest that the level of the test is very accurate using the

formula.

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Next, we performed simulations to compare the power of our proposed test based on Rn

for the composite hypothesis of normality with selected tests for different symmetric al-

ternatives. We selected all the test-competitors considered in Romao et al. (2010), an ex-

tensive empirical power study of tests for normality. These tests are Kolmogorov-Smirnov

(K − S), Anderson-Darling (AD∗), the Zhang-Wu ZC and ZA tests, the Glenn-Leemis-Barr

test (PS), the D’Agostino-Pearson K2 test, the Jarque-Bera test (JB), the Doornik-Hansen

test (DH), the Gel-Gastwirth robust Jarque-Bera test (RJB), the Hosking L-moments

based test (TLmom), the Hosking test based on trimmed L-moments (T(1)Lmom, T

(2)Lmom and

T(3)Lmom), the Bontemps-Meddahi tests (BM3−4 and BM3−6), the Brys-Hubert-Struyf MC-LR

tests (TMC−LR), the Bonett-Seier test (Tw), the Brys-Hubert-Struyf-Bonett-Seier joint test

(TMC−LR − Tw), the Cabana-Cabana tests (TS,5 and TK,5), the Shapiro-Wilk test (W ), the

Shapiro-Francia test (WSF ), the Rahman-Govindarajulu modification of the Shapiro-Wilk

test (WRG), D’Agostino’s D test (Da), the Filliben correlation test (r), the Chen-Shapiro test

(CS), Zhang’s Q test, the del Barrio-Cuesta-Albertos-Matran-Rodrıguez-Rodrıguez quantile

correlation test (BCMR), the β23 Coin test, the Epps-Pulley test (TEP ), the Martinez-

Iglewicz test (In) and the Gel-Miao-Gastwirth test (RsJ). We refer the reader to Romao et

al. (2010) for details and references on these tests. We also added the test of Spiegelhal-

ter (1977) (S), which is also a directional test for symmetric alternatives.

Different symmetric alternatives have been chosen. We first studied 142 GEP alternatives to

cover a large spectrum of tail behaviours. They are listed in Tables 3 to 6. For each of the

following kurtosis β2 of 1.1, 1.5, 1.84, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 4, 4.5, 5, 6, 10, 20, 50,

we considered several GEP with different parameters θ1, θ2, θ3. For each kurtosis, we plotted

these GEP densities in Figures 1 to 3. It is interesting to see that for the same kurtosis, the

tail behaviour can vary significantly. The kurtosis have been chosen to cover the densities

from very short-tailed to very heavy-tailed, and the parameters have been chosen to cover

all the possible shapes for each given kurtosis. We think that this set of alternatives is very

comprehensive and will give us the best indication of the performance of the tests to detect

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non-normality for symmetric alternatives.

We also studied 28 symmetric alternatives considered in Romao et al. (2010) and usually in

the empirical power studies of tests for normality. They are listed in Tables 7 to 10. We find

the Beta, Location Contaminated Normal (LoConN), Tukey, Johnson SB (JSB), Johnson

SU (JSU), Student, logistic and Laplace, with their kurtosis β2 ranging from 1.5 to infinity.

This set of alternatives is relatively limited when covering the full spectrum of tail behaviours,

but it is still interesting to observe the performance of our test against the usual chosen al-

ternative in empirical studies not using GEP densities. Note also that we reparametrized the

classical Johnson SU distribution JSU(γ, δ, ξ, λ) to obtain our JSU(ν, τ, µ, σ) distribution

by letting ν = −γ, τ = 1/δ, µ = ξ−λ√

exp(δ−2) sinh(γ/δ) and σ = λ/c. Doing so, the mean

and standard deviation of our JSU distribution is respectively µ and σ. See Jonhson (1949)

for further details.

Note that we have included, in these tables, results for the GEP (1,−1.1, 0) alternative.

Our reason for doing so is that the Student-t(10) density is virtually undistinguishable from

that GEP density. It is interesting to note that all tests perform the same for both of these

alternatives, that is, for the Student density and its approximation by a GEP density. Again,

this suggests that a test of normality against GEP alternatives should fare well against non-

GEP alternatives that can be approximated by a member of the GEP family of distributions.

In Tables 3 to 10, we report the empirical power (as measured by the proportion of simulated

samples for which the composite hypothesis of normality was rejected) under the considered

alternatives. For our simulation study, we generated 1,000,000 samples under each alternative

(except for the GEP alternatives for which we generated 100,000 samples) and for sample

sizes of n = 20, 50, 100 and 200 at a significance level of 5%. We also report three measures

of performance for each test in each simulation. First, we report the average power against

all alternatives. Then, for each alternative, we also consider for each test the difference of its

power with the best one among all the tests for this specific alternative. These differences

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are labeled in the tables as “Deviation to the Best” and, for each test, we report its mean

and maximum. The highest the first measure, and the smallest the last two measures, the

best is the performance of a test. We also report the rank of each test among the 34 tests,

for each one of the three quality measures.

When GEP alternatives are considered, our test Rn has the best rank for the three measures

for n = 50, n = 100 and n = 200, except for n = 50 where its rank is 6th for the maximum

deviation to the best, see Tables 3 to 6. For n = 20, its rank is 6th for the average of the

powers and the mean of the deviation to the best and 10th for the maximum deviation to

the best. These results clearly suggest that the test based on Rn has the best performance

among the 34 tests against the GEP alternatives for n ≥ 50.

When the 28 usual alternatives are considered, our test Rn performs relatively well, see

Tables 7 to 10. For n = 20 and n = 50, its rank is 14th for the three measures, for n = 100,

its rank is 10th, and for n = 200, its rank is 2nd for the two first measures and 5th for

the third one. Is is more difficult to make some conclusions with this more limited set of

alternatives, but it seems that our test becomes better as n increases, ranking among the

bests for n = 200.

In summary, our simulations suggest that our test, specifically designed to detect non-

normality in the tails for symmetric densities, has the best results for symmetric alternatives.

The performance seems to improve with the sample size, but the dominance of the test is

already visible at n = 50.

5 Conclusion

We have proposed a new test of normality based on using Rao’s score test on a GEP family

of distributions which includes the normal as a special case. This test is tailored to detect

17

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departures from normality in the tails of the distribution. Our goal was to provide a simple

and intuitive test of normality against symmetric alternatives with non-normal tails. We

have obtained a test which is the most powerful against GEP alternatives compared to the

best normality tests previously available.

Appendix : Proofs

We first state a lemma needed for the proof of Proposition 1.

Lemma 1. If X ∼ N(µ, σ2), the vector rn(µ, σ) defined by (4) satisfies

∂µrn(µ, σ) = oP (1)13 and

∂σrn(µ, σ) = σ−1v0 + oP (1)13.

Proof of Lemma 1.

In order to prove this result, we work from the fact that

rn(µ, σ) =1

n

n∑i=1

d0(Yi).

Note that we can write d0(y) = (d1(y), d2(y), d3(y))T and that the following properties hold:

(i) dj(y) is an even function,

(ii) dj(Y ) is almost everywhere differentiable, implying that d0(Y ) and rn(µ, σ) are differ-

entiable with probability 1,

(iii) E0

[dj(Y )

]= E0

[Y dj(Y )

]= 0,

(iv) E0

[Y 2|dj(Y )|

]<∞,

18

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(v) E0

[|d′j(Y )|

]<∞,

(vi) E0

[|Y d′j(Y )|

]<∞,

where d′j(y) = ddydj(y), for j = 1, 2, 3. On the other hand, by the weak law of large numbers,

we have

∂µrn(µ, σ) = E0

[∂

∂µd0(Y )

]+ oP (1)13

= E0

[∂

∂µ

(d0

(X − µσ

))]+ oP (1)13,

a similar result holding for the partial derivative with respect to σ. Note that properties (v)

and (vi) are used here to ensure the weak law of large numbers can be applied. In order to

calculate the last expectation, first note that

E0

[∂

∂µdj

(X − µσ

)]= −σ−1 E0

[d′j(Y )

].

Now, recall Stein’s famous Lemma which, in the case of the standard normal distribution,

reduces to

E0

[h′(Y )

]= E0

[Y h(Y )

],

whenever these expectations exist (cf. Stein, 1981). Using this result with properties (iii)

and (v) leads to

E0

[d′j(Y )

]= E0

[Y dj(Y )

]= 0,

for j = 1, 2, 3 so that the first result is proved. In the same way, it is easy to verify that

E0

[∂

∂σdj

(X − µσ

)]= −σ−1 E0

[Y d′j(Y )

].

Using Stein’s Lemma one more time with h(Y ) = Y dj(Y ) and h′(Y ) = dj(Y ) +Y d′j(Y ), and

19

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using properties (iii), (iv) and (vi), we have

E0

[Y d′j(Y )

]= E0

[dj(Y ) + Y d′j(Y )

]= E0

[Y 2dj(Y )

],

for j = 1, 2, 3 so that the second result also holds. �

Proof of Proposition 1.

Under normality, it is well known that

Xn − µ = OP (n−1/2). (10)

As a result, it is easy to verify that

S2n − σ2 = −σ2

(1− 1

n

n∑i=1

Y 2i

)− (Xn − µ)2 = −σ2

(1− Tn

)+ oP (n−1/2),

so that, upon applying the δ-method, we get

Sn − σ =1

2σ(S2

n − σ2) + oP (n−1/2) = −σ2

(1− Tn

)+ oP (n−1/2). (11)

Note that 1−Tn = OP (n−1/2) by the CLT. Now, to obtain the wanted result, we use a Taylor

expansion in probability of rn(Xn, Sn) along with (10), (11) and Lemma 1. This leads to

rn(Xn, Sn) = rn(µ, σ) + (Xn − µ)∂

∂µrn(µ, σ) + (Sn − σ)

∂σrn(µ, σ) + oP (n−

12 )13

= rn(µ, σ)− 1

2

(1− Tn

)v0 + oP (n−

12 )13.

To conclude, it is sufficient to evaluate the vector v0 numerically, where v0 is given in

equation (7). �

Proof of Theorem 1. First, note that a proof of this theorem could also be obtained

20

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using general results on score tests exposed in Cox & Hinkley (1979).

Now, note that Proposition 1 allows us to write

n1/2rn(Xn, Sn) = n1/2MZn + oP (1)13, (12)

where Zn = 1n

∑ni=1Zi, and where the matrix M and random vectors Zi are defined as

M =(I3 ; −1

2v0

)and Zi =

d0(Yi)

1− Y 2i

,

and where d0(y) and v0 are defined respectively in (3) and (7).

On the other hand, under the null hypothesis, the central limit theorem implies that

n1/2ZnD−→ N4

0,

J0 v0

vT0 2

, (13)

since

E0

[(1− Y 2)d0(Y )

]= −E0

[Y 2d0(Y )

]= v0.

Finally, (12) and (13) together imply that

n1/2rn(Xn, Sn)D−→ N3

(0,J0 −

1

2v0v

T0

).

The convergence in distribution of Rn follows directly. �

21

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Acknowledgements

This research of A. Desgagne, P. Lafaye de Micheaux and A. Leblanc was supported by

the Natural Sciences and Engineering Research Council of Canada. The authors are also

grateful for the good time spent together during their doctoral years, when this project

started following discussions at a student seminar in the Department de Mathematiques et

de Statistique of the Universite de Montreal.

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Table 1: Set of parameters an, bn, cn for the two formulas (8) and (9), for 0.01 ≤ p-value,α ≤ 0.50

n an bn cn n an bn cn

10 -90.771 91.196 -0.02 270 -15.022 15.936 -0.12

20 -21.640 22.203 -0.08 280 -14.979 15.906 -0.12

30 -12.320 13.022 -0.13 290 -16.889 17.774 -0.11

40 -10.463 11.218 -0.15 300 -16.848 17.744 -0.11

50 -8.805 9.650 -0.17 310 -16.820 17.725 -0.11

60 -8.957 9.807 -0.17 320 -16.778 17.692 -0.11

70 -9.072 9.928 -0.17 330 -19.069 19.940 -0.10

80 -8.279 9.200 -0.18 340 -19.028 19.908 -0.10

90 -9.154 10.043 -0.17 350 -18.998 19.886 -0.10

100 -9.163 10.071 -0.17 360 -18.970 19.864 -0.10

110 -9.153 10.082 -0.17 370 -18.936 19.838 -0.10

120 -10.101 10.999 -0.16 380 -18.893 19.805 -0.10

130 -10.083 10.999 -0.16 390 -21.672 22.539 -0.09

140 -10.059 10.993 -0.16 400 -21.639 22.513 -0.09

150 -11.128 12.028 -0.15 410 -21.609 22.489 -0.09

160 -11.098 12.015 -0.15 420 -21.580 22.465 -0.09

170 -11.070 12.002 -0.15 430 -21.548 22.440 -0.09

180 -11.031 11.980 -0.15 440 -25.013 25.860 -0.08

190 -12.240 13.153 -0.14 450 -24.976 25.829 -0.08

200 -12.198 13.127 -0.14 460 -24.956 25.813 -0.08

210 -12.160 13.104 -0.14 470 -24.928 25.790 -0.08

220 -13.551 14.457 -0.13 480 -24.897 25.765 -0.08

230 -13.507 14.428 -0.13 490 -24.874 25.745 -0.08

240 -13.468 14.402 -0.13 500 -24.848 25.724 -0.08

250 -15.096 15.988 -0.12 1000 -54.342 55.166 -0.04

260 -15.058 15.963 -0.12 ∞ 64.056 -63.392 0.04

28

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Table 2: Empirical power under the null of Rn based on 100,000 simulations and on quantilesgiven by formula in Table 1

n and α× 100 1 2 3 4 5 6 7 8 9 10 25 30 35 40 45 50

10 1.03 2.00 3.00 4.01 5.00 5.94 6.85 7.84 9.06 9.83 24.84 30.15 35.14 40.13 45.03 49.78

20 1.03 2.03 2.97 4.09 5.02 5.96 7.07 8.04 9.09 10.11 25.29 30.51 35.21 39.85 44.56 49.04

30 0.98 1.88 2.88 3.91 4.86 5.87 6.98 8.01 9.06 10.17 25.59 30.57 35.37 39.98 44.40 48.76

40 1.00 1.98 2.91 3.93 4.99 6.05 7.11 7.98 9.27 10.35 25.71 30.69 35.38 39.77 44.43 48.77

50 1.04 1.98 2.95 3.88 5.05 5.86 7.11 8.24 9.33 10.33 25.57 30.45 34.92 39.39 43.87 48.08

60 1.06 1.90 2.89 3.87 4.84 5.95 6.97 8.08 9.14 10.20 25.81 30.63 35.22 39.64 43.94 48.62

70 0.98 1.98 2.80 3.90 4.89 5.92 6.96 8.02 9.10 10.19 25.66 30.49 35.31 39.88 44.35 48.51

80 1.04 1.92 2.95 3.86 4.92 5.95 7.09 8.18 9.29 10.30 25.69 30.54 35.07 39.23 43.56 47.60

90 1.03 1.90 2.76 4.00 4.95 6.05 7.18 8.05 9.11 10.18 25.51 30.49 35.41 39.80 44.03 48.34

100 1.01 1.92 2.95 3.91 4.89 6.05 6.99 8.22 9.15 10.36 25.62 30.56 35.38 39.78 44.14 48.31

110 0.98 1.94 2.89 3.80 4.98 6.05 7.15 8.18 9.23 10.27 25.74 30.20 35.42 39.67 43.91 47.95

120 1.00 1.97 2.88 3.87 5.00 6.07 7.01 7.97 9.08 10.34 25.72 30.57 35.47 40.08 44.26 48.62

130 1.03 1.95 2.96 3.88 4.90 5.96 7.07 8.07 9.20 10.15 25.79 30.52 35.17 39.75 44.09 48.24

140 1.06 1.93 2.87 3.86 5.01 6.10 7.14 7.93 9.02 10.10 25.65 30.44 35.24 39.60 43.99 48.19

150 1.08 1.90 2.89 3.83 4.82 5.95 6.88 8.24 9.23 10.02 25.67 30.51 35.44 39.85 44.51 48.56

160 1.01 1.93 2.92 3.89 4.96 5.98 7.05 8.08 9.15 10.15 25.75 30.68 35.30 39.68 44.52 48.52

170 1.00 1.90 2.97 3.93 5.12 6.08 6.92 8.02 9.05 10.14 25.72 30.58 35.33 39.63 44.20 48.38

180 1.00 1.93 2.93 3.96 5.04 5.99 7.12 8.24 9.30 10.35 25.85 30.30 35.42 39.92 44.11 48.33

190 1.00 1.98 2.92 3.83 4.96 5.82 6.94 7.85 9.01 10.36 25.56 30.58 35.15 39.92 44.31 48.51

200 1.00 1.89 2.86 3.91 4.97 6.09 6.95 8.23 9.25 10.27 25.58 30.63 35.24 39.77 44.22 48.26

210 1.01 1.90 2.97 3.99 4.90 5.97 7.04 8.11 9.15 10.16 25.60 30.51 35.25 39.86 44.01 48.43

220 1.06 1.89 2.93 3.91 5.02 5.90 6.90 8.09 9.17 10.18 25.56 30.64 35.37 40.12 44.25 48.50

230 1.05 1.91 2.91 4.02 4.95 5.87 7.04 8.09 9.10 10.08 25.56 30.37 35.15 39.83 44.24 48.66

240 1.06 1.91 3.04 3.94 4.94 6.06 7.08 8.08 9.12 10.39 25.88 30.46 35.19 39.98 44.01 48.26

250 1.03 1.93 2.92 3.90 4.92 5.96 7.00 8.12 9.19 10.17 25.73 30.72 35.50 39.99 44.57 48.76

260 0.96 1.88 2.96 3.82 4.97 5.92 6.96 7.99 9.01 10.16 25.62 30.59 35.29 39.94 44.28 48.65

270 1.02 1.93 2.78 3.90 4.95 6.05 7.09 8.12 9.14 10.23 25.54 30.53 35.45 39.90 44.27 48.38

280 1.01 1.98 2.99 3.93 4.96 6.02 7.07 8.12 9.09 10.14 25.58 30.07 35.34 39.89 44.31 48.61

290 1.01 1.97 2.98 4.00 4.93 5.99 6.93 7.96 9.09 10.21 25.66 30.53 35.34 39.94 44.63 48.78

300 1.03 1.97 2.96 3.96 4.89 6.01 6.94 8.14 9.05 10.23 25.68 30.68 35.41 39.75 44.34 48.56

310 1.00 1.99 2.99 4.05 4.97 5.98 7.03 8.10 9.19 10.24 25.79 30.29 35.39 40.01 44.21 48.62

320 0.98 1.96 2.96 3.98 4.99 5.97 7.06 8.08 9.20 10.31 25.61 30.59 35.25 39.75 44.18 48.64

330 1.01 2.00 3.02 3.92 4.89 6.02 7.07 8.13 9.00 10.17 25.71 30.54 35.45 40.10 44.70 48.98

340 0.98 1.95 2.95 4.06 4.97 5.99 7.01 8.02 8.92 10.16 25.68 30.57 35.27 40.02 44.28 48.78

350 0.96 1.97 2.88 3.96 4.89 5.93 6.95 8.05 9.16 10.17 25.86 30.43 35.40 39.87 44.36 48.60

360 1.04 1.95 2.93 4.02 4.95 6.01 6.92 8.08 9.17 10.22 25.45 30.46 35.31 39.96 44.11 48.61

370 1.00 1.91 2.90 3.91 4.98 5.96 7.04 8.02 9.08 10.14 25.39 30.37 35.12 39.83 44.00 48.61

380 1.07 1.89 3.03 3.81 5.18 5.92 7.02 8.00 9.08 10.13 25.51 30.35 35.26 39.78 44.16 48.44

390 1.04 1.98 2.92 3.94 4.83 5.91 6.87 7.97 8.98 10.03 25.70 30.65 35.57 40.17 44.64 48.52

400 0.96 1.90 2.91 3.91 4.84 5.93 6.92 7.98 9.05 10.21 25.48 30.43 35.21 39.80 44.51 48.67

410 0.96 1.93 2.91 3.89 4.92 5.86 6.94 7.94 9.05 10.22 25.58 30.40 35.27 39.93 44.45 48.56

420 0.95 1.90 2.92 3.97 4.87 5.93 6.96 7.95 9.16 10.00 25.39 30.40 35.17 39.74 44.16 48.41

430 0.95 1.88 2.97 3.84 4.86 6.06 6.95 7.97 8.99 10.22 25.61 30.41 35.29 39.86 44.25 48.48

440 0.97 1.99 2.86 3.86 4.81 5.84 6.94 7.98 9.05 10.07 25.63 30.59 35.50 40.38 44.79 48.96

450 1.00 1.91 2.85 3.89 4.85 6.04 7.03 8.08 9.08 10.11 25.56 30.22 35.28 39.99 44.42 48.78

460 1.11 1.86 2.87 3.88 5.01 6.04 6.97 8.02 9.02 10.15 25.66 30.57 35.34 39.95 44.50 48.75

470 0.99 1.96 2.95 4.00 4.92 5.77 6.97 8.07 9.25 10.18 25.54 30.40 35.23 39.80 44.25 48.66

480 1.05 1.95 2.96 3.91 5.15 5.90 7.02 7.99 9.02 10.18 25.86 30.42 35.34 39.60 44.59 48.66

490 0.95 1.90 2.96 3.92 4.90 5.92 6.92 8.22 8.95 10.01 25.42 30.37 35.10 40.15 44.21 48.84

500 0.95 1.91 2.91 3.94 5.07 5.94 6.91 7.92 8.95 10.01 25.86 30.81 35.31 39.82 44.35 48.67

1000 0.94 1.92 2.86 3.87 4.85 6.11 6.96 8.05 9.10 10.04 25.44 30.23 35.13 39.77 44.59 48.60

29

Page 30: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le3:

Pow

erag

ainst

GE

Pdis

trib

uti

onfo

r5%

leve

lte

sts.M

=10

0,00

0.n

=20

(beg

innin

g)

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

1.1

GE

P(1

,-40

.5,0

)10

010

010

010

010

089

.74.

210

04.

210

010

010

099

.74.

210

082

.596

.096

.68.

184

.810

010

010

034

.010

010

061

.110

090

.010

013

.954

.499

.894

.4

1.1

GE

P(4

,-18

.1,0

)10

010

010

010

010

089

.94.

110

04.

110

010

010

099

.44.

110

083

.196

.196

.98.

384

.810

010

010

030

.810

010

065

.310

091

.310

014

.554

.299

.993

.3

1.1

GE

P(6

,-14

.6,0

)10

010

010

010

010

089

.84.

210

04.

110

010

099

.999

.14.

210

083

.595

.997

.08.

784

.610

010

010

030

.310

010

067

.410

092

.010

014

.754

.910

092

.8

1.1

GE

P(8

,-12

.9,0

)10

010

010

010

010

089

.94.

110

04.

010

010

099

.998

.64.

110

083

.595

.997

.28.

884

.810

010

010

029

.910

010

069

.110

092

.510

014

.855

.310

092

.5

1.1

GE

P(1

5,-1

0.9,

0)10

010

010

010

010

090

.14.

110

03.

910

010

099

.897

.34.

010

084

.895

.697

.79.

184

.510

010

010

030

.410

010

072

.210

093

.810

014

.256

.810

092

.2

1.1

GE

P(3

0,-1

0,0)

99.9

100

100

100

100

90.3

4.0

100

3.8

100

100

99.6

95.9

3.9

100

86.3

95.5

98.1

9.1

84.4

100

100

100

31.3

100

100

74.3

100

94.5

100

14.1

57.6

100

92.1

1.1

GE

P(6

0,-9

.8,0

)99

.910

010

010

010

090

.54.

010

03.

710

010

099

.695

.33.

910

087

.495

.598

.49.

384

.310

010

010

031

.810

010

075

.410

094

.810

013

.957

.710

092

.1

1.5

GE

P(1

,-8.

3,0)

89.8

96.2

83.8

67.8

96.1

61.7

1.5

70.0

1.9

72.4

80.8

75.3

62.5

1.6

54.1

65.1

85.8

79.1

5.8

70.5

89.1

76.2

94.0

3.3

72.5

90.9

22.9

87.3

55.9

82.1

4.5

51.5

64.6

78.7

1.5

GE

P(2

,-5.

7,0)

74.2

88.1

77.1

59.2

87.9

58.3

1.1

64.0

1.3

70.0

68.7

57.4

42.8

1.1

45.1

53.7

82.5

66.3

5.0

68.6

80.7

62.4

90.0

2.1

58.0

83.4

23.5

77.6

58.6

74.5

3.6

52.8

66.0

72.6

1.5

GE

P(4

,-3.

7,0)

55.4

75.2

71.3

53.0

74.8

54.5

1.0

57.3

1.0

68.6

53.4

37.2

24.4

1.0

36.9

43.3

75.5

49.7

4.4

65.1

71.5

49.9

85.6

2.5

45.4

74.9

26.2

67.2

63.4

64.2

2.7

53.2

70.4

63.9

1.5

GE

P(6

,-2.

9,0)

47.7

69.3

69.9

52.3

69.0

52.6

0.8

54.7

0.8

67.9

45.7

28.7

17.7

0.8

33.9

40.0

71.1

43.0

3.8

63.2

68.3

46.1

84.1

3.1

41.6

71.9

28.7

63.8

65.9

59.4

2.2

52.3

73.8

59.7

1.5

GE

P(8

,-2.

5,0)

43.8

66.5

69.8

52.4

66.2

51.6

0.8

53.0

0.7

67.8

41.7

24.5

14.6

0.8

32.3

38.7

68.3

39.8

3.7

61.8

67.2

44.4

84.0

3.6

39.9

70.9

30.7

62.5

67.7

56.7

2.1

51.8

76.3

57.7

1.5

GE

P(1

5,-2

.1,0

)38

.662

.970

.253

.662

.550

.20.

850

.80.

667

.336

.119

.511

.30.

830

.337

.664

.536

.33.

559

.666

.342

.984

.03.

938

.469

.934

.861

.370

.052

.91.

850

.379

.854

.6

1.5

GE

P(3

0,-1

.9,0

)36

.962

.271

.455

.561

.849

.60.

850

.00.

667

.633

.817

.610

.20.

829

.637

.662

.835

.03.

458

.466

.843

.184

.84.

038

.570

.537

.661

.671

.451

.51.

649

.782

.053

.7

1.5

GE

P(6

0,-1

.8,0

)36

.261

.971

.956

.361

.549

.50.

749

.80.

667

.632

.917

.09.

80.

729

.537

.762

.734

.83.

358

.167

.043

.085

.14.

038

.670

.738

.961

.972

.351

.01.

649

.782

.853

.2

1.84

GE

P(0

.6,-

8.4,

0)68

.874

.049

.334

.673

.532

.01.

433

.31.

638

.952

.050

.742

.51.

420

.848

.971

.550

.74.

345

.157

.340

.367

.81.

936

.760

.813

.754

.229

.647

.03.

242

.134

.860

.7

1.84

GE

P(0

.8,-

6,0)

62.7

68.7

45.5

31.4

68.2

29.9

1.4

30.5

1.6

36.6

47.6

45.4

37.3

1.4

18.5

44.7

69.3

45.7

4.0

43.3

52.6

35.6

64.2

1.8

32.2

56.2

13.1

49.4

28.7

43.6

2.8

41.9

33.6

57.0

1.84

GE

P(1

,-5,

0)55

.462

.041

.627

.961

.427

.61.

227

.41.

334

.442

.139

.131

.61.

215

.940

.166

.140

.03.

740

.947

.630

.760

.11.

627

.651

.012

.944

.128

.239

.82.

540

.932

.452

.0

1.84

GE

P(1

.5,-

3.9,

0)39

.446

.432

.720

.945

.922

.60.

921

.01.

029

.130

.825

.920

.10.

911

.130

.156

.827

.53.

035

.936

.121

.050

.52.

218

.439

.412

.032

.726

.730

.72.

038

.530

.240

.3

1.84

GE

P(2

,-3.

2,0)

30.1

36.9

27.8

17.3

36.5

19.2

0.8

17.1

0.8

25.5

23.4

18.7

14.0

0.8

8.6

24.3

48.8

20.5

2.5

32.1

29.6

16.0

44.6

3.5

13.8

32.7

11.5

26.4

25.4

25.4

1.5

35.6

28.8

33.0

1.84

GE

P(3

,-2.

1,0)

20.7

27.1

23.0

13.8

26.8

16.0

0.6

13.4

0.6

22.0

16.1

11.4

8.5

0.6

6.3

18.6

38.9

14.3

2.0

28.0

23.2

11.5

38.2

5.8

9.8

26.0

11.0

20.2

24.8

19.5

1.2

31.2

28.3

25.9

1.84

GE

P(4

,-1.

4,0)

16.4

22.4

20.6

12.1

22.0

14.2

0.5

11.4

0.4

19.7

12.3

8.4

6.2

0.5

5.2

16.2

33.4

11.5

1.7

25.3

20.1

9.3

35.6

7.3

7.9

22.7

11.0

17.2

24.3

16.4

0.8

28.0

28.2

22.5

1.84

GE

P(6

,-0.

7,0)

12.6

18.5

19.0

10.9

18.1

12.4

0.4

9.8

0.3

17.6

9.1

6.1

4.8

0.4

4.2

14.1

27.3

9.2

1.4

22.7

17.7

7.7

33.2

8.5

6.5

20.1

11.1

14.9

24.1

13.8

0.6

24.0

29.0

19.7

1.84

GE

P(8

,-0.

4,0)

11.1

16.9

18.6

10.5

16.5

12.0

0.4

9.1

0.3

17.1

7.9

5.2

4.4

0.4

3.8

13.4

25.0

8.3

1.3

21.7

16.9

7.1

32.5

9.0

5.9

19.2

11.2

14.1

24.6

12.6

0.6

22.5

30.3

18.5

1.84

GE

P(1

5.2,

0,0)

9.3

15.2

18.5

10.4

14.9

10.9

0.3

8.2

0.2

16.0

6.5

4.6

4.1

0.3

3.3

12.6

21.8

7.4

1.1

19.8

16.2

6.6

32.5

9.5

5.4

18.6

12.1

13.5

25.1

11.2

0.4

20.2

32.2

16.9

2G

EP

(0.5

,-8.

9,0)

60.4

63.3

39.3

27.3

62.7

24.6

1.7

24.9

1.7

30.5

43.2

43.2

36.9

1.7

14.9

43.2

64.1

40.7

4.2

36.7

46.5

31.3

56.9

1.8

28.2

49.7

12.0

43.5

23.2

36.9

2.8

38.2

27.5

53.2

2G

EP

(0.8

,-5.

1,0)

49.8

52.9

32.8

22.3

52.3

20.8

1.5

20.4

1.5

26.6

35.2

34.2

28.9

1.5

11.7

36.6

58.6

32.6

3.7

32.7

38.4

24.3

49.8

1.6

21.7

41.5

11.0

35.5

21.3

31.1

2.6

36.2

24.8

45.1

2G

EP

(1,-

4.2,

0)41

.744

.828

.118

.844

.218

.01.

317

.31.

323

.629

.627

.923

.11.

39.

431

.253

.326

.53.

329

.632

.419

.544

.41.

817

.335

.510

.429

.520

.026

.72.

234

.423

.338

.7

2G

EP

(1.5

,-3.

2,0)

26.4

29.6

20.5

13.2

29.2

13.2

1.0

12.0

0.9

18.5

19.3

16.5

13.4

1.0

6.1

21.8

40.6

16.3

2.6

23.8

22.6

12.1

34.6

3.2

10.6

25.1

9.4

20.0

17.8

19.1

1.6

29.5

20.6

26.1

2G

EP

(2,-

2.5,

0)19

.222

.216

.510

.221

.810

.90.

89.

20.

815

.314

.011

.59.

20.

84.

617

.033

.012

.02.

120

.517

.58.

829

.24.

87.

619

.68.

715

.216

.615

.01.

326

.119

.320

.4

2G

EP

(3,-

1.4,

0)12

.515

.312

.97.

815

.08.

50.

66.

70.

512

.39.

17.

05.

60.

63.

313

.024

.18.

11.

616

.612

.96.

023

.96.

85.

214

.77.

911

.015

.411

.00.

920

.818

.215

.3

2G

EP

(4,-

0.7,

0)9.

712

.211

.26.

512

.07.

00.

55.

40.

410

.36.

95.

34.

60.

52.

511

.119

.76.

71.

414

.610

.84.

721

.57.

83.

912

.47.

59.

114

.49.

00.

717

.717

.613

.1

2G

EP

(6,0

,0)

7.4

9.7

9.8

5.5

9.5

5.9

0.4

4.4

0.3

8.7

5.1

4.4

4.2

0.4

2.0

9.3

15.8

5.6

1.1

12.3

9.1

3.7

19.3

8.4

3.1

10.5

7.1

7.5

13.7

7.2

0.5

14.9

17.5

11.3

2G

EP

(8,0

.3,0

)6.

48.

89.

55.

28.

65.

60.

44.

00.

38.

24.

54.

14.

10.

41.

89.

014

.35.

51.

011

.78.

63.

418

.88.

62.

89.

97.

17.

013

.76.

50.

513

.517

.910

.8

2G

EP

(15,

0.7,

0)5.

67.

69.

04.

87.

45.

00.

33.

40.

37.

44.

04.

04.

30.

31.

68.

512

.15.

10.

810

.47.

83.

018

.18.

72.

59.

17.

26.

313

.65.

60.

411

.718

.710

.1

2G

EP

(0,8

65,8

73)

22.2

27.4

25.1

26.1

27.8

28.5

30.5

31.5

35.2

32.5

20.4

13.9

10.4

30.3

32.3

5.7

27.4

20.4

27.1

28.2

26.1

31.5

15.7

28.1

32.0

25.1

17.9

27.8

30.0

26.3

28.0

29.2

28.2

26.0

2.25

GE

P(0

.5,-

7.2,

0)46

.145

.526

.618

.845

.016

.02.

415

.62.

120

.530

.331

.827

.92.

39.

134

.350

.426

.64.

225

.831

.820

.641

.12.

018

.534

.49.

829

.415

.724

.42.

931

.318

.539

.9

2.25

GE

P(0

.7,-

4.7,

0)38

.738

.022

.716

.037

.513

.52.

113

.01.

817

.825

.226

.223

.02.

17.

529

.645

.121

.53.

822

.726

.616

.635

.91.

915

.029

.09.

224

.514

.420

.92.

628

.917

.033

.1

2.25

GE

P(0

.8,-

3.8,

0)32

.631

.819

.513

.631

.411

.91.

911

.11.

715

.921

.421

.518

.71.

96.

225

.540

.417

.63.

520

.322

.513

.631

.82.

112

.124

.78.

520

.413

.418

.02.

327

.215

.828

.0

2.25

GE

P(1

,-3.

3,0)

26.4

26.0

16.5

11.5

25.6

10.1

1.8

9.4

1.6

14.0

17.6

17.3

15.0

1.8

5.3

21.4

35.0

14.0

3.3

17.9

18.6

11.0

27.6

2.6

9.8

20.7

8.2

16.8

12.4

15.3

2.2

25.0

14.7

22.8

2.25

GE

P(1

.5,-

2.4,

0)14

.815

.110

.87.

414

.86.

61.

45.

81.

19.

710

.19.

38.

01.

43.

313

.823

.08.

22.

512

.711

.56.

219

.44.

55.

412

.96.

810

.210

.010

.11.

618

.812

.013

.5

2.25

GE

P(2

,-1.

7,0)

10.1

10.7

8.2

5.6

10.5

5.1

1.1

4.3

0.9

7.5

7.0

6.2

5.5

1.1

2.5

10.7

16.9

6.2

2.0

10.1

8.5

4.5

15.6

5.4

3.9

9.6

6.2

7.4

8.8

7.7

1.2

14.8

10.8

10.1

2.25

GE

P(3

,-0.

5,0)

6.6

7.2

6.1

4.1

7.0

3.6

1.0

2.9

0.8

5.5

4.8

4.5

4.2

1.0

1.9

8.1

11.2

4.9

1.7

7.5

6.1

3.0

12.0

6.3

2.6

6.9

5.1

5.2

7.3

5.5

1.0

10.6

9.2

7.4

2.25

GE

P(3

.7,0

,0)

5.6

6.1

5.3

3.5

5.9

3.1

0.8

2.5

0.7

4.7

4.3

4.2

4.2

0.8

1.6

7.2

9.4

4.6

1.4

6.5

5.2

2.6

10.8

6.4

2.2

5.9

4.8

4.4

6.7

4.8

0.9

9.2

8.8

6.7

2.25

GE

P(4

,0.2

,0)

5.3

5.6

5.2

3.3

5.5

2.9

0.8

2.3

0.7

4.5

4.0

4.2

4.3

0.8

1.6

7.0

8.7

4.5

1.4

6.1

5.0

2.4

10.2

6.5

2.1

5.7

4.6

4.2

6.5

4.4

0.9

8.5

8.7

6.6

2.25

GE

P(8

,1.2

,0)

4.1

4.4

4.0

2.5

4.2

2.1

0.6

1.7

0.6

3.4

4.0

4.8

5.2

0.6

1.2

5.9

6.2

4.4

1.0

4.6

3.9

1.9

8.6

6.2

1.7

4.4

3.9

3.2

5.7

3.5

0.8

6.3

8.3

6.1

2.25

GE

P(0

,785

,782

)22

.527

.525

.126

.127

.928

.630

.631

.635

.332

.720

.614

.310

.630

.432

.35.

727

.520

.427

.028

.526

.131

.715

.728

.332

.225

.018

.228

.030

.426

.327

.929

.328

.326

.2

2.5

GE

P(0

.5,-

6.1,

0)36

.233

.719

.814

.833

.311

.83.

211

.32.

715

.122

.424

.322

.13.

27.

128

.138

.717

.84.

718

.923

.415

.430

.72.

513

.925

.48.

921

.711

.717

.73.

225

.113

.929

.5

2.5

GE

P(0

.7,-

3.9,

0)28

.426

.116

.011

.925

.79.

83.

28.

92.

512

.617

.718

.917

.33.

25.

723

.032

.513

.64.

315

.918

.411

.925

.32.

710

.820

.18.

116

.910

.214

.22.

922

.512

.023

.1

2.5

GE

P(0

.8,-

3.2,

0)22

.520

.713

.410

.020

.48.

32.

87.

42.

310

.814

.314

.713

.42.

84.

819

.227

.310

.74.

013

.615

.09.

521

.43.

18.

616

.57.

513

.79.

211

.82.

620

.010

.918

.3

2.5

GE

P(1

,-2.

7,0)

17.3

16.0

11.0

8.3

15.7

6.9

2.7

6.2

2.1

9.1

11.3

11.4

10.4

2.6

4.1

15.6

22.2

8.4

3.6

11.6

12.0

7.6

17.9

3.5

6.9

13.2

6.9

10.9

8.3

9.9

2.5

17.3

9.7

13.9

2.5

GE

P(1

.5,-

1.8,

0)9.

18.

87.

05.

48.

64.

52.

23.

81.

86.

06.

36.

05.

62.

23.

09.

612

.65.

32.

97.

67.

24.

511

.64.

94.

18.

05.

56.

46.

26.

42.

011

.37.

67.

7

2.5

GE

P(2

,-1,

0)6.

36.

35.

44.

36.

23.

41.

93.

21.

64.

64.

74.

74.

51.

92.

57.

68.

94.

62.

65.

65.

43.

59.

25.

13.

26.

04.

94.

95.

25.

21.

88.

46.

55.

9

2.5

GE

P(2

.8,0

,0)

4.7

4.6

4.1

3.3

4.5

2.6

1.6

2.4

1.5

3.7

3.9

4.4

4.4

1.6

2.1

6.0

6.3

4.4

2.2

4.3

4.1

2.7

7.1

5.1

2.5

4.5

4.0

3.7

4.4

4.0

1.7

6.4

5.7

5.0

2.5

GE

P(3

,0.2

,0)

4.7

4.4

3.7

3.0

4.3

2.4

1.6

2.3

1.4

3.4

4.0

4.4

4.4

1.6

2.0

5.9

5.8

4.3

2.0

4.0

3.8

2.4

6.7

5.1

2.3

4.1

3.9

3.3

4.2

3.8

1.7

5.8

5.5

4.8

2.5

GE

P(4

,0.9

,0)

4.2

4.0

3.3

2.7

3.9

2.1

1.5

2.1

1.5

3.1

4.3

4.9

5.0

1.5

1.9

5.2

4.9

4.4

1.9

3.4

3.4

2.4

6.0

4.8

2.3

3.7

3.5

3.1

4.0

3.5

1.8

4.9

5.4

4.9

2.5

GE

P(8

,1.9

,0)

4.2

3.8

2.7

2.2

3.7

1.6

1.2

1.9

1.5

2.7

5.7

6.8

6.6

1.2

1.8

4.8

3.9

4.6

1.6

2.5

2.9

2.2

4.9

4.5

2.1

3.0

2.8

2.7

3.6

3.1

2.1

3.9

5.1

5.4

2.5

GE

P(0

,730

,739

)22

.527

.324

.926

.027

.728

.330

.331

.335

.132

.420

.614

.210

.530

.132

.25.

727

.420

.526

.828

.125

.931

.415

.627

.931

.925

.017

.827

.730

.026

.327

.729

.128

.126

.0

2.75

GE

P(0

.5,-

5.3,

0)28

.625

.315

.912

.724

.99.

84.

49.

13.

511

.816

.818

.817

.94.

46.

423

.330

.012

.85.

514

.818

.112

.623

.43.

311

.819

.78.

417

.09.

113

.43.

820

.610

.523

.0

2.75

GE

P(0

.6,-

4.1,

0)25

.422

.214

.311

.521

.99.

04.

38.

33.

510

.815

.016

.715

.84.

36.

020

.926

.910

.95.

313

.516

.111

.321

.13.

410

.517

.68.

015

.18.

512

.13.

719

.29.

720

.3

2.75

GE

P(0

.7,-

3.3,

0)21

.518

.812

.610

.418

.58.

14.

27.

53.

39.

613

.014

.313

.34.

25.

518

.423

.49.

25.

112

.014

.010

.018

.63.

69.

315

.27.

613

.17.

810

.83.

617

.49.

017

.1

2.75

GE

P(0

.8,-

2.8,

0)17

.715

.511

.09.

115

.37.

34.

16.

63.

28.

710

.911

.811

.14.

15.

015

.719

.87.

64.

910

.711

.88.

516

.13.

87.

912

.97.

311

.17.

39.

43.

415

.68.

214

.0

2.75

GE

P(1

,-2.

3,0)

11.9

10.8

8.6

7.2

10.7

6.0

3.9

5.3

3.1

6.9

7.7

8.0

7.6

3.9

4.4

11.7

14.0

5.7

4.5

8.4

8.9

6.5

12.5

4.5

6.1

9.8

6.4

8.3

6.3

7.4

3.2

12.2

7.0

9.5

2.75

GE

P(1

.5,-

1.4,

0)6.

46.

15.

75.

06.

04.

43.

54.

02.

94.

84.

84.

74.

63.

53.

67.

37.

54.

53.

95.

65.

64.

47.

85.

04.

26.

05.

35.

34.

85.

13.

17.

35.

45.

6

2.75

GE

P(2

,-0.

5,0)

5.0

4.9

4.6

4.1

4.8

3.8

3.2

3.4

2.8

4.1

4.4

4.5

4.4

3.2

3.3

6.0

5.6

4.5

3.5

4.4

4.5

3.7

6.3

4.7

3.6

4.8

4.6

4.3

4.3

4.4

3.0

5.8

4.8

4.6

2.75

GE

P(2

.3,0

,0)

4.6

4.5

4.2

3.8

4.5

3.4

3.1

3.3

2.9

3.9

4.4

4.7

4.6

3.1

3.3

5.3

5.1

4.4

3.4

4.0

4.1

3.6

5.6

4.7

3.4

4.3

4.2

4.0

4.2

4.2

3.0

5.2

4.7

4.6

2.75

GE

P(3

,0.8

,0)

4.5

4.2

3.7

3.4

4.2

3.1

3.0

3.1

3.0

3.7

4.9

5.4

5.3

3.0

3.2

5.0

4.4

4.6

3.2

3.4

3.8

3.4

4.9

4.4

3.3

3.8

3.8

3.6

4.1

4.0

3.2

4.4

4.7

4.8

2.75

GE

P(4

,1.6

,0)

4.9

4.4

3.4

3.2

4.4

2.7

2.7

3.2

3.2

3.8

6.1

6.6

6.3

2.7

3.2

4.8

4.3

4.9

3.0

3.2

3.6

3.4

4.4

4.3

3.4

3.6

3.6

3.5

4.2

4.0

3.6

4.4

4.9

5.1

2.75

GE

P(0

,690

,706

)22

.327

.625

.026

.027

.928

.530

.531

.535

.332

.620

.614

.110

.530

.332

.35.

727

.520

.427

.128

.226

.031

.615

.728

.132

.125

.018

.027

.930

.026

.327

.829

.228

.325

.9

3G

EP

(0.5

,-4.

6,0)

23.5

20.1

13.9

11.8

19.9

9.3

5.8

8.6

4.7

10.2

13.5

15.5

15.1

5.8

6.7

19.7

23.2

9.6

6.5

12.5

15.2

11.8

18.8

4.1

11.2

16.4

8.4

14.5

7.9

11.2

4.5

17.0

8.6

18.8

3G

EP

(0.6

,-3.

6,0)

20.2

17.1

12.3

10.7

16.9

8.4

5.6

7.7

4.5

9.1

11.7

13.4

13.1

5.6

6.3

17.5

20.1

8.1

6.3

11.2

13.3

10.4

16.5

4.2

9.8

14.3

7.9

12.7

7.2

10.0

4.4

15.6

7.9

16.1

3G

EP

(0.7

,-2.

9,0)

16.7

14.3

11.1

9.8

14.1

7.8

5.7

7.1

4.5

8.2

10.0

11.0

11.0

5.6

6.0

15.1

17.0

6.8

6.2

10.1

11.6

9.3

14.3

4.5

8.8

12.5

7.7

11.2

6.7

8.9

4.3

13.8

7.1

13.4

3G

EP

(0.8

,-2.

5,0)

13.4

11.7

9.7

8.6

11.6

7.2

5.6

6.5

4.4

7.3

8.3

9.0

8.8

5.6

5.7

12.9

13.8

5.9

6.1

8.9

9.8

8.1

12.2

4.8

7.7

10.5

7.3

9.5

6.2

7.8

4.3

11.8

6.5

10.8

3G

EP

(1,-

1.9,

0)8.

88.

37.

87.

18.

26.

45.

55.

84.

56.

26.

06.

26.

05.

55.

49.

19.

55.

05.

77.

37.

66.

69.

15.

26.

48.

16.

67.

45.

66.

54.

48.

95.

77.

3

3G

EP

(1.5

,-1,

0)5.

45.

45.

75.

35.

35.

45.

25.

14.

64.

94.

54.

54.

45.

15.

05.

85.

74.

85.

15.

65.

45.

26.

15.

35.

15.

65.

45.

45.

05.

14.

65.

84.

95.

2

3G

EP

(2,0

,0)

5.1

5.1

5.2

4.9

5.1

5.1

5.1

5.1

5.0

5.0

5.1

5.1

5.0

5.1

5.2

5.0

5.1

4.9

5.1

5.0

5.0

5.2

5.2

5.2

5.1

5.0

5.1

5.0

5.2

5.0

5.0

5.2

5.1

5.2

30

Page 31: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le3:

Pow

erag

ainst

GE

Pdis

trib

uti

onfo

r5%

leve

lte

sts.M

=10

0,00

0.n

=20

(con

tinuin

g)

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

3G

EP

(3,1

.4,0

)5.

55.

44.

64.

55.

44.

64.

95.

15.

45.

56.

86.

96.

34.

85.

14.

65.

15.

44.

84.

54.

75.

14.

45.

15.

14.

64.

54.

85.

65.

15.

75.

25.

55.

8

3G

EP

(0,6

65,6

61)

22.3

27.5

25.1

26.1

27.8

28.7

30.7

31.3

35.4

32.6

20.6

14.2

10.5

30.4

32.5

5.7

27.4

20.6

27.1

28.2

26.1

31.6

15.7

28.2

32.1

25.1

17.8

28.0

30.0

26.3

27.8

29.1

28.2

26.0

3.5

GE

P(0

.5,-

3.8,

0)17

.114

.812

.711

.614

.710

.18.

79.

37.

29.

19.

411

.011

.38.

68.

515

.415

.56.

88.

711

.312

.911

.713

.76.

211

.413

.68.

912

.87.

59.

66.

612

.97.

014

.7

3.5

GE

P(0

.6,-

2.8,

0)14

.312

.611

.510

.812

.59.

68.

78.

97.

28.

48.

39.

49.

68.

68.

413

.312

.76.

08.

710

.211

.510

.811

.86.

410

.611

.98.

411

.56.

98.

76.

711

.26.

312

.4

3.5

GE

P(0

.7,-

2.3,

0)11

.410

.610

.610

.110

.69.

59.

08.

77.

48.

07.

07.

77.

88.

98.

410

.810

.55.

68.

89.

710

.310

.110

.26.

89.

910

.78.

210

.36.

98.

06.

99.

96.

210

.5

3.5

GE

P(0

.8,-

1.9,

0)9.

08.

99.

69.

28.

99.

18.

98.

67.

77.

66.

06.

26.

28.

88.

48.

98.

55.

48.

69.

09.

19.

48.

66.

99.

39.

37.

89.

37.

07.

67.

38.

55.

88.

7

3.5

GE

P(1

,-1.

4,0)

6.8

7.4

8.6

8.3

7.4

8.8

9.0

8.6

8.3

7.3

5.3

5.1

4.9

8.9

8.7

6.3

6.8

5.5

8.6

8.2

7.9

8.7

6.9

7.2

8.7

8.0

7.4

8.3

7.1

7.3

7.9

7.0

5.9

7.2

3.5

GE

P(1

.6,0

,0)

6.5

7.4

7.9

7.8

7.4

9.0

9.6

9.4

10.0

8.7

7.0

6.4

5.7

9.5

9.5

4.7

7.1

6.4

8.8

8.2

7.5

9.1

5.4

7.6

9.1

7.3

7.0

8.0

8.6

7.5

9.8

7.3

7.4

7.6

3.5

GE

P(2

,0.8

,0)

7.2

7.9

8.0

7.9

7.9

9.2

9.8

9.7

10.7

9.2

8.2

7.4

6.6

9.7

10.0

4.6

7.7

7.0

9.0

8.4

7.8

9.5

5.2

7.8

9.6

7.5

7.0

8.3

9.1

7.9

10.4

7.8

8.1

8.1

3.5

GE

P(3

,2.3

,0)

9.2

9.7

8.0

8.3

9.8

9.3

10.2

10.6

12.2

11.3

11.7

10.4

8.6

10.1

10.8

4.7

9.5

7.8

9.3

8.7

8.5

10.6

5.4

8.7

10.8

8.1

7.0

9.2

10.9

9.0

12.1

9.6

10.1

10.0

3.5

GE

P(0

,620

,609

)22

.227

.325

.126

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.728

.630

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.535

.232

.420

.414

.210

.430

.332

.45.

727

.320

.327

.128

.426

.131

.615

.728

.032

.025

.118

.127

.930

.226

.127

.729

.028

.125

.9

4G

EP

(0.4

,-4.

5,0)

16.4

14.7

14.0

13.4

14.7

12.2

11.4

11.4

9.7

10.1

8.9

10.4

10.7

11.4

10.9

14.3

13.5

6.7

11.2

12.4

13.9

13.7

13.1

8.2

13.5

14.4

10.0

14.1

8.4

10.3

8.6

12.1

7.2

15.3

4G

EP

(0.5

,-3.

2,0)

14.1

13.1

13.1

12.6

13.1

12.0

11.6

11.5

10.0

9.7

7.9

8.9

9.2

11.6

11.1

12.4

11.7

6.2

11.2

11.8

12.8

13.1

11.6

8.4

13.0

13.1

9.6

13.1

8.4

9.9

9.0

11.0

7.0

13.4

4G

EP

(0.6

,-2.

3,0)

11.6

11.6

12.6

12.2

11.5

12.1

12.0

11.6

10.5

9.7

6.8

7.4

7.7

11.9

11.4

10.3

10.1

6.2

11.4

11.6

12.0

12.8

10.3

8.9

12.7

12.2

9.8

12.4

8.6

9.5

9.5

10.1

7.0

12.0

4G

EP

(0.7

,-1.

9,0)

9.5

10.1

11.7

11.5

10.1

12.0

12.3

11.6

11.0

9.4

6.0

6.2

6.2

12.3

11.6

8.3

8.6

6.1

11.5

11.2

11.1

12.1

8.9

9.1

12.1

11.1

9.4

11.5

8.8

9.3

10.1

8.9

6.9

10.2

4G

EP

(0.8

,-1.

5,0)

8.1

9.4

11.4

11.2

9.4

12.2

12.7

11.9

11.8

9.9

5.8

5.4

5.1

12.6

12.1

6.6

8.2

6.4

11.9

11.2

10.5

12.0

7.9

9.5

12.1

10.4

9.3

11.1

9.4

9.3

10.9

8.6

7.5

9.4

4G

EP

(1,-

1.1,

0)7.

79.

311

.111

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312

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45.

34.

913

.212

.95.

58.

67.

312

.111

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.37.

310

.012

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.29.

211

.010

.59.

712

.18.

98.

59.

3

4G

EP

(1.4

,0,0

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311

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47.

86.

614

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710

.88.

813

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.313

.97.

111

.514

.110

.99.

712

.112

.911

.314

.611

.111

.211

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4G

EP

(2,1

.5,0

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.212

.911

.912

.213

.114

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.215

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.28.

315

.115

.94.

713

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.113

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.115

.37.

212

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.611

.69.

713

.115

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.416

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.513

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4G

EP

(3,3

.1,0

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.715

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.415

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.317

.320

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.516

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.110

.316

.217

.85.

115

.511

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.617

.57.

814

.517

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.118

.716

.216

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4G

EP

(0,5

80,5

84)

22.4

27.5

25.2

26.1

27.9

28.6

30.6

31.7

35.4

32.8

20.8

14.2

10.5

30.4

32.5

5.9

27.6

20.6

27.1

28.4

26.1

31.7

15.7

28.3

32.1

25.2

18.2

28.0

30.3

26.4

28.1

29.3

28.3

26.2

4.5

GE

P(0

.4,-

4,0)

14.6

14.1

14.7

14.5

14.1

14.1

14.1

13.6

12.4

11.1

7.8

8.8

9.2

14.0

13.4

12.2

12.0

6.9

13.3

13.5

14.3

15.1

11.9

10.4

15.1

14.5

10.9

14.8

9.9

11.2

10.9

11.6

7.9

15.0

4.5

GE

P(0

.5,-

2.7,

0)12

.513

.114

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17.

57.

814

.814

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013

.813

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.214

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.211

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.18.

214

.0

4.5

GE

P(0

.6,-

2,0)

10.8

12.3

14.3

14.1

12.3

15.0

15.5

14.7

14.1

11.8

6.5

6.5

6.5

15.4

14.7

8.6

10.0

7.3

14.3

13.9

13.4

15.1

10.2

11.6

15.2

13.4

11.0

14.1

11.1

11.4

12.7

10.4

8.6

12.8

4.5

GE

P(0

.7,-

1.5,

0)9.

511

.714

.113

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.416

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.76.

45.

65.

416

.115

.66.

810

.28.

114

.814

.213

.115

.49.

312

.115

.513

.011

.113

.912

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.79.

611

.9

4.5

GE

P(0

.8,-

1.2,

0)9.

211

.914

.114

.112

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.916

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15.

75.

216

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610

.88.

815

.314

.613

.215

.98.

812

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.112

.911

.314

.113

.312

.315

.111

.310

.811

.8

4.5

GE

P(1

,-0.

8,0)

10.0

12.8

14.2

14.3

12.9

16.6

17.7

17.3

18.5

15.5

8.9

6.7

5.7

17.6

17.8

5.0

12.3

9.9

15.8

15.5

13.6

16.7

8.5

13.9

16.9

13.2

11.2

14.6

15.1

13.2

16.6

12.8

12.7

12.7

4.5

GE

P(1

.3,0

,0)

12.4

15.2

15.0

15.4

15.4

17.7

19.1

19.2

21.2

18.6

12.4

9.5

7.6

18.9

19.6

4.7

15.0

11.5

17.0

16.8

14.9

18.7

8.9

15.7

19.0

14.4

11.9

16.2

17.7

15.0

19.0

15.7

15.6

14.6

4.5

GE

P(2

,2.1

,0)

15.4

18.3

16.2

17.0

18.5

19.0

20.6

21.3

24.3

22.2

16.5

12.6

9.7

20.4

21.9

5.2

18.4

13.5

18.2

18.6

16.9

21.3

9.8

18.2

21.6

16.2

12.7

18.3

20.8

17.2

21.6

19.4

19.2

17.7

4.5

GE

P(3

,3.9

,0)

18.5

21.7

17.7

18.7

22.0

20.6

22.5

23.8

27.7

26.0

20.5

15.3

11.4

22.3

24.4

5.5

21.8

15.6

19.6

20.6

19.1

24.2

10.6

20.9

24.6

18.1

13.6

20.7

24.0

19.8

24.2

23.2

22.6

20.5

4.5

GE

P(0

,550

,557

)22

.427

.424

.825

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.828

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.531

.435

.232

.620

.614

.310

.530

.332

.25.

727

.520

.426

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.226

.031

.615

.628

.132

.024

.918

.027

.830

.126

.227

.829

.328

.326

.0

5G

EP

(0.5

,-2.

4,0)

12.1

13.9

15.9

15.9

13.9

16.8

17.5

16.8

16.3

13.6

7.0

6.9

7.0

17.4

16.8

9.1

11.5

8.3

16.1

15.8

15.1

17.2

11.0

13.2

17.3

15.0

12.2

15.9

12.7

12.8

14.2

12.1

10.0

14.9

5G

EP

(0.6

,-1.

7,0)

11.1

13.9

16.4

16.4

14.0

18.0

18.9

18.0

18.1

15.0

7.0

6.1

5.9

18.8

18.4

7.4

11.7

9.1

17.1

16.5

15.4

18.1

10.5

14.4

18.2

15.2

12.5

16.4

14.2

13.8

16.0

12.3

11.1

14.4

5G

EP

(0.7

,-1.

3,0)

10.8

14.2

16.4

16.5

14.3

18.6

19.7

19.1

19.7

16.4

7.7

5.9

5.2

19.6

19.4

6.0

12.8

10.3

17.8

17.4

15.6

18.6

10.2

15.5

18.8

15.2

12.8

16.6

15.8

14.5

17.4

13.5

12.8

14.2

5G

EP

(0.8

,-1,

0)11

.915

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.115

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46.

85.

720

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314

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.115

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.615

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.015

.014

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.9

5G

EP

(1,-

0.6,

0)13

.617

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07.

122

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117

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5G

EP

(1.1

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.519

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523

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219

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.120

.223

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.322

.118

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.720

.620

.118

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5G

EP

(2,2

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)19

.623

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623

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.326

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.025

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5G

EP

(3,4

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.223

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.127

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926

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5G

EP

(0,5

00,6

00)

22.3

27.3

25.0

25.9

27.7

28.5

30.4

31.4

35.1

32.3

20.3

14.1

10.5

30.2

32.2

5.6

27.3

20.4

26.9

28.1

25.9

31.5

15.6

27.9

32.0

24.9

17.8

27.7

30.0

26.1

27.9

28.9

28.1

26.0

6G

EP

(0.4

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9,0)

14.1

17.0

19.4

19.4

17.1

21.0

21.9

21.1

21.0

17.4

7.9

7.1

7.1

21.8

21.3

9.1

13.9

10.4

20.1

19.6

18.5

21.4

12.9

17.0

21.6

18.3

14.3

19.5

16.5

16.1

17.8

14.7

13.0

18.0

6G

EP

(0.5

,-1.

9,0)

13.8

17.6

19.9

20.0

17.7

21.9

23.1

22.6

22.7

19.1

8.4

6.6

6.3

23.0

22.8

7.6

14.9

11.4

20.9

20.5

19.1

22.5

12.8

18.4

22.7

18.7

14.8

20.2

18.2

17.4

19.4

15.9

14.7

17.9

6G

EP

(0.6

,-1.

3,0)

14.1

18.6

20.7

20.9

18.7

23.4

24.8

24.4

25.2

21.6

9.5

6.7

5.7

24.6

24.8

6.2

17.1

13.4

22.3

22.1

20.0

23.8

12.8

20.1

24.1

19.5

15.6

21.4

20.7

19.0

21.3

17.8

17.3

18.5

6G

EP

(0.8

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8,0)

17.3

22.2

22.7

23.2

22.5

25.9

27.5

27.9

30.1

26.6

14.0

9.4

7.2

27.3

28.4

5.2

21.8

16.8

24.6

25.1

22.6

27.3

13.8

23.8

27.6

21.9

16.4

24.2

25.1

22.4

24.7

23.0

22.4

21.4

6G

EP

(1,0

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22.2

27.3

24.9

25.9

27.7

28.5

30.4

31.2

35.0

32.2

20.5

14.2

10.4

30.2

32.0

5.6

27.3

20.3

26.8

28.1

25.8

31.4

15.6

28.0

31.8

24.9

18.0

27.6

30.0

26.0

27.5

29.1

28.1

25.9

6G

EP

(2,3

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231

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6G

EP

(3,6

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433

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6G

EP

(0,1

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29.1

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727

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10G

EP

(0.4

,-1.

7,0)

24.5

31.4

32.0

32.9

31.7

35.6

37.5

37.8

39.9

35.8

15.3

9.0

6.8

37.3

38.6

7.0

29.6

23.1

33.7

34.9

32.0

37.5

21.3

33.6

37.8

31.2

22.3

33.9

33.8

31.5

29.6

31.2

30.2

30.2

10G

EP

(0.5

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0)28

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240

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235

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10G

EP

(0.7

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447

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10G

EP

(1.4

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35.7

42.8

38.0

40.0

43.2

41.5

43.8

45.7

50.6

48.6

29.6

18.8

13.0

43.6

46.9

7.0

42.0

32.8

39.2

42.1

40.3

46.7

26.5

43.5

47.1

39.1

25.5

42.5

44.4

40.7

32.6

45.1

42.8

39.9

10G

EP

(0,6

,12.

2)32

.238

.834

.736

.239

.238

.340

.642

.246

.744

.427

.017

.612

.540

.443

.26.

638

.529

.936

.338

.736

.542

.923

.939

.543

.335

.323

.938

.640

.936

.831

.541

.239

.336

.5

10G

EP

(0,1

7,-1

2.6)

32.7

39.2

35.2

36.8

39.6

38.9

41.1

42.6

47.2

44.9

27.2

17.5

12.2

40.9

43.7

6.7

38.8

30.2

36.6

39.2

37.1

43.2

24.2

40.1

43.7

35.9

23.8

39.2

41.3

37.4

31.9

41.5

39.6

36.8

20G

EP

(0.3

,-1.

3,0)

45.0

52.8

48.1

50.1

53.2

51.2

53.8

55.6

60.0

58.2

32.9

18.2

11.3

53.5

57.0

8.2

51.2

42.2

48.5

52.2

50.4

56.7

35.9

53.8

57.2

49.2

32.0

52.6

53.8

50.8

33.2

54.6

51.9

49.3

20G

EP

(0.5

,0,0

)64

.771

.159

.963

.671

.561

.064

.067

.274

.474

.958

.241

.228

.563

.769

.114

.467

.157

.557

.763

.665

.471

.748

.869

.472

.163

.938

.867

.568

.065

.329

.173

.567

.865

.9

20G

EP

(0.8

,2.3

,0)

54.1

61.6

53.8

56.5

62.0

56.2

58.9

61.3

67.1

66.6

44.7

28.2

18.5

58.6

62.9

10.0

59.3

49.5

53.1

57.7

57.6

63.9

42.0

61.3

64.4

56.1

35.2

59.7

60.8

58.0

31.9

64.1

60.0

57.1

20G

EP

(0,1

4,-1

7.6)

43.3

50.8

45.2

47.5

51.3

48.4

50.9

52.8

57.9

56.4

34.6

21.6

14.5

50.7

54.2

8.0

49.7

40.5

45.6

49.5

48.0

54.3

33.5

51.4

54.8

46.7

30.2

50.1

51.7

48.3

32.5

53.3

50.4

47.7

20G

EP

(0,6

,2.6

)39

.847

.142

.044

.047

.545

.447

.749

.554

.452

.532

.020

.213

.747

.550

.87.

546

.236

.843

.046

.144

.450

.730

.547

.651

.243

.128

.346

.548

.344

.932

.449

.447

.044

.1

50G

EP

(0.3

,-0.

7,0)

70.9

77.0

67.4

70.7

77.3

68.0

70.7

73.6

79.9

80.2

61.1

40.9

26.2

70.4

75.4

15.2

73.1

64.7

64.5

70.4

72.3

77.5

57.2

75.7

77.9

71.0

43.8

74.1

73.9

72.3

25.4

78.9

73.8

72.1

50G

EP

(0.4

,0,0

)81

.685

.974

.878

.686

.173

.776

.479

.586

.687

.675

.759

.042

.776

.281

.523

.780

.173

.670

.076

.780

.785

.066

.483

.885

.379

.448

.482

.281

.079

.719

.487

.380

.781

.1

50G

EP

(0.6

,1.2

,0)

76.9

82.3

72.6

76.1

82.6

72.4

75.1

78.0

84.0

84.9

67.8

47.4

31.3

74.9

79.8

18.0

77.5

70.1

68.8

74.9

77.6

82.3

63.3

80.9

82.7

76.4

47.1

79.3

78.5

77.5

22.1

83.8

78.0

77.4

50G

EP

(0,5

.1,2

.1)

44.1

51.4

45.8

48.0

51.8

49.0

51.4

53.4

58.3

56.9

35.1

21.9

14.6

51.1

54.6

8.1

50.3

41.2

46.4

50.0

48.5

54.7

34.3

51.9

55.2

47.2

30.3

50.6

52.4

48.9

32.2

53.9

51.1

48.1

50G

EP

(0,8

,-7.

2)50

.958

.251

.754

.158

.654

.657

.059

.164

.463

.440

.124

.716

.156

.860

.69.

356

.447

.151

.655

.754

.861

.140

.158

.461

.653

.633

.956

.858

.155

.331

.660

.457

.154

.3

50G

EP

(0,1

0,-1

3.8)

57.0

64.1

56.8

59.5

64.5

59.2

61.7

64.1

69.4

68.8

45.6

28.1

18.1

61.4

65.5

10.5

61.9

52.7

56.0

60.8

60.5

66.3

45.4

64.0

66.8

59.1

37.2

62.5

63.4

60.8

30.3

66.3

62.5

60.0

50G

EP

(0,1

0.8,

-17.

3)62

.769

.561

.364

.470

.063

.065

.768

.274

.173

.951

.432

.720

.765

.469

.811

.966

.657

.559

.764

.865

.571

.250

.169

.171

.764

.139

.867

.567

.965

.728

.771

.767

.265

.1

Mea

n

2027

.530

.827

.625

.230

.824

.815

.426

.216

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.623

.119

.316

.315

.324

.017

.231

.922

.714

.827

.428

.426

.728

.316

.926

.228

.917

.128

.326

.926

.412

.226

.927

.928

.5

Ran

k

2012

2.5

1120

2.5

2131

18.5

295

2325

3032

2226

124

3313

716

8.5

2818

.54

278.

514

.517

3414

.510

6

Dev

.to

the

Bes

t-

Mea

n

209.

05.

78.

911

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711

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.319

.77.

813

.417

.220

.221

.212

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.24.

613

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18.

19.

88.

219

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.37.

619

.48.

29.

510

.124

.39.

68.

68.

0

Ran

k

2012

2.5

1120

2.5

2131

18.5

295

2325

3032

2226

124

3313

716

8.5

2818

.54

278.

514

1734

1510

6

Dev

.to

the

Bes

t-

Max

2048

.923

.225

.839

.423

.642

.096

.040

.796

.335

.152

.268

.175

.396

.155

.666

.922

.450

.391

.928

.920

.942

.126

.192

.946

.517

.873

.324

.144

.434

.191

.745

.839

.231

.9

Ran

k

2021

47

145

1632

1534

1223

2628

3324

253

2230

92

178

3120

127

618

1129

1913

10

31

Page 32: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le4:

Pow

erag

ainst

GE

Pdis

trib

uti

onfo

r5%

leve

lte

sts.M

=10

0,00

0.n

=50

(beg

innin

g)

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

1.1

GE

P(1

,-40

.5,0

)10

010

010

010

010

037

.110

010

01.

310

010

010

010

010

010

089

.610

010

014

.099

.510

010

010

097

.610

010

090

.310

010

010

010

.183

.310

010

0

1.1

GE

P(4

,-18

.1,0

)10

010

010

010

010

037

.010

010

01.

210

010

010

010

010

010

090

.310

010

018

.099

.510

010

010

092

.410

010

094

.410

010

010

09.

686

.010

010

0

1.1

GE

P(6

,-14

.6,0

)10

010

010

010

010

037

.110

010

01.

110

010

010

010

010

010

091

.310

010

021

.799

.410

010

010

089

.210

010

095

.510

010

010

09.

586

.710

010

0

1.1

GE

P(8

,-12

.9,0

)10

010

010

010

010

037

.610

010

01.

110

010

010

010

010

010

092

.110

010

025

.299

.410

010

010

087

.310

010

096

.410

010

010

09.

587

.210

010

0

1.1

GE

P(1

5,-1

0.9,

0)10

010

010

010

010

038

.110

010

01.

010

010

010

010

010

010

094

.110

010

032

.099

.310

010

010

084

.410

010

097

.510

010

010

09.

488

.010

010

0

1.1

GE

P(3

0,-1

0,0)

100

100

100

100

100

38.7

100

100

1.0

100

100

100

100

100

100

95.9

100

100

36.0

99.3

100

100

100

83.8

100

100

98.2

100

100

100

9.2

88.5

100

100

1.1

GE

P(6

0,-9

.8,0

)10

010

010

010

010

038

.910

010

01.

010

010

010

010

010

010

096

.710

010

037

.099

.310

010

010

083

.910

010

098

.510

010

010

09.

188

.710

010

0

1.5

GE

P(1

,-8.

3,0)

100

100

99.9

99.7

100

98.8

50.7

97.5

0.3

99.9

100

100

100

55.2

98.9

86.3

99.6

100

14.9

97.4

100

100

100

1.0

100

100

39.1

100

95.8

100

0.5

86.2

80.5

99.1

1.5

GE

P(2

,-5.

7,0)

99.9

100

99.8

99.3

100

99.4

43.7

98.4

0.2

99.9

100

99.9

99.7

48.9

99.3

75.7

99.5

99.9

17.0

97.6

100

99.8

100

0.5

99.7

100

44.9

99.9

98.0

100

0.3

88.8

87.8

98.8

1.5

GE

P(4

,-3.

7,0)

97.3

99.9

99.7

99.3

99.9

99.7

37.3

98.9

0.1

99.9

99.3

97.4

93.1

42.7

99.3

62.9

99.4

99.2

17.3

97.7

99.8

98.8

100

2.8

98.4

99.9

56.8

99.7

99.4

99.8

0.2

91.8

96.1

98.4

1.5

GE

P(6

,-2.

9,0)

93.8

99.6

99.8

99.5

99.7

99.7

35.4

98.9

0.1

99.9

98.1

92.0

82.4

40.6

99.1

60.2

99.2

98.2

16.7

97.7

99.7

98.4

100

6.8

97.7

99.9

65.8

99.6

99.7

99.7

0.1

92.9

98.6

97.9

1.5

GE

P(8

,-2.

5,0)

91.5

99.4

99.8

99.6

99.5

99.6

34.7

98.8

0.1

99.9

96.7

87.5

74.0

40.0

98.8

59.9

99.0

97.3

16.4

97.5

99.7

98.1

100

9.5

97.3

99.8

71.5

99.5

99.7

99.5

0.1

93.2

99.3

97.6

1.5

GE

P(1

5,-2

.1,0

)87

.499

.299

.999

.899

.399

.633

.398

.50.

199

.893

.878

.761

.338

.398

.559

.798

.695

.815

.797

.499

.898

.110

013

.097

.399

.980

.799

.699

.999

.20.

193

.799

.997

.1

1.5

GE

P(3

0,-1

.9,0

)85

.899

.299

.999

.999

.299

.633

.198

.40.

199

.892

.274

.455

.438

.398

.459

.698

.495

.215

.897

.299

.898

.310

013

.897

.699

.985

.599

.799

.999

.00.

193

.610

096

.8

1.5

GE

P(6

0,-1

.8,0

)85

.099

.299

.999

.999

.299

.633

.098

.40.

199

.891

.372

.453

.338

.098

.359

.398

.394

.715

.597

.199

.898

.410

014

.197

.799

.987

.299

.799

.999

.00.

193

.610

096

.5

1.84

GE

P(0

.6,-

8.4,

0)99

.899

.989

.284

.099

.978

.27.

065

.70.

390

.699

.099

.599

.68.

671

.678

.698

.099

.18.

079

.098

.095

.597

.10.

494

.598

.420

.197

.766

.993

.20.

480

.038

.496

.9

1.84

GE

P(0

.8,-

6,0)

99.3

99.8

86.6

80.6

99.8

77.8

5.5

63.5

0.3

89.8

98.4

99.1

99.0

6.9

69.5

73.8

97.8

98.6

7.5

78.0

96.6

92.5

96.0

0.3

91.1

97.2

20.1

96.0

67.1

91.9

0.3

80.6

38.4

96.4

1.84

GE

P(1

,-5,

0)97

.899

.383

.176

.499

.377

.14.

161

.50.

288

.597

.297

.997

.45.

267

.067

.097

.497

.66.

876

.994

.287

.994

.60.

585

.995

.419

.893

.267

.690

.10.

280

.938

.495

.5

1.84

GE

P(1

.5,-

3.9,

0)88

.094

.975

.367

.595

.276

.12.

256

.80.

285

.691

.490

.086

.32.

961

.249

.995

.891

.65.

774

.285

.873

.190

.03.

869

.788

.520

.583

.570

.084

.30.

181

.840

.390

.8

1.84

GE

P(2

,-3.

2,0)

75.4

87.7

70.1

62.2

88.2

75.4

1.4

52.9

0.1

82.9

83.0

77.3

69.4

1.9

56.4

38.8

93.4

83.0

4.6

72.2

78.4

61.3

86.5

11.3

57.3

82.3

21.6

75.1

72.3

78.9

0.1

81.6

42.9

83.9

1.84

GE

P(3

,-2.

1,0)

56.3

75.1

65.6

57.7

75.9

74.6

0.8

48.2

0.1

78.2

66.4

53.8

42.3

1.2

50.4

28.7

87.1

67.7

3.6

69.2

69.4

48.5

83.0

26.7

44.1

74.6

23.7

64.8

75.4

70.3

0.1

78.8

48.2

72.1

1.84

GE

P(4

,-1.

4,0)

44.7

66.5

64.2

56.6

67.4

73.7

0.6

45.1

0.0

74.2

53.9

38.5

27.5

0.8

46.2

24.0

80.7

57.2

3.1

66.6

65.0

41.8

81.9

38.0

37.4

70.8

26.0

59.6

77.5

64.3

0.0

74.8

53.3

64.7

1.84

GE

P(6

,-0.

7,0)

33.6

57.9

65.1

58.0

58.9

72.5

0.4

41.8

0.0

69.4

39.4

23.6

15.3

0.6

41.8

21.1

71.7

46.4

2.5

63.5

62.2

37.0

82.6

48.7

32.4

68.7

30.3

56.1

80.1

57.2

0.0

68.0

62.0

57.0

1.84

GE

P(8

,-0.

4,0)

29.3

54.1

67.0

60.3

55.0

72.0

0.4

40.6

0.0

67.1

32.8

18.2

11.4

0.5

39.9

19.7

66.8

41.5

2.3

62.2

62.3

35.9

84.0

52.5

31.4

68.9

33.9

55.7

81.7

53.8

0.0

64.1

68.3

53.5

1.84

GE

P(1

5.2,

0,0)

23.8

50.3

71.3

65.9

51.2

71.2

0.3

38.9

0.0

63.9

24.4

12.1

7.5

0.4

37.4

17.9

60.2

35.4

2.0

59.7

64.0

35.9

87.1

55.9

31.0

70.7

41.4

56.8

84.1

49.0

0.0

58.7

78.1

49.4

2G

EP

(0.5

,-8.

9,0)

99.0

99.4

77.5

70.9

99.5

64.2

3.7

49.3

0.6

79.8

96.7

98.5

98.7

4.5

54.6

74.2

96.0

97.1

6.7

65.5

93.2

87.6

91.4

0.7

85.7

94.3

16.5

92.4

52.1

83.1

0.5

76.0

26.9

95.3

2G

EP

(0.8

,-5.

1,0)

95.9

97.7

69.8

62.7

97.9

61.3

2.3

44.4

0.4

76.0

93.4

95.7

95.6

2.9

49.0

64.4

94.9

94.2

5.7

61.6

86.7

77.3

86.7

0.7

74.5

88.7

15.7

85.2

51.1

78.0

0.4

75.7

25.8

93.2

2G

EP

(1,-

4.2,

0)90

.494

.363

.456

.294

.559

.31.

540

.90.

373

.089

.091

.090

.02.

044

.755

.193

.590

.05.

059

.180

.067

.382

.31.

563

.982

.915

.377

.750

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.40.

375

.425

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2G

EP

(1.5

,-3.

2,0)

68.3

78.3

50.7

43.8

79.1

54.7

0.7

33.0

0.2

65.3

73.2

70.9

65.5

1.0

35.2

35.5

87.3

72.4

3.6

52.5

62.4

44.8

71.6

8.5

41.0

67.0

15.1

58.6

50.9

61.2

0.2

73.4

25.4

75.1

2G

EP

(2,-

2.5,

0)50

.863

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.351

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428

.30.

159

.157

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.243

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629

.425

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.22.

947

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.117

.629

.156

.915

.347

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.052

.30.

169

.026

.460

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2G

EP

(3,-

1.4,

0)31

.845

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.132

.546

.747

.80.

222

.80.

150

.337

.427

.720

.80.

322

.817

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.939

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942

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.718

.846

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.135

.651

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059

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2G

EP

(4,-

0.7,

0)23

.237

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.230

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119

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044

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219

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638

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.657

.237

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051

.831

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2G

EP

(6,0

,0)

15.9

29.2

36.5

30.7

29.9

43.6

0.1

17.1

0.0

38.9

16.4

9.5

6.6

0.1

16.1

12.9

43.9

22.7

1.2

35.0

33.2

14.3

57.6

43.5

11.8

39.6

19.0

27.7

54.5

28.6

0.0

42.8

37.4

30.4

2G

EP

(8,0

.3,0

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.326

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.532

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.60.

116

.00.

036

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.97.

25.

30.

114

.911

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.41.

133

.332

.713

.559

.445

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.421

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.955

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038

.042

.228

.0

2G

EP

(15,

0.7,

0)10

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.335

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.141

.20.

014

.30.

032

.78.

85.

14.

30.

113

.010

.232

.415

.90.

830

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.063

.545

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.541

.025

.027

.558

.122

.10.

032

.751

.025

.4

2G

EP

(0,8

65,8

73)

43.6

55.0

46.0

45.6

56.0

49.1

55.4

57.3

66.1

62.2

42.4

29.8

21.9

56.0

60.3

7.3

62.8

51.0

42.1

59.3

52.5

59.4

26.9

60.0

60.2

48.8

27.1

54.2

60.9

52.3

55.1

63.8

69.4

58.2

2.25

GE

P(0

.5,-

7.2,

0)93

.994

.955

.349

.795

.244

.12.

630

.01.

159

.087

.192

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.03.

032

.963

.689

.088

.06.

045

.077

.066

.874

.61.

264

.079

.412

.975

.333

.961

.51.

067

.315

.690

.8

2.25

GE

P(0

.7,-

4.7,

0)87

.489

.447

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.489

.940

.02.

125

.50.

953

.880

.587

.187

.92.

327

.555

.286

.281

.95.

240

.067

.455

.067

.81.

451

.870

.712

.165

.131

.754

.90.

865

.314

.386

.9

2.25

GE

P(0

.8,-

3.8,

0)78

.681

.641

.236

.682

.337

.11.

622

.30.

849

.773

.479

.579

.51.

923

.746

.282

.974

.04.

536

.658

.644

.861

.52.

241

.862

.411

.555

.730

.549

.10.

764

.013

.681

.2

2.25

GE

P(1

,-3.

3,0)

67.0

71.0

35.0

31.1

71.9

34.1

1.4

19.3

0.6

45.1

64.7

69.1

67.8

1.6

20.0

37.4

78.1

64.0

3.9

33.0

49.0

34.8

54.6

4.0

31.8

53.2

11.0

45.9

29.3

43.1

0.6

61.6

12.8

72.0

2.25

GE

P(1

.5,-

2.4,

0)36

.942

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.120

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.60.

712

.60.

333

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.037

.333

.30.

812

.220

.558

.836

.72.

524

.129

.316

.840

.111

.914

.733

.79.

926

.026

.628

.90.

350

.411

.840

.9

2.25

GE

P(2

,-1.

7,0)

22.9

27.9

17.4

15.7

28.6

22.1

0.4

9.4

0.2

26.3

24.9

21.3

17.6

0.5

8.5

14.1

44.5

23.8

1.8

19.0

20.7

10.2

32.8

16.9

8.7

24.5

9.3

17.7

24.3

21.4

0.2

40.6

11.3

26.1

2.25

GE

P(3

,-0.

5,0)

11.9

16.0

13.3

11.7

16.5

17.3

0.2

6.3

0.1

18.4

12.3

9.1

7.1

0.2

5.3

9.8

28.0

13.8

1.2

13.9

13.8

5.6

27.2

21.0

4.6

17.2

8.7

11.2

22.3

14.2

0.1

27.2

11.4

15.5

2.25

GE

P(3

.7,0

,0)

9.2

13.1

12.4

10.8

13.5

15.9

0.1

5.3

0.1

15.7

9.0

6.6

5.4

0.2

4.4

8.6

22.6

11.2

1.0

12.2

12.3

4.5

25.7

21.7

3.6

15.5

8.7

9.7

21.5

12.0

0.1

22.5

11.7

13.2

2.25

GE

P(4

,0.2

,0)

8.1

11.8

12.0

10.4

12.2

15.3

0.1

4.9

0.1

14.5

7.7

5.8

4.9

0.1

4.0

8.3

20.6

10.2

1.0

11.6

11.5

4.1

25.2

22.1

3.3

14.7

8.4

9.1

21.2

11.1

0.1

20.5

12.0

12.1

2.25

GE

P(8

,1.2

,0)

5.0

7.6

11.3

9.3

7.8

12.0

0.0

3.1

0.1

9.3

4.1

4.6

5.1

0.1

2.4

6.2

11.3

6.5

0.6

8.3

9.4

2.7

25.6

21.2

2.1

12.4

8.9

7.0

20.5

7.1

0.0

11.9

15.1

9.9

2.25

GE

P(0

,785

,782

)43

.654

.846

.245

.755

.849

.555

.657

.666

.362

.342

.529

.721

.856

.260

.57.

363

.251

.442

.159

.652

.559

.726

.960

.160

.448

.927

.354

.461

.352

.155

.164

.169

.858

.5

2.5

GE

P(0

.5,-

6.1,

0)83

.483

.738

.435

.384

.329

.63.

719

.11.

940

.872

.282

.584

.93.

920

.253

.877

.273

.06.

329

.858

.047

.555

.92.

244

.761

.111

.556

.121

.841

.91.

956

.59.

284

.2

2.5

GE

P(0

.7,-

3.9,

0)70

.870

.830

.728

.371

.625

.53.

115

.41.

734

.960

.970

.873

.03.

415

.843

.270

.761

.05.

625

.045

.935

.046

.92.

632

.649

.210

.443

.619

.634

.61.

752

.58.

375

.3

2.5

GE

P(0

.8,-

3.2,

0)57

.056

.824

.622

.857

.821

.82.

712

.61.

529

.950

.257

.959

.02.

912

.533

.663

.148

.74.

920

.835

.825

.539

.23.

923

.539

.19.

733

.517

.828

.51.

548

.47.

562

.7

2.5

GE

P(1

,-2.

7,0)

43.0

42.9

19.5

18.3

43.8

18.6

2.3

10.2

1.3

25.3

39.3

44.1

43.6

2.5

9.7

25.0

53.4

36.2

4.3

17.2

27.1

17.9

32.3

5.6

16.3

30.2

8.9

24.8

16.3

23.0

1.2

42.8

6.9

46.5

2.5

GE

P(1

.5,-

1.8,

0)18

.318

.810

.810

.719

.311

.71.

55.

80.

915

.317

.517

.315

.51.

64.

912

.529

.615

.83.

010

.113

.27.

319

.99.

66.

415

.57.

211

.512

.412

.90.

826

.75.

417

.8

2.5

GE

P(2

,-1,

0)10

.311

.07.

67.

611

.48.

71.

04.

00.

710

.59.

78.

77.

61.

13.

28.

618

.79.

72.

26.

98.

64.

215

.310

.73.

610

.56.

27.

110

.68.

90.

617

.95.

19.

8

2.5

GE

P(2

.8,0

,0)

6.2

6.8

5.5

5.6

7.0

6.2

0.6

2.6

0.5

7.0

5.5

5.1

4.8

0.7

2.0

6.7

10.8

6.5

1.6

4.5

5.9

2.6

12.3

10.3

2.2

7.4

5.3

4.7

8.6

6.4

0.4

10.9

4.8

6.1

2.5

GE

P(3

,0.2

,0)

5.5

6.1

5.1

5.1

6.3

5.6

0.6

2.4

0.5

6.3

5.0

4.6

4.4

0.6

1.9

6.3

9.6

6.0

1.5

4.1

5.3

2.3

11.8

10.2

2.0

6.7

5.2

4.2

8.4

5.8

0.4

9.7

4.8

5.8

2.5

GE

P(4

,0.9

,0)

4.5

4.8

4.2

4.1

4.9

4.4

0.4

1.7

0.5

4.6

4.2

4.9

5.1

0.4

1.3

5.5

6.5

5.0

1.1

3.1

4.2

1.7

10.6

9.3

1.4

5.5

4.7

3.3

7.3

4.5

0.4

6.8

5.0

5.4

2.5

GE

P(8

,1.9

,0)

4.4

4.3

3.6

3.1

4.4

2.7

0.2

1.0

0.4

2.8

5.6

8.0

8.6

0.2

0.9

4.6

3.7

4.3

0.8

1.7

3.8

1.4

10.4

7.2

1.2

4.7

3.8

2.8

5.8

3.5

0.5

3.8

6.1

6.9

2.5

GE

P(0

,730

,739

)43

.654

.945

.745

.555

.848

.855

.057

.265

.962

.142

.829

.822

.155

.660

.27.

362

.851

.341

.759

.252

.459

.326

.859

.860

.248

.727

.154

.161

.052

.154

.863

.869

.458

.4

2.75

GE

P(0

.5,-

5.3,

0)70

.568

.328

.727

.269

.321

.55.

714

.33.

428

.356

.369

.373

.45.

914

.645

.363

.056

.37.

721

.143

.334

.841

.03.

632

.846

.010

.941

.614

.828

.53.

345

.55.

975

.9

2.75

GE

P(0

.6,-

4.1,

0)63

.360

.424

.923

.861

.519

.45.

412

.73.

225

.350

.062

.266

.05.

612

.840

.058

.449

.47.

219

.037

.329

.336

.23.

827

.539

.810

.335

.613

.625

.13.

242

.75.

370

.0

2.75

GE

P(0

.7,-

3.3,

0)53

.750

.221

.320

.451

.317

.45.

111

.23.

122

.342

.553

.056

.25.

311

.133

.752

.641

.16.

716

.730

.923

.631

.24.

122

.133

.49.

729

.312

.621

.63.

139

.34.

960

.8

2.75

GE

P(0

.8,-

2.8,

0)43

.239

.917

.817

.340

.815

.04.

89.

82.

919

.034

.642

.945

.05.

19.

427

.145

.332

.26.

314

.124

.718

.426

.04.

717

.227

.09.

123

.211

.217

.92.

935

.24.

549

.0

2.75

GE

P(1

,-2.

3,0)

25.5

23.1

12.4

12.4

23.7

11.5

4.2

7.5

2.6

13.8

21.4

25.1

25.3

4.4

6.8

16.8

30.8

18.4

5.5

10.4

15.5

11.2

18.1

6.2

10.5

17.3

8.1

14.4

9.4

12.5

2.7

25.8

3.8

26.9

2.75

GE

P(1

.5,-

1.4,

0)9.

48.

96.

97.

19.

26.

73.

04.

52.

27.

58.

38.

47.

93.

24.

18.

512

.97.

74.

05.

77.

45.

210

.47.

04.

98.

46.

16.

76.

67.

12.

212

.43.

28.

5

2.75

GE

P(2

,-0.

5,0)

5.8

5.6

4.7

5.0

5.8

4.6

2.4

3.3

2.1

5.2

5.1

5.2

4.9

2.6

3.0

6.1

7.4

5.4

3.2

3.8

4.9

3.4

7.6

6.1

3.2

5.7

5.1

4.3

5.2

5.3

1.9

7.3

3.2

4.9

2.75

GE

P(2

.3,0

,0)

4.9

4.8

4.1

4.3

5.0

3.9

2.1

2.9

2.1

4.5

4.5

4.8

4.7

2.2

2.6

5.5

5.9

5.0

2.9

3.2

4.3

3.0

6.8

5.7

2.8

4.9

4.6

3.7

4.6

4.7

1.9

6.0

3.4

4.3

2.75

GE

P(3

,0.8

,0)

4.5

4.2

3.1

3.3

4.3

2.8

1.6

2.2

1.9

3.7

4.7

5.5

5.6

1.7

2.0

5.0

4.3

4.7

2.4

2.2

3.5

2.4

5.9

4.8

2.2

3.9

3.6

3.1

3.9

4.0

1.8

4.3

4.1

4.3

2.75

GE

P(4

,1.6

,0)

4.9

4.4

2.4

2.6

4.5

1.9

1.2

1.6

1.8

3.5

6.2

7.7

7.9

1.2

1.7

4.6

3.8

4.5

2.0

1.6

3.2

2.2

5.3

4.1

2.1

3.4

3.0

2.7

3.4

3.8

2.0

3.8

5.1

5.4

2.75

GE

P(0

,690

,706

)43

.454

.946

.145

.755

.949

.055

.457

.366

.262

.142

.629

.922

.056

.060

.47.

363

.051

.242

.059

.452

.659

.526

.860

.160

.348

.927

.254

.461

.152

.254

.964

.069

.658

.5

3G

EP

(0.5

,-4.

6,0)

57.7

53.2

23.0

22.4

54.2

17.3

8.0

12.9

5.1

20.1

42.0

55.9

60.9

8.3

12.7

38.2

48.6

41.4

9.3

16.9

33.0

27.5

29.8

5.2

26.1

34.9

11.0

31.9

11.0

20.1

5.1

35.2

4.1

67.1

3G

EP

(0.6

,-3.

6,0)

49.2

44.3

20.3

19.9

45.4

15.7

7.9

11.8

5.2

17.3

35.4

47.5

52.2

8.3

11.7

32.6

42.7

34.1

9.0

15.2

27.9

23.3

25.6

5.5

22.2

29.7

10.6

26.9

10.1

17.1

5.2

31.7

3.7

58.9

3G

EP

(0.7

,-2.

9,0)

39.2

34.4

17.0

16.9

35.3

13.7

7.5

10.5

5.0

14.9

28.3

37.8

41.2

7.8

10.3

26.2

35.9

26.1

8.4

13.2

22.3

18.6

21.0

5.8

17.9

23.9

9.8

21.4

9.2

14.3

5.1

27.8

3.4

47.3

3G

EP

(0.8

,-2.

5,0)

29.2

25.1

14.4

14.3

25.9

12.2

7.3

9.7

5.0

12.4

21.6

28.0

30.0

7.6

9.2

20.3

28.3

18.8

8.1

11.7

17.6

14.9

17.0

6.1

14.3

18.9

9.2

17.0

8.4

11.9

5.1

22.9

3.3

34.6

3G

EP

(1,-

1.9,

0)15

.213

.210

.310

.413

.59.

36.

67.

94.

78.

611

.814

.214

.56.

87.

412

.016

.210

.07.

28.

711

.29.

711

.16.

49.

412

.08.

010

.86.

88.

34.

814

.52.

916

.5

3G

EP

(1.5

,-1,

0)6.

26.

06.

66.

56.

26.

35.

66.

04.

95.

65.

25.

45.

35.

85.

86.

36.

55.

45.

96.

16.

36.

06.

35.

76.

06.

66.

26.

15.

35.

64.

96.

53.

66.

4

3G

EP

(2,0

,0)

5.0

5.0

5.0

5.0

5.1

5.0

4.8

5.2

5.1

5.2

4.9

5.1

4.9

5.0

5.1

5.0

5.0

5.0

5.0

4.9

5.1

5.0

5.0

5.0

5.0

5.1

5.1

5.0

5.2

5.0

5.0

5.1

5.1

5.1

32

Page 33: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le4:

Pow

erag

ainst

GE

Pdis

trib

uti

onfo

r5%

leve

lte

sts.M

=10

0,00

0.n

=50

(con

tinuin

g)

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

3G

EP

(3,1

.4,0

)6.

25.

83.

73.

86.

03.

53.

84.

15.

45.

77.

88.

58.

24.

04.

44.

55.

75.

54.

13.

94.

64.

74.

24.

54.

74.

33.

74.

45.

05.

35.

65.

78.

26.

5

3G

EP

(0,6

65,6

61)

43.5

54.7

46.0

45.6

55.7

49.0

55.2

57.3

65.9

62.0

42.5

29.8

21.9

55.8

60.2

7.3

62.8

51.2

42.1

59.3

52.6

59.4

27.0

59.8

60.1

48.9

27.0

54.4

60.9

52.2

54.6

63.7

69.2

58.4

3.5

GE

P(0

.5,-

3.8,

0)38

.633

.120

.119

.934

.116

.313

.615

.110

.112

.923

.434

.840

.314

.014

.828

.227

.322

.813

.116

.524

.323

.318

.59.

422

.924

.912

.324

.310

.113

.210

.221

.23.

751

.8

3.5

GE

P(0

.6,-

2.8,

0)29

.825

.418

.117

.826

.215

.513

.814

.710

.411

.217

.826

.230

.414

.214

.522

.821

.316

.813

.015

.620

.620

.514

.89.

620

.320

.811

.920

.69.

811

.410

.817

.53.

741

.5

3.5

GE

P(0

.7,-

2.3,

0)21

.518

.416

.215

.719

.014

.513

.814

.310

.89.

912

.818

.120

.714

.214

.017

.115

.812

.012

.814

.717

.217

.811

.99.

717

.617

.211

.317

.39.

610

.011

.213

.94.

030

.2

3.5

GE

P(0

.8,-

1.9,

0)14

.213

.114

.713

.913

.514

.013

.914

.111

.39.

08.

811

.512

.914

.313

.911

.911

.38.

512

.514

.314

.415

.69.

59.

815

.714

.211

.014

.79.

79.

011

.810

.74.

721

.1

3.5

GE

P(1

,-1.

4,0)

7.9

8.8

12.4

11.5

9.0

12.7

13.7

13.5

12.4

8.6

5.6

6.1

6.2

14.1

13.5

7.2

7.7

6.6

11.9

13.4

11.6

13.4

6.6

9.8

13.6

11.0

9.9

12.0

9.9

8.3

12.8

7.9

6.8

12.9

3.5

GE

P(1

.6,0

,0)

7.7

9.2

10.2

9.1

9.5

11.3

13.2

13.2

15.2

11.3

8.3

7.4

6.5

13.7

13.8

4.7

10.5

8.2

10.8

13.4

10.3

12.7

4.9

10.7

13.0

9.2

8.1

10.8

12.3

9.3

15.6

10.4

14.3

11.6

3.5

GE

P(2

,0.8

,0)

9.4

10.9

9.6

8.7

11.3

11.0

13.2

13.3

16.6

13.3

11.3

10.2

9.0

13.6

14.1

4.5

13.1

9.5

10.5

13.7

10.5

13.2

4.7

11.6

13.5

9.3

7.7

11.1

13.6

10.3

17.1

12.9

17.9

13.2

3.5

GE

P(3

,2.3

,0)

14.3

15.8

8.9

8.5

16.3

10.2

12.7

13.4

19.2

18.2

19.6

17.5

15.1

13.2

15.1

5.1

19.3

12.9

10.1

14.1

12.2

15.1

5.1

13.9

15.4

10.5

6.3

12.6

16.1

13.2

20.2

19.2

25.7

17.7

3.5

GE

P(0

,620

,609

)43

.554

.645

.645

.355

.648

.754

.957

.065

.961

.842

.630

.022

.155

.659

.97.

462

.950

.941

.759

.252

.359

.226

.859

.759

.948

.527

.053

.960

.951

.954

.863

.869

.458

.3

4G

EP

(0.4

,-4.

5,0)

34.2

30.1

23.1

22.5

31.0

19.9

19.4

20.0

15.3

12.8

17.8

28.1

34.3

19.8

19.7

26.3

20.9

18.6

17.5

20.5

25.5

26.8

16.6

13.9

26.7

25.3

14.7

26.1

13.0

13.3

15.6

17.5

5.4

51.0

4G

EP

(0.5

,-3.

2,0)

27.2

24.1

21.4

20.6

24.9

19.3

19.6

19.9

16.0

12.1

13.8

21.3

26.0

20.0

19.7

21.7

16.7

14.6

17.4

20.1

22.8

24.6

14.0

14.2

24.6

22.3

14.3

23.4

13.3

12.5

16.3

14.8

5.8

42.6

4G

EP

(0.6

,-2.

3,0)

20.0

18.9

20.4

19.4

19.6

19.4

20.4

20.5

17.2

11.8

10.1

14.5

17.6

20.8

20.2

16.5

13.0

11.2

17.6

20.5

20.8

23.2

11.9

14.8

23.3

20.1

14.2

21.5

14.1

12.0

17.8

12.5

6.9

33.8

4G

EP

(0.7

,-1.

9,0)

13.9

15.1

19.3

18.1

15.6

19.4

21.0

20.9

18.5

11.9

7.4

9.4

10.9

21.4

20.7

11.6

10.9

9.3

17.7

20.8

18.8

21.7

9.8

15.4

21.9

17.9

13.8

19.7

15.1

12.1

19.2

11.1

8.9

25.5

4G

EP

(0.8

,-1.

5,0)

10.4

13.3

18.7

17.1

13.6

19.6

21.7

21.6

20.3

13.1

6.2

6.5

6.9

22.1

21.6

8.2

10.7

8.9

17.8

21.5

17.8

21.2

8.6

16.1

21.5

16.7

13.8

18.7

16.4

12.7

20.9

11.0

11.5

20.8

4G

EP

(1,-

1.1,

0)9.

212

.917

.615

.813

.219

.321

.922

.022

.415

.17.

05.

75.

322

.322

.25.

912

.910

.117

.522

.016

.920

.67.

717

.121

.115

.512

.817

.818

.413

.622

.913

.216

.318

.3

4G

EP

(1.4

,0,0

)13

.217

.417

.415

.817

.819

.923

.323

.928

.021

.914

.010

.88.

923

.924

.94.

821

.615

.117

.724

.518

.523

.27.

321

.223

.716

.512

.319

.623

.617

.128

.321

.627

.621

.5

4G

EP

(2,1

.5,0

)18

.422

.817

.916

.723

.620

.524

.425

.332

.228

.022

.017

.614

.425

.027

.25.

228

.819

.718

.226

.421

.226

.28.

124

.926

.818

.611

.822

.327

.620

.932

.629

.036

.226

.2

4G

EP

(3,3

.1,0

)25

.130

.018

.217

.930

.920

.825

.427

.237

.035

.532

.126

.020

.926

.130

.26.

037

.325

.618

.428

.924

.630

.29.

330

.030

.921

.311

.225

.832

.625

.537

.837

.845

.632

.5

4G

EP

(0,5

80,5

84)

43.4

54.8

45.8

45.6

55.9

48.9

55.3

57.2

66.1

62.1

42.5

29.9

22.1

55.9

60.2

7.2

62.8

51.0

41.9

59.3

52.6

59.5

26.8

60.0

60.2

48.8

27.0

54.2

60.8

52.1

54.7

63.8

69.3

58.2

4.5

GE

P(0

.4,-

4,0)

27.4

26.0

25.5

24.4

26.9

24.0

25.2

25.4

21.4

14.4

12.0

19.3

24.4

25.7

25.1

21.9

16.1

15.3

21.5

25.4

26.5

29.5

15.0

19.3

29.7

25.5

16.9

27.4

18.0

15.1

21.6

15.4

8.9

46.3

4.5

GE

P(0

.5,-

2.7,

0)21

.522

.225

.123

.822

.924

.526

.326

.323

.115

.19.

213

.717

.126

.826

.017

.414

.213

.122

.026

.325

.128

.613

.520

.229

.024

.016

.926

.119

.215

.523

.414

.510

.739

.4

4.5

GE

P(0

.6,-

2,0)

16.2

19.3

24.4

22.9

19.9

25.0

27.5

27.4

25.1

16.2

7.3

9.4

11.2

28.0

27.2

12.6

13.5

11.7

22.4

27.3

23.9

28.0

12.0

21.4

28.4

22.6

16.9

25.1

21.1

16.1

25.7

14.1

13.3

32.4

4.5

GE

P(0

.7,-

1.5,

0)12

.817

.823

.822

.018

.225

.428

.428

.427

.418

.26.

86.

46.

828

.928

.48.

814

.912

.222

.528

.423

.127

.610

.722

.628

.021

.516

.524

.323

.217

.227

.915

.617

.527

.3

4.5

GE

P(0

.8,-

1.2,

0)12

.218

.323

.821

.718

.826

.129

.629

.830

.421

.48.

26.

15.

530

.130

.06.

418

.614

.423

.129

.923

.428

.210

.324

.628

.821

.616

.324

.626

.019

.130

.519

.022

.925

.5

4.5

GE

P(1

,-0.

8,0)

14.8

21.5

24.1

22.2

22.1

27.0

30.8

31.3

34.4

26.2

12.7

8.8

7.1

31.4

32.1

5.2

24.8

18.1

23.7

31.9

24.9

30.0

10.6

27.6

30.6

22.5

16.3

26.1

29.8

22.1

34.0

25.0

30.8

26.7

4.5

GE

P(1

.3,0

,0)

20.4

27.3

25.2

23.5

28.1

28.4

33.0

33.9

39.9

33.4

21.1

15.1

11.9

33.6

35.4

5.2

33.5

24.0

24.6

34.9

27.8

33.6

11.3

32.4

34.3

25.1

16.2

29.3

34.8

26.5

38.8

33.7

40.8

31.8

4.5

GE

P(2

,2.1

,0)

28.7

36.1

27.0

26.2

37.1

30.2

35.6

37.4

46.7

42.9

32.8

25.2

19.6

36.3

40.2

6.1

44.2

32.1

25.9

39.1

32.8

39.4

13.3

39.2

40.2

29.4

16.6

34.4

41.6

32.9

45.1

44.9

52.0

39.5

4.5

GE

P(3

,3.9

,0)

35.9

44.3

29.2

29.4

45.3

31.9

38.3

40.8

52.5

50.9

42.4

32.6

25.1

39.0

44.9

7.3

52.5

39.3

27.2

43.4

38.1

45.2

15.4

45.8

46.0

33.9

16.8

39.7

47.4

39.0

50.6

53.6

60.4

46.6

4.5

GE

P(0

,550

,557

)43

.855

.346

.245

.956

.249

.355

.557

.566

.362

.442

.830

.122

.156

.160

.57.

263

.351

.342

.159

.652

.859

.826

.960

.260

.549

.127

.254

.561

.252

.455

.064

.269

.858

.7

5G

EP

(0.5

,-2.

4,0)

19.7

23.8

28.6

27.1

24.5

29.4

32.3

32.3

30.1

19.9

7.8

9.7

12.0

32.8

32.2

15.0

16.5

14.9

26.2

32.2

28.6

33.1

14.4

26.4

33.6

27.0

19.1

29.9

26.1

19.9

29.9

17.5

17.0

39.2

5G

EP

(0.6

,-1.

7,0)

16.5

22.8

29.1

27.1

23.3

30.7

34.3

34.2

33.2

22.7

7.4

6.9

7.6

34.8

34.4

10.3

18.8

15.4

27.1

34.3

28.6

33.9

13.7

28.6

34.4

26.6

19.4

30.1

29.0

21.7

33.2

19.7

21.9

34.6

5G

EP

(0.7

,-1.

3,0)

15.9

23.7

29.1

27.0

24.3

31.9

35.7

36.3

36.9

26.9

9.4

6.5

5.8

36.3

36.5

7.1

23.9

18.3

27.8

36.4

29.1

34.7

13.4

31.1

35.3

26.8

19.5

30.6

32.5

24.2

36.3

24.3

28.9

32.3

5G

EP

(0.8

,-1,

0)18

.026

.729

.727

.727

.333

.037

.438

.040

.932

.013

.68.

66.

638

.039

.05.

829

.922

.328

.538

.730

.736

.613

.734

.137

.328

.119

.632

.236

.027

.739

.430

.236

.032

.9

5G

EP

(1,-

0.6,

0)23

.132

.131

.429

.632

.934

.739

.841

.046

.338

.921

.214

.210

.740

.442

.55.

438

.428

.529

.941

.934

.040

.514

.839

.141

.331

.020

.035

.641

.232

.143

.738

.645

.437

.4

5G

EP

(1.1

,0,0

)28

.337

.532

.831

.538

.536

.341

.743

.350

.644

.728

.320

.015

.142

.445

.45.

845

.334

.031

.044

.837

.143

.816

.143

.244

.533

.820

.538

.845

.336

.047

.045

.752

.842

.0

5G

EP

(2,2

.7,0

)38

.047

.736

.436

.248

.739

.345

.747

.958

.154

.941

.030

.623

.346

.451

.37.

256

.143

.633

.550

.343

.850

.919

.551

.351

.839

.921

.445

.552

.943

.852

.757

.163

.650

.9

5G

EP

(3,4

.6,0

)43

.954

.539

.039

.655

.641

.648

.851

.563

.361

.549

.136

.627

.649

.555

.78.

162

.949

.835

.254

.348

.756

.122

.157

.056

.944

.422

.150

.558

.149

.557

.164

.170

.056

.7

5G

EP

(0,5

00,6

00)

43.7

54.8

46.0

45.6

55.8

49.0

55.3

57.4

66.1

62.1

42.6

30.0

22.0

55.9

60.4

7.2

63.0

51.0

42.0

59.4

52.5

59.4

27.0

60.0

60.2

48.9

27.1

54.3

61.0

52.3

54.7

64.0

69.5

58.2

6G

EP

(0.4

,-2.

9,0)

23.3

30.3

35.4

33.8

31.0

36.9

40.6

40.9

39.1

27.6

8.3

8.7

10.8

41.1

40.9

14.9

22.8

20.2

32.5

40.9

35.7

41.3

18.5

35.2

41.9

33.7

23.2

37.3

35.1

27.2

37.4

24.2

25.4

45.9

6G

EP

(0.5

,-1.

9,0)

22.4

31.6

37.0

35.2

32.3

39.3

43.5

44.1

43.7

32.5

9.4

6.9

7.4

44.1

44.3

11.4

28.0

23.1

34.3

44.2

37.4

43.4

18.9

39.0

44.0

35.1

24.0

39.1

39.6

30.6

41.4

28.9

32.5

43.8

6G

EP

(0.6

,-1.

3,0)

23.4

34.1

38.3

36.3

34.8

41.4

46.1

47.0

48.6

38.7

13.2

7.5

6.1

46.7

47.7

8.0

35.4

28.0

36.0

47.6

39.2

45.6

19.5

43.0

46.3

36.5

24.6

41.0

44.4

34.9

45.1

36.0

41.3

42.9

6G

EP

(0.8

,-0.

8,0)

31.5

42.9

41.2

39.8

43.8

44.9

50.4

51.7

57.2

50.1

25.9

15.7

11.0

51.0

53.6

6.1

49.5

38.8

38.5

53.0

44.7

51.5

22.1

50.8

52.2

41.4

25.5

46.4

52.4

42.9

50.4

49.8

56.2

48.1

6G

EP

(1,0

,0)

43.3

54.5

45.6

45.3

55.5

48.7

54.8

56.9

65.6

61.8

42.5

30.0

22.2

55.4

59.9

7.4

62.8

50.8

41.6

59.0

52.2

59.1

26.5

59.7

59.9

48.4

26.8

54.0

60.8

51.8

54.9

63.8

69.4

57.9

6G

EP

(2,3

.9,0

)52

.063

.350

.651

.164

.353

.159

.962

.572

.570

.152

.337

.927

.960

.566

.18.

670

.959

.445

.465

.059

.466

.231

.567

.066

.955

.429

.161

.067

.559

.658

.072

.276

.965

.8

6G

EP

(3,6

.2,0

)55

.267

.053

.154

.268

.055

.362

.565

.275

.373

.556

.240

.629

.663

.169

.19.

474

.363

.147

.067

.862

.669

.233

.870

.469

.958

.530

.364

.270

.763

.059

.775

.679

.769

.0

6G

EP

(0,1

40,1

29.1

)44

.455

.646

.646

.356

.749

.755

.957

.866

.762

.843

.330

.222

.356

.560

.87.

363

.652

.042

.460

.053

.360

.127

.660

.660

.849

.627

.655

.061

.752

.955

.164

.670

.259

.0

10G

EP

(0.4

,-1.

7,0)

46.4

59.9

58.7

58.1

60.7

61.6

66.8

68.4

71.6

65.1

27.7

12.7

8.1

67.4

69.8

10.4

62.6

53.8

53.8

69.3

62.1

68.3

38.0

67.6

68.9

59.1

36.1

63.7

68.3

60.0

53.6

63.1

68.0

65.8

10G

EP

(0.5

,-1,

0)56

.168

.663

.363

.369

.465

.971

.473

.278

.574

.743

.223

.514

.371

.975

.39.

173

.764

.557

.774

.668

.574

.143

.774

.674

.765

.438

.470

.075

.068

.153

.874

.478

.671

.7

10G

EP

(0.7

,0,0

)77

.786

.073

.875

.586

.574

.079

.781

.789

.489

.275

.759

.345

.680

.185

.014

.289

.383

.265

.583

.982

.086

.158

.586

.986

.579

.342

.983

.186

.182

.347

.690

.892

.186

.3

10G

EP

(1.4

,5.7

,0)

67.4

77.9

68.2

69.0

78.7

69.7

75.4

77.4

84.5

82.8

62.3

43.8

31.6

75.8

80.2

10.4

83.1

75.0

61.3

79.2

75.3

80.2

50.6

81.1

80.7

72.2

40.6

76.6

80.8

75.3

51.4

84.3

87.0

79.4

10G

EP

(0,6

,12.

2)62

.073

.363

.664

.274

.165

.571

.373

.280

.978

.757

.239

.928

.671

.876

.39.

579

.370

.157

.475

.370

.676

.045

.276

.876

.667

.437

.572

.176

.970

.653

.180

.383

.775

.2

10G

EP

(0,1

7,-1

2.6)

62.8

73.7

64.3

64.9

74.5

66.1

71.9

74.0

81.3

79.2

57.6

40.3

28.9

72.4

76.8

9.6

79.7

70.6

57.9

75.9

71.2

76.6

45.8

77.4

77.2

68.0

38.0

72.6

77.4

71.2

52.8

80.8

84.0

75.9

20G

EP

(0.3

,-1.

3,0)

79.7

87.8

81.3

82.2

88.3

82.1

86.2

87.6

91.8

91.0

68.3

42.0

25.3

86.5

89.6

13.9

90.5

85.4

74.5

88.9

86.3

89.5

67.6

90.1

89.9

84.3

50.3

87.2

89.7

86.7

40.8

91.2

92.8

88.4

20G

EP

(0.5

,0,0

)95

.397

.891

.493

.197

.989

.693

.194

.398

.098

.294

.485

.473

.193

.396

.228

.198

.096

.783

.395

.696

.197

.285

.397

.597

.395

.157

.796

.496

.696

.023

.098

.898

.697

.4

20G

EP

(0.8

,2.3

,0)

88.7

93.8

86.6

88.0

94.1

86.1

89.9

91.2

95.6

95.5

84.4

66.5

49.9

90.2

93.3

18.0

95.4

92.3

79.1

92.6

91.8

94.0

76.7

94.5

94.2

90.3

54.1

92.4

93.8

92.1

32.4

96.2

96.8

93.8

20G

EP

(0,1

4,-1

7.6)

77.7

86.0

77.6

78.7

86.5

78.3

83.1

84.7

90.3

89.4

71.0

50.8

36.3

83.4

87.1

12.5

89.4

83.8

70.6

86.3

83.6

87.3

63.1

88.0

87.7

81.4

47.3

84.6

87.5

83.9

43.0

90.5

92.0

86.8

20G

EP

(0,6

,2.6

)73

.682

.873

.974

.983

.474

.979

.981

.888

.086

.866

.847

.233

.780

.384

.311

.687

.080

.267

.183

.580

.384

.458

.285

.284

.977

.744

.581

.484

.980

.446

.288

.190

.183

.9

50G

EP

(0.3

,-0.

7,0)

97.3

98.8

95.2

96.2

98.8

94.0

96.2

96.9

98.9

99.1

95.8

85.5

70.1

96.3

98.0

30.4

98.9

98.2

89.4

97.7

97.9

98.5

91.5

98.7

98.6

97.4

64.2

98.1

98.2

97.9

14.6

99.3

99.3

98.5

50G

EP

(0.4

,0,0

)99

.499

.898

.098

.699

.896

.698

.198

.599

.799

.899

.296

.691

.198

.199

.347

.399

.699

.693

.299

.099

.499

.696

.799

.799

.799

.268

.899

.599

.499

.46.

999

.999

.899

.6

50G

EP

(0.6

,1.2

,0)

98.7

99.5

97.3

98.0

99.5

96.1

97.8

98.2

99.5

99.6

97.9

90.9

78.5

97.8

99.0

37.1

99.5

99.2

92.6

98.8

99.0

99.3

95.0

99.4

99.4

98.7

68.1

99.1

99.1

99.0

9.4

99.7

99.7

99.3

50G

EP

(0,5

.1,2

.1)

78.6

86.7

78.5

79.6

87.3

79.1

83.8

85.3

90.7

90.0

71.5

51.6

36.8

84.1

87.7

12.7

90.1

84.5

71.6

86.9

84.5

87.9

64.3

88.7

88.3

82.3

48.0

85.4

88.1

84.7

41.8

91.0

92.5

87.4

50G

EP

(0,8

,-7.

2)85

.691

.784

.685

.892

.084

.788

.689

.994

.393

.879

.058

.442

.288

.991

.915

.093

.889

.977

.991

.289

.792

.373

.492

.992

.588

.052

.690

.492

.290

.034

.794

.795

.591

.9

50G

EP

(0,1

0,-1

3.8)

90.2

94.7

88.9

90.0

95.0

88.5

91.7

92.8

96.3

96.2

85.1

66.4

48.9

91.9

94.5

18.3

96.0

93.4

82.1

93.9

93.1

94.9

80.2

95.3

95.1

91.8

56.8

93.6

94.8

93.3

28.2

96.7

97.1

94.6

50G

EP

(0,1

0.8,

-17.

3)93

.997

.092

.193

.397

.291

.394

.195

.097

.897

.990

.473

.956

.194

.296

.522

.297

.796

.185

.996

.095

.897

.085

.797

.397

.194

.860

.696

.196

.695

.922

.598

.398

.496

.8

Mea

n

5046

.751

.744

.743

.252

.342

.031

.542

.626

.448

.944

.140

.236

.632

.143

.625

.054

.747

.423

.147

.048

.646

.843

.333

.946

.348

.726

.948

.247

.345

.518

.750

.643

.154

.8

Ran

k

5014

417

213

2429

2331

618

2526

2819

322

1033

128

1320

2715

730

911

1634

522

1

Dev

.to

the

Bes

t-

Mea

n

5014

.69.

616

.718

.19.

119

.329

.918

.834

.912

.417

.221

.124

.729

.217

.736

.36.

713

.938

.314

.412

.714

.518

.127

.515

.112

.634

.513

.114

.115

.842

.610

.718

.26.

5

Ran

k

5014

417

20.5

324

2923

316

1825

2628

1932

210

3312

813

20.5

2715

730

911

1634

522

1

Dev

.to

the

Bes

t-

Max

5063

.340

.147

.249

.639

.463

.095

.865

.899

.947

.662

.775

.079

.695

.064

.780

.531

.151

.792

.855

.134

.151

.247

.999

.556

.132

.283

.036

.063

.147

.499

.933

.579

.638

.1

Ran

k

5021

89

137

1931

2333

.511

1824

25.5

3022

271

1529

164

1412

3217

228

520

1033

.53

25.5

6

33

Page 34: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le5:

Pow

erag

ainst

GE

Pdis

trib

uti

onfo

r5%

leve

lte

sts.M

=10

0,00

0.n

=10

0(b

egin

nin

g)

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

1.1

GE

P(1

,-40

.5,0

)10

010

010

010

010

00.

810

010

010

010

010

010

010

010

010

093

.010

010

038

.910

010

010

010

010

010

010

099

.210

010

010

03.

897

.410

010

0

1.1

GE

P(4

,-18

.1,0

)10

010

010

010

010

00.

910

010

010

010

010

010

010

010

010

094

.610

010

063

.510

010

010

010

010

010

010

099

.910

010

010

03.

297

.810

010

0

1.1

GE

P(6

,-14

.6,0

)10

010

010

010

010

00.

810

010

010

010

010

010

010

010

010

096

.010

010

075

.510

010

010

010

010

010

010

010

010

010

010

02.

998

.110

010

0

1.1

GE

P(8

,-12

.9,0

)10

010

010

010

010

00.

910

010

010

010

010

010

010

010

010

097

.010

010

083

.110

010

010

010

010

010

010

010

010

010

010

02.

898

.210

010

0

1.1

GE

P(1

5,-1

0.9,

0)10

010

010

010

010

00.

910

010

010

010

010

010

010

010

010

098

.910

010

090

.810

010

010

010

099

.910

010

010

010

010

010

03.

098

.410

010

0

1.1

GE

P(3

0,-1

0,0)

100

100

100

100

100

0.9

100

100

100

100

100

100

100

100

100

99.6

100

100

93.7

100

100

100

100

99.9

100

100

100

100

100

100

3.0

98.6

100

100

1.1

GE

P(6

0,-9

.8,0

)10

010

010

010

010

00.

910

010

010

010

010

010

010

010

010

099

.810

010

094

.410

010

010

010

099

.910

010

010

010

010

010

02.

898

.710

010

0

1.5

GE

P(1

,-8.

3,0)

100

100

100

100

100

91.9

99.7

99.9

86.8

100

100

100

100

99.7

100

92.3

100

100

60.6

100

100

100

100

0.3

100

100

55.1

100

100

100

0.0

98.5

79.3

100

1.5

GE

P(2

,-5.

7,0)

100

100

100

100

100

95.6

100

100

88.1

100

100

100

100

100

100

88.0

100

100

72.8

100

100

100

100

0.2

100

100

67.3

100

100

100

0.0

99.1

90.9

100

1.5

GE

P(4

,-3.

7,0)

100

100

100

100

100

97.2

100

100

87.4

100

100

100

100

100

100

81.8

100

100

75.7

100

100

100

100

8.6

100

100

85.4

100

100

100

0.0

99.6

99.1

100

1.5

GE

P(6

,-2.

9,0)

100

100

100

100

100

97.4

100

100

84.4

100

100

100

99.7

100

100

83.1

100

100

74.9

100

100

100

100

21.2

100

100

93.5

100

100

100

0.0

99.7

99.9

100

1.5

GE

P(8

,-2.

5,0)

100

100

100

100

100

97.4

100

100

82.0

100

100

99.8

98.8

100

100

84.9

100

100

74.1

100

100

100

100

28.5

100

100

96.9

100

100

100

0.0

99.8

100

100

1.5

GE

P(1

5,-2

.1,0

)99

.810

010

010

010

097

.510

010

078

.110

010

099

.195

.610

010

086

.310

010

072

.810

010

010

010

037

.710

010

099

.210

010

010

00.

099

.810

010

0

1.5

GE

P(3

0,-1

.9,0

)99

.810

010

010

010

097

.510

010

076

.010

099

.998

.693

.010

010

086

.310

010

072

.710

010

010

010

040

.710

010

099

.710

010

010

00.

099

.810

010

0

1.5

GE

P(6

0,-1

.8,0

)99

.710

010

010

010

097

.510

010

075

.310

099

.998

.291

.910

010

086

.210

010

072

.510

010

010

010

041

.510

010

099

.810

010

010

00.

099

.810

010

0

1.84

GE

P(0

.6,-

8.4,

0)10

010

099

.799

.510

094

.674

.689

.324

.199

.910

010

010

073

.396

.091

.110

010

028

.897

.210

010

010

00.

210

010

026

.010

090

.499

.90.

097

.026

.910

0

1.84

GE

P(0

.8,-

6,0)

100

100

99.5

99.2

100

95.1

74.0

89.7

20.9

99.8

100

100

100

72.6

95.6

89.7

100

100

28.3

97.2

100

100

100

0.2

100

100

26.6

100

91.2

99.9

0.0

97.1

27.5

99.9

1.84

GE

P(1

,-5,

0)10

010

099

.198

.610

095

.873

.289

.917

.599

.810

010

010

071

.795

.386

.010

010

028

.197

.310

010

099

.91.

110

010

027

.510

092

.599

.90.

097

.428

.599

.9

1.84

GE

P(1

.5,-

3.9,

0)99

.810

097

.796

.410

097

.171

.490

.710

.299

.810

099

.999

.969

.694

.569

.999

.910

025

.797

.499

.799

.299

.716

.299

.099

.830

.799

.795

.299

.60.

098

.133

.099

.8

1.84

GE

P(2

,-3.

2,0)

98.6

99.8

96.6

95.0

99.8

98.0

70.6

91.4

6.2

99.6

99.7

99.3

98.4

68.5

94.0

57.3

99.9

99.7

23.0

97.6

99.1

97.1

99.4

42.2

96.5

99.4

34.8

98.8

96.9

99.3

0.0

98.3

38.3

99.5

1.84

GE

P(3

,-2.

1,0)

91.0

98.5

95.8

94.2

98.5

98.8

68.8

92.0

2.9

99.2

97.0

92.0

84.7

66.4

92.9

45.4

99.6

97.8

19.1

97.4

97.5

92.4

99.0

74.8

90.9

98.4

42.0

96.8

98.5

98.2

0.0

98.3

49.4

97.9

1.84

GE

P(4

,-1.

4,0)

81.9

96.5

96.1

94.8

96.5

99.0

67.3

92.0

1.7

98.6

91.2

78.8

64.8

64.8

91.7

40.6

98.9

94.1

16.6

97.3

96.5

89.3

99.0

86.3

87.3

97.8

48.9

95.5

99.0

96.9

0.0

97.8

59.7

95.7

1.84

GE

P(6

,-0.

7,0)

69.9

93.8

97.4

96.8

93.8

99.1

66.7

92.0

0.9

97.7

79.4

56.6

39.0

64.1

90.4

36.7

97.0

87.8

14.7

97.0

96.6

87.8

99.4

92.6

85.3

98.0

61.4

95.3

99.5

95.0

0.0

96.0

76.0

92.7

1.84

GE

P(8

,-0.

4,0)

62.7

92.2

98.3

98.1

92.3

99.2

65.5

91.9

0.7

96.9

70.1

43.3

27.1

62.7

89.5

34.4

95.3

83.0

13.4

96.7

97.0

87.9

99.6

94.3

85.3

98.3

70.1

95.8

99.7

93.6

0.0

94.5

85.5

90.9

1.84

GE

P(1

5.2,

0,0)

54.1

90.7

99.2

99.2

90.7

99.1

64.5

91.4

0.4

95.7

56.7

28.5

15.9

61.7

88.0

29.7

92.0

75.5

12.2

96.3

98.1

90.1

99.9

95.3

87.4

99.0

84.4

97.0

99.8

91.2

0.0

91.5

95.9

88.5

2G

EP

(0.5

,-8.

9,0)

100

100

97.8

97.0

100

85.0

52.9

75.0

10.2

98.7

100

100

100

51.1

85.7

90.2

99.9

100

20.3

90.7

100

100

99.7

0.4

100

100

20.0

100

76.9

99.1

0.1

95.5

15.1

99.9

2G

EP

(0.8

,-5.

1,0)

100

100

95.2

93.6

100

85.4

48.8

73.5

6.4

98.3

100

100

100

47.0

82.8

85.7

99.9

100

18.5

89.5

99.8

99.6

99.2

1.3

99.5

99.8

19.9

99.8

78.0

98.5

0.1

95.8

15.0

99.8

2G

EP

(1,-

4.2,

0)99

.910

092

.289

.910

086

.045

.271

.94.

297

.999

.910

010

043

.280

.378

.499

.899

.916

.888

.899

.398

.598

.45.

198

.299

.420

.299

.279

.597

.70.

096

.115

.399

.7

2G

EP

(1.5

,-3.

2,0)

96.6

99.0

84.1

80.6

99.0

86.7

38.4

69.3

1.5

96.3

98.7

98.5

97.6

36.3

74.5

54.4

99.5

98.7

12.8

86.7

94.6

89.0

95.0

31.2

87.4

95.8

22.2

93.8

82.8

94.4

0.0

96.3

17.0

98.4

2G

EP

(2,-

2.5,

0)86

.795

.279

.075

.795

.287

.433

.867

.10.

694

.194

.091

.086

.231

.769

.940

.598

.694

.410

.284

.988

.377

.392

.254

.874

.791

.024

.486

.685

.390

.50.

095

.619

.594

.2

2G

EP

(3,-

1.4,

0)64

.784

.275

.873

.084

.388

.528

.864

.50.

289

.577

.764

.351

.926

.664

.128

.694

.581

.07.

482

.879

.962

.689

.977

.458

.884

.628

.676

.788

.982

.90.

092

.225

.282

.7

2G

EP

(4,-

0.7,

0)50

.075

.176

.574

.275

.388

.626

.062

.50.

184

.861

.042

.329

.723

.860

.024

.288

.368

.96.

080

.976

.355

.989

.984

.451

.882

.133

.072

.490

.676

.70.

086

.832

.074

.2

2G

EP

(6,0

,0)

35.0

65.4

80.3

78.9

65.4

88.8

23.0

60.1

0.0

78.6

40.0

21.6

13.1

20.9

54.9

19.5

78.3

54.0

4.6

78.6

75.6

51.3

92.1

88.0

46.7

82.1

41.8

70.7

92.8

68.6

0.0

77.7

44.6

65.3

2G

EP

(8,0

.3,0

)29

.562

.084

.283

.662

.088

.922

.259

.40.

075

.430

.814

.58.

520

.253

.217

.272

.347

.04.

277

.477

.652

.194

.588

.547

.384

.249

.472

.494

.264

.70.

072

.255

.562

.2

2G

EP

(15,

0.7,

0)22

.858

.590

.290

.658

.488

.620

.757

.70.

070

.619

.98.

45.

418

.649

.913

.763

.637

.83.

475

.382

.355

.397

.387

.849

.888

.564

.076

.895

.759

.10.

064

.174

.358

.4

2G

EP

(0,8

65,8

73)

70.7

82.7

69.6

68.9

82.8

72.5

80.2

81.0

89.1

87.6

70.6

53.3

40.3

80.0

83.6

13.0

90.4

82.8

57.1

84.8

80.0

84.0

49.2

87.3

84.7

76.2

36.5

81.0

86.2

80.4

74.5

91.0

94.1

85.7

2.25

GE

P(0

.5,-

7.2,

0)10

010

084

.882

.810

064

.425

.649

.62.

590

.499

.810

010

024

.360

.186

.899

.399

.812

.871

.299

.098

.495

.81.

498

.199

.114

.999

.052

.891

.20.

591

.76.

299

.6

2.25

GE

P(0

.7,-

4.7,

0)99

.999

.977

.975

.199

.962

.321

.645

.71.

587

.999

.599

.999

.920

.354

.381

.298

.999

.510

.967

.597

.095

.092

.52.

894

.497

.314

.196

.951

.987

.70.

491

.35.

999

.3

2.25

GE

P(0

.8,-

3.8,

0)99

.399

.670

.367

.599

.660

.018

.142

.10.

985

.398

.799

.699

.717

.048

.972

.898

.698

.79.

263

.693

.188

.888

.26.

187

.794

.013

.792

.751

.083

.60.

391

.05.

498

.9

2.25

GE

P(1

,-3.

3,0)

96.5

98.0

62.6

59.6

98.0

58.5

15.3

39.2

0.6

81.9

96.7

98.2

98.3

14.1

44.0

61.1

97.7

96.4

8.0

60.3

86.1

78.7

82.6

11.9

76.8

88.0

13.2

85.2

51.0

78.6

0.2

90.3

5.4

97.6

2.25

GE

P(1

.5,-

2.4,

0)70

.479

.144

.242

.679

.253

.08.

730

.40.

269

.579

.878

.874

.77.

931

.032

.090

.276

.84.

649

.859

.945

.265

.732

.642

.364

.713

.157

.450

.461

.30.

184

.65.

479

.9

2.25

GE

P(2

,-1.

7,0)

46.5

58.4

36.1

35.5

58.5

49.6

5.7

25.2

0.1

59.0

57.6

51.3

44.0

5.1

23.8

20.8

78.3

56.2

3.3

43.4

44.7

28.6

57.3

43.4

25.9

50.8

13.4

41.4

50.1

48.9

0.0

74.7

5.6

57.0

2.25

GE

P(3

,-0.

5,0)

23.3

35.7

31.2

31.1

35.7

45.9

3.5

19.8

0.0

44.4

29.0

20.2

14.6

3.0

16.8

13.2

56.1

32.4

2.1

35.7

32.5

16.7

51.4

51.5

14.4

39.2

14.6

28.7

50.6

34.7

0.0

55.2

6.9

34.1

2.25

GE

P(3

.7,0

,0)

16.9

28.5

30.3

30.5

28.5

43.9

2.8

17.4

0.0

37.9

19.4

12.3

8.6

2.4

14.2

11.2

45.5

24.7

1.7

32.4

29.1

13.5

50.9

52.7

11.6

36.1

15.6

25.1

50.7

29.0

0.0

45.3

7.6

28.1

2.25

GE

P(4

,0.2

,0)

14.6

26.0

30.5

30.9

25.9

43.4

2.5

16.8

0.0

35.4

16.3

9.8

6.9

2.2

13.3

10.5

41.4

21.8

1.7

31.2

28.3

12.6

51.5

52.7

10.7

35.4

16.2

24.2

51.0

27.0

0.0

41.5

8.2

26.0

2.25

GE

P(8

,1.2

,0)

7.3

16.8

37.3

37.7

16.6

40.2

1.4

12.6

0.0

22.7

5.3

4.6

5.0

1.2

8.8

6.6

22.0

11.0

0.9

25.1

28.7

10.3

61.1

50.0

8.3

37.4

22.4

23.3

54.6

17.1

0.0

22.7

15.3

21.1

2.25

GE

P(0

,785

,782

)70

.782

.769

.668

.982

.872

.580

.181

.089

.087

.570

.853

.340

.279

.983

.313

.190

.282

.657

.284

.779

.883

.949

.287

.184

.676

.036

.980

.986

.180

.374

.790

.894

.085

.6

2.5

GE

P(0

.5,-

6.1,

0)99

.799

.864

.362

.599

.844

.113

.530

.61.

972

.798

.299

.799

.812

.737

.581

.896

.197

.910

.049

.292

.989

.982

.93.

189

.193

.312

.892

.832

.972

.61.

584

.62.

798

.8

2.5

GE

P(0

.7,-

3.9,

0)97

.898

.352

.250

.698

.339

.49.

925

.61.

465

.795

.198

.498

.99.

330

.171

.494

.094

.78.

142

.282

.776

.672

.74.

775

.184

.011

.482

.230

.363

.31.

282

.22.

297

.7

2.5

GE

P(0

.8,-

3.2,

0)91

.392

.741

.840

.992

.735

.47.

621

.81.

158

.989

.194

.795

.77.

124

.357

.990

.687

.76.

936

.469

.360

.562

.38.

058

.671

.710

.768

.328

.454

.50.

979

.42.

195

.0

2.5

GE

P(1

,-2.

7,0)

78.0

80.5

32.7

32.4

80.6

31.6

5.6

18.0

0.9

51.7

79.1

85.6

86.3

5.2

18.9

43.2

84.2

75.0

5.7

30.8

53.8

43.4

51.7

12.3

41.3

57.2

10.0

52.3

26.8

45.5

0.7

74.5

1.8

86.5

2.5

GE

P(1

.5,-

1.8,

0)35

.539

.117

.618

.839

.123

.22.

511

.10.

532

.941

.841

.738

.52.

39.

618

.056

.836

.23.

419

.024

.815

.431

.820

.514

.128

.78.

522

.822

.925

.90.

352

.91.

540

.4

2.5

GE

P(2

,-1,

0)17

.921

.212

.514

.021

.218

.21.

37.

60.

422

.321

.218

.415

.41.

25.

611

.036

.519

.52.

313

.315

.07.

724

.722

.16.

718

.67.

913

.120

.217

.00.

235

.51.

519

.2

2.5

GE

P(2

.8,0

,0)

8.4

11.2

9.4

10.9

11.1

14.1

0.7

5.0

0.2

13.7

9.0

7.1

5.9

0.6

3.2

7.3

19.6

10.4

1.5

9.0

9.7

3.9

20.7

21.0

3.3

12.7

7.2

8.0

17.9

10.7

0.1

19.8

1.6

9.8

2.5

GE

P(3

,0.2

,0)

7.4

9.9

9.1

10.5

9.8

13.5

0.6

4.6

0.2

12.3

7.5

6.0

5.1

0.5

2.8

6.8

17.1

9.2

1.4

8.2

8.9

3.4

20.3

20.7

2.9

12.1

7.3

7.3

17.6

9.8

0.1

17.3

1.7

8.9

2.5

GE

P(4

,0.9

,0)

5.2

6.9

8.5

9.6

6.8

11.2

0.3

3.4

0.1

8.2

4.6

4.8

5.0

0.3

1.8

5.5

9.9

6.4

1.0

6.2

7.5

2.4

20.2

18.4

2.0

10.6

7.2

5.9

16.3

7.1

0.1

10.2

2.1

7.9

2.5

GE

P(8

,1.9

,0)

5.0

6.2

10.1

10.6

6.0

8.0

0.1

1.9

0.1

3.7

6.3

10.7

12.2

0.1

0.9

4.3

4.0

4.5

0.7

3.5

8.0

2.1

26.8

13.5

1.7

11.5

8.4

5.7

15.1

4.4

0.1

4.3

4.1

12.0

2.5

GE

P(0

,730

,739

)70

.882

.769

.468

.882

.972

.380

.080

.989

.087

.670

.853

.740

.579

.883

.412

.890

.382

.757

.284

.779

.883

.949

.287

.184

.675

.936

.780

.886

.280

.474

.590

.994

.085

.5

2.75

GE

P(0

.5,-

5.3,

0)97

.797

.946

.245

.797

.930

.310

.620

.93.

352

.192

.398

.099

.010

.225

.075

.488

.291

.710

.233

.279

.274

.563

.85.

073

.279

.912

.279

.120

.250

.63.

473

.51.

297

.5

2.75

GE

P(0

.6,-

4.1,

0)95

.095

.040

.139

.895

.027

.59.

218

.52.

947

.588

.396

.097

.68.

821

.469

.385

.387

.59.

329

.371

.065

.157

.05.

363

.772

.011

.370

.718

.645

.03.

071

.01.

196

.3

2.75

GE

P(0

.7,-

3.3,

0)88

.788

.633

.433

.788

.624

.68.

116

.22.

842

.081

.591

.894

.37.

918

.260

.280

.779

.98.

525

.560

.253

.548

.66.

152

.061

.810

.959

.716

.938

.82.

967

.11.

093

.6

2.75

GE

P(0

.8,-

2.8,

0)78

.177

.727

.527

.977

.821

.87.

114

.02.

636

.572

.584

.387

.56.

915

.349

.274

.168

.77.

821

.848

.141

.240

.27.

339

.850

.310

.347

.415

.632

.42.

662

.10.

888

.2

2.75

GE

P(1

,-2.

3,0)

49.4

48.1

18.2

18.8

48.1

16.9

5.5

10.3

2.3

26.1

49.1

58.4

60.3

5.4

10.4

28.3

55.3

41.4

6.4

15.5

28.1

22.2

26.4

9.4

21.3

30.4

8.9

27.2

13.1

21.6

2.3

48.7

0.7

61.2

2.75

GE

P(1

.5,-

1.4,

0)15

.114

.78.

69.

614

.79.

73.

25.

81.

712

.416

.116

.715

.63.

14.

910

.622

.912

.94.

37.

510

.37.

013

.410

.26.

611

.96.

69.

59.

110

.21.

522

.00.

815

.4

2.75

GE

P(2

,-0.

5,0)

7.1

7.2

5.5

6.7

7.1

6.4

2.1

3.7

1.5

7.3

7.1

6.8

6.1

2.1

2.8

6.6

11.1

7.0

3.1

4.4

5.9

3.6

10.0

8.7

3.4

7.2

5.4

5.2

7.1

6.4

1.2

11.1

1.1

6.1

2.75

GE

P(2

.3,0

,0)

5.5

5.7

4.5

5.6

5.5

5.3

1.7

3.0

1.4

5.7

5.3

5.2

4.9

1.7

2.2

5.5

7.7

5.7

2.7

3.4

4.8

2.7

8.9

7.8

2.6

5.9

4.8

4.1

6.4

5.5

1.1

7.9

1.6

4.6

2.75

GE

P(3

,0.8

,0)

4.6

4.5

3.4

4.4

4.3

3.6

1.1

2.2

1.2

4.1

4.8

5.7

5.9

1.0

1.4

4.8

4.5

4.7

2.1

2.0

3.7

1.9

8.2

6.1

1.8

4.7

4.1

3.1

5.2

4.4

1.0

4.6

2.9

4.4

2.75

GE

P(4

,1.6

,0)

5.6

5.1

2.9

3.7

5.0

2.4

0.7

1.5

1.2

3.4

7.4

9.7

10.3

0.7

1.0

4.4

3.9

4.7

1.6

1.2

3.6

1.8

8.2

4.7

1.7

4.4

3.5

2.9

4.1

4.0

1.0

3.9

5.2

7.0

2.75

GE

P(0

,690

,706

)70

.882

.769

.769

.082

.872

.580

.281

.089

.187

.670

.853

.540

.380

.083

.412

.990

.482

.857

.384

.880

.084

.149

.287

.284

.876

.136

.781

.086

.380

.374

.791

.094

.185

.8

3G

EP

(0.5

,-4.

6,0)

91.9

91.4

35.7

35.9

91.5

23.0

12.1

17.6

5.5

35.1

80.6

93.3

96.2

11.8

20.1

68.3

74.7

79.9

11.6

24.6

63.4

59.4

45.7

7.1

58.4

64.0

12.8

63.6

13.5

33.3

6.0

59.6

0.7

95.4

3G

EP

(0.6

,-3.

6,0)

85.1

83.6

30.1

30.3

83.7

20.5

11.3

15.7

5.4

30.0

72.4

87.6

91.9

11.1

17.6

60.6

68.6

71.0

11.1

21.4

53.3

48.9

38.3

7.2

48.0

54.0

11.9

53.4

12.2

27.7

6.0

55.1

0.6

92.7

3G

EP

(0.7

,-2.

9,0)

72.5

69.9

24.8

25.0

69.9

17.9

10.4

14.0

5.3

24.8

61.2

77.7

83.2

10.3

15.3

49.1

59.7

57.6

10.4

18.2

41.6

38.0

30.1

7.3

37.2

42.5

11.1

41.6

10.7

22.4

5.8

48.5

0.6

86.5

3G

EP

(0.8

,-2.

5,0)

56.2

52.2

20.1

20.2

52.2

15.7

9.7

12.5

5.1

20.2

48.0

63.2

68.4

9.6

13.2

36.7

48.9

42.1

9.7

15.6

30.7

27.8

23.0

7.7

27.2

31.7

10.5

30.7

9.9

17.7

5.8

41.0

0.6

73.6

3G

EP

(1,-

1.9,

0)27

.023

.813

.313

.323

.711

.58.

39.

85.

012

.424

.532

.034

.38.

29.

818

.227

.418

.58.

211

.116

.415

.113

.17.

614

.917

.18.

916

.47.

710

.75.

424

.80.

737

.0

3G

EP

(1.5

,-1,

0)7.

26.

97.

27.

06.

77.

26.

36.

65.

06.

26.

67.

16.

96.

26.

46.

97.

96.

15.

96.

87.

06.

86.

56.

26.

87.

26.

56.

95.

86.

05.

08.

02.

48.

2

3G

EP

(2,0

,0)

5.2

5.1

4.9

4.9

4.9

5.0

5.1

5.1

5.0

5.1

5.1

5.3

5.0

5.0

4.8

5.0

5.0

5.2

4.9

5.0

4.9

4.9

5.0

5.0

5.0

5.0

4.8

4.9

5.0

5.2

5.0

5.2

5.1

5.0

34

Page 35: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le5:

Pow

erag

ainst

GE

Pdis

trib

uti

onfo

r5%

leve

lte

sts.M

=10

0,00

0.n

=10

0(c

onti

nuin

g)

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

3G

EP

(3,1

.4,0

)7.

56.

83.

03.

46.

73.

13.

53.

45.

56.

210

.011

.411

.03.

53.

54.

57.

06.

03.

73.

44.

54.

24.

54.

54.

34.

33.

24.

24.

55.

45.

47.

211

.78.

0

3G

EP

(0,6

65,6

61)

70.8

82.8

69.5

68.8

82.9

72.4

80.2

80.9

89.0

87.6

71.0

53.7

40.7

80.0

83.5

12.7

90.3

82.7

57.5

84.7

79.8

84.0

49.3

87.2

84.7

76.0

36.5

80.9

86.1

80.4

74.7

90.9

94.1

85.6

3.5

GE

P(0

.5,-

3.8,

0)70

.667

.229

.128

.567

.421

.620

.321

.112

.716

.649

.672

.481

.420

.122

.254

.442

.449

.517

.023

.442

.843

.324

.413

.643

.241

.915

.443

.812

.816

.414

.833

.21.

488

.7

3.5

GE

P(0

.6,-

2.8,

0)56

.251

.525

.624

.451

.620

.120

.020

.412

.913

.437

.158

.468

.319

.821

.144

.033

.036

.516

.222

.034

.235

.518

.513

.335

.633

.214

.635

.212

.613

.515

.226

.51.

780

.9

3.5

GE

P(0

.7,-

2.3,

0)39

.034

.222

.620

.934

.219

.119

.919

.813

.611

.125

.040

.648

.919

.820

.431

.422

.722

.815

.821

.126

.828

.713

.613

.329

.125

.613

.927

.812

.711

.316

.219

.32.

365

.1

3.5

GE

P(0

.8,-

1.9,

0)23

.520

.819

.917

.720

.718

.319

.919

.614

.69.

515

.424

.329

.119

.719

.820

.014

.513

.415

.220

.421

.123

.79.

813

.224

.119

.813

.122

.113

.010

.017

.113

.53.

544

.4

3.5

GE

P(1

,-1.

4,0)

9.6

10.7

16.4

13.7

10.4

16.9

19.5

18.7

16.5

9.1

6.5

8.2

9.1

19.4

18.8

8.9

8.3

7.5

14.0

19.7

15.6

18.6

6.3

13.4

19.2

14.1

11.8

16.5

14.1

9.2

19.0

8.6

7.8

21.6

3.5

GE

P(1

.6,0

,0)

9.5

12.1

12.5

10.0

12.0

14.7

18.5

17.7

21.4

15.3

10.7

8.9

7.7

18.3

18.2

4.8

16.2

11.1

12.1

19.3

13.6

16.8

4.7

16.1

17.4

11.6

9.0

14.3

17.7

12.1

23.3

16.5

23.9

16.3

3.5

GE

P(2

,0.8

,0)

13.5

16.0

11.7

9.6

15.9

14.1

18.4

17.8

23.9

19.7

17.0

14.6

12.3

18.1

18.6

5.0

22.2

14.7

11.6

19.9

14.6

17.9

5.2

18.3

18.6

12.3

7.9

15.2

19.7

14.5

25.7

22.5

32.0

19.4

3.5

GE

P(3

,2.3

,0)

23.4

26.7

10.3

9.1

26.7

12.4

17.0

16.6

28.5

29.7

33.6

30.3

25.7

16.8

19.5

6.5

35.7

23.4

10.6

20.5

18.2

21.3

7.1

23.4

22.1

15.3

5.7

18.6

23.3

20.3

30.6

36.3

47.8

29.7

3.5

GE

P(0

,620

,609

)70

.983

.070

.069

.283

.272

.880

.581

.289

.287

.971

.053

.840

.780

.283

.712

.890

.582

.957

.484

.880

.384

.249

.387

.484

.976

.437

.081

.386

.380

.774

.791

.094

.185

.9

4G

EP

(0.4

,-4.

5,0)

62.5

60.3

33.4

31.9

60.5

27.6

29.3

29.0

20.8

14.1

35.8

61.2

73.4

29.1

29.7

52.8

28.7

40.3

22.5

30.5

43.1

46.0

21.4

21.8

46.3

41.1

18.8

44.5

19.4

15.8

24.0

23.4

3.9

87.4

4G

EP

(0.5

,-3.

2,0)

49.5

47.0

32.0

29.8

47.1

27.8

30.3

29.7

22.4

13.3

25.6

46.6

59.0

30.1

30.0

43.2

21.6

29.7

22.7

30.9

37.7

41.2

17.5

22.6

41.8

35.6

18.9

39.2

20.9

15.0

25.8

18.8

5.1

80.7

4G

EP

(0.6

,-2.

3,0)

34.7

33.5

30.2

27.3

33.6

28.1

31.4

30.6

24.3

12.5

16.6

30.9

40.4

31.2

30.7

32.0

15.7

20.0

22.6

31.8

33.0

37.4

13.9

23.4

38.0

30.6

18.6

34.5

22.5

14.3

28.1

14.8

6.9

68.6

4G

EP

(0.7

,-1.

9,0)

21.0

23.2

28.3

24.6

23.0

28.2

32.2

31.3

26.6

13.6

9.7

16.4

21.5

32.0

31.0

20.2

12.1

13.4

22.4

32.4

28.7

33.7

11.2

24.2

34.5

26.2

18.0

30.3

24.3

15.0

30.6

12.7

10.6

50.5

4G

EP

(0.8

,-1.

5,0)

13.5

18.4

26.9

22.8

18.2

28.4

33.1

31.9

29.3

15.8

6.7

8.6

10.3

32.8

31.7

11.7

13.1

11.5

22.1

33.4

26.6

31.7

9.6

25.6

32.6

23.8

17.4

28.1

26.7

16.3

33.3

13.9

16.2

37.0

4G

EP

(1,-

1.1,

0)11

.218

.125

.921

.417

.928

.934

.333

.333

.820

.97.

65.

85.

334

.033

.16.

919

.414

.422

.135

.125

.831

.28.

728

.532

.222

.616

.527

.330

.419

.237

.120

.026

.430

.1

4G

EP

(1.4

,0,0

)20

.328

.325

.321

.428

.329

.636

.335

.743

.235

.122

.316

.312

.736

.136

.75.

838

.426

.822

.139

.129

.435

.19.

636

.936

.125

.514

.930

.838

.627

.645

.638

.949

.434

.8

4G

EP

(2,1

.5,0

)31

.139

.525

.622

.739

.630

.038

.037

.950

.346

.738

.230

.524

.237

.740

.46.

952

.537

.922

.342

.734

.740

.511

.944

.341

.730

.013

.835

.944

.635

.252

.353

.163

.944

.4

4G

EP

(3,3

.1,0

)44

.654

.127

.126

.054

.330

.539

.940

.458

.660

.356

.947

.438

.239

.645

.710

.166

.751

.522

.947

.743

.048

.716

.953

.849

.937

.611

.944

.051

.945

.960

.667

.876

.757

.2

4G

EP

(0,5

80,5

84)

70.9

83.0

69.8

69.1

83.2

72.6

80.2

81.0

89.1

87.7

71.1

53.7

40.5

80.0

83.6

12.9

90.4

82.9

57.5

84.9

80.1

84.2

49.3

87.4

84.9

76.3

36.7

81.1

86.4

80.7

74.7

91.0

94.0

85.8

4.5

GE

P(0

.4,-

4,0)

48.3

49.1

38.5

35.8

49.4

35.3

39.2

38.5

31.1

16.0

20.4

41.1

54.9

39.0

38.3

44.8

18.8

29.4

28.5

39.8

43.7

48.6

19.5

31.7

49.2

40.8

22.8

45.5

29.7

19.0

34.9

17.9

9.5

83.0

4.5

GE

P(0

.5,-

2.7,

0)35

.938

.537

.433

.938

.736

.140

.739

.833

.417

.213

.427

.338

.040

.439

.534

.816

.022

.828

.641

.140

.245

.717

.033

.346

.537

.122

.542

.032

.219

.937

.716

.612

.874

.5

4.5

GE

P(0

.6,-

2,0)

25.2

30.8

36.8

32.6

30.8

37.4

42.6

41.7

37.0

20.0

8.7

15.2

21.3

42.3

41.2

23.7

16.2

18.1

29.0

43.2

37.9

43.8

15.3

35.4

44.7

34.6

22.3

39.8

35.6

21.9

41.4

17.6

18.2

61.6

4.5

GE

P(0

.7,-

1.5,

0)18

.127

.135

.931

.326

.938

.544

.443

.641

.224

.87.

17.

79.

644

.143

.013

.821

.118

.129

.445

.436

.642

.714

.038

.143

.732

.922

.238

.339

.425

.145

.422

.326

.948

.3

4.5

GE

P(0

.8,-

1.2,

0)17

.128

.036

.031

.127

.839

.846

.245

.646

.231

.39.

46.

15.

545

.945

.28.

529

.622

.329

.847

.837

.043

.413

.941

.844

.533

.021

.838

.843

.729

.149

.330

.538

.242

.8

4.5

GE

P(1

,-0.

8,0)

22.3

34.3

36.8

32.3

34.2

41.7

48.9

48.5

53.2

41.4

18.4

11.2

8.1

48.6

48.8

6.4

43.0

31.7

30.7

51.4

40.1

46.8

14.8

48.0

47.9

35.6

21.6

41.8

49.7

35.7

55.1

43.7

53.4

44.6

4.5

GE

P(1

.3,0

,0)

34.3

46.2

38.6

35.1

46.4

43.6

52.1

52.1

61.5

54.7

35.8

25.2

18.8

51.8

53.7

7.2

59.4

45.5

32.0

56.4

46.1

52.8

17.8

56.4

53.9

41.2

21.2

47.7

57.0

45.0

61.8

60.0

69.4

53.4

4.5

GE

P(2

,2.1

,0)

50.1

62.2

42.2

40.4

62.5

46.6

56.5

57.2

71.0

69.6

57.4

45.4

35.3

56.1

61.0

10.1

74.7

61.2

34.4

63.3

55.8

61.9

24.0

67.1

63.1

50.3

20.8

57.2

66.1

57.1

70.1

75.7

82.9

66.4

4.5

GE

P(3

,3.9

,0)

61.9

73.5

46.4

46.4

73.7

49.6

60.7

61.9

78.2

79.2

71.1

58.2

46.4

60.3

67.4

13.3

83.6

72.1

36.5

69.2

64.6

70.1

30.5

75.3

71.3

59.0

20.1

65.7

72.9

67.0

76.7

84.6

89.7

75.9

4.5

GE

P(0

,550

,557

)70

.983

.069

.869

.283

.172

.680

.381

.089

.187

.771

.153

.740

.580

.183

.612

.990

.582

.757

.584

.980

.284

.349

.387

.384

.976

.336

.881

.186

.280

.674

.391

.094

.185

.8

5G

EP

(0.5

,-2.

4,0)

31.3

39.1

43.4

39.5

39.2

44.4

49.9

49.1

44.4

25.8

8.8

15.7

23.1

49.7

48.4

28.8

20.8

23.9

34.4

50.8

45.7

51.8

19.6

44.0

52.7

41.9

26.2

47.5

43.8

28.3

48.1

22.8

24.1

71.1

5G

EP

(0.6

,-1.

7,0)

24.7

36.3

44.0

39.5

36.3

46.6

52.9

52.3

49.8

32.2

7.8

8.4

11.1

52.6

51.5

18.4

27.3

24.8

35.4

54.1

45.5

51.9

19.3

47.9

53.0

41.5

26.3

47.4

48.9

32.9

52.7

29.2

34.2

60.8

5G

EP

(0.7

,-1.

3,0)

24.0

38.0

44.7

40.1

37.9

48.7

55.6

55.3

56.1

41.0

11.4

6.6

5.9

55.3

55.0

10.7

39.0

30.6

36.7

57.8

46.8

53.4

19.5

52.8

54.5

42.6

26.3

48.7

54.5

38.9

57.6

40.1

48.0

54.1

5G

EP

(0.8

,-1,

0)28

.543

.546

.041

.843

.450

.858

.358

.162

.150

.320

.110

.77.

258

.058

.37.

751

.139

.638

.361

.049

.656

.321

.057

.957

.545

.226

.451

.559

.245

.162

.051

.960

.854

.3

5G

EP

(1,-

0.6,

0)38

.753

.548

.544

.953

.653

.461

.661

.669

.361

.735

.722

.515

.761

.363

.17.

365

.052

.339

.965

.455

.161

.724

.665

.062

.850

.426

.756

.865

.753

.567

.065

.774

.160

.8

5G

EP

(1.1

,0,0

)48

.462

.450

.948

.362

.655

.564

.264

.674

.570

.349

.535

.225

.863

.966

.98.

874

.461

.841

.569

.060

.666

.628

.170

.667

.755

.626

.962

.070

.460

.470

.775

.182

.167

.9

5G

EP

(2,2

.7,0

)64

.676

.857

.256

.477

.160

.470

.171

.083

.182

.470

.055

.743

.469

.875

.013

.086

.376

.245

.576

.571

.276

.337

.480

.677

.366

.427

.472

.479

.072

.577

.187

.191

.479

.8

5G

EP

(3,4

.6,0

)73

.184

.162

.362

.984

.364

.374

.675

.987

.988

.478

.864

.751

.374

.380

.416

.091

.383

.648

.881

.878

.082

.444

.886

.383

.273

.628

.278

.984

.379

.881

.492

.094

.985

.9

5G

EP

(0,5

00,6

00)

71.1

83.0

69.9

69.2

83.2

72.5

80.3

80.9

89.2

87.8

71.2

53.8

40.7

80.1

83.5

13.1

90.5

82.9

57.5

84.8

80.1

84.1

49.5

87.3

84.8

76.2

36.8

81.1

86.2

80.6

74.5

91.1

94.2

85.8

6G

EP

(0.4

,-2.

9,0)

37.6

50.5

54.7

51.2

50.6

56.7

62.8

62.4

58.7

40.2

8.5

11.8

18.4

62.6

61.6

30.3

33.5

35.0

44.3

64.1

57.5

63.8

28.5

59.0

64.7

53.6

32.8

59.5

59.0

42.2

58.6

36.4

39.9

78.0

6G

EP

(0.5

,-1.

9,0)

35.8

51.3

56.5

52.6

51.3

59.6

65.9

65.9

64.5

48.6

10.6

7.6

9.6

65.7

65.3

21.9

43.5

39.3

46.2

67.9

59.3

65.5

29.4

63.9

66.4

55.2

33.0

61.1

64.6

48.5

63.1

45.6

51.6

72.9

6G

EP

(0.6

,-1.

3,0)

38.4

55.5

58.5

54.9

55.5

62.6

69.7

69.6

71.3

59.6

18.7

8.2

6.2

69.4

69.6

13.4

57.8

48.5

48.6

72.0

62.3

68.4

31.6

69.3

69.4

57.9

33.9

64.0

70.2

56.5

67.1

58.8

66.2

69.1

6G

EP

(0.8

,-0.

8,0)

52.6

68.6

63.0

60.6

68.7

67.1

74.6

75.2

81.3

75.7

44.2

25.6

16.4

74.4

76.5

9.5

78.1

67.2

52.3

78.4

70.1

75.5

38.2

78.7

76.4

65.6

34.9

71.4

78.9

68.9

72.3

78.6

84.6

74.7

6G

EP

(1,0

,0)

70.3

82.6

69.1

68.5

82.8

72.2

80.0

80.6

88.8

87.4

70.5

53.3

40.0

79.7

83.2

12.9

90.1

82.6

56.8

84.5

79.7

83.8

48.5

87.0

84.5

75.8

36.5

80.7

85.9

80.2

74.7

90.7

93.9

85.5

6G

EP

(2,3

.9,0

)80

.790

.275

.976

.390

.377

.484

.885

.793

.393

.282

.366

.852

.184

.688

.616

.895

.190

.062

.289

.486

.689

.758

.892

.290

.383

.538

.887

.391

.087

.676

.195

.597

.291

.6

6G

EP

(3,6

.2,0

)84

.292

.479

.380

.292

.580

.387

.388

.394

.995

.085

.470

.055

.087

.190

.918

.696

.392

.364

.991

.789

.492

.063

.594

.192

.486

.840

.390

.092

.990

.376

.496

.798

.093

.4

6G

EP

(0,1

40,1

29.1

)72

.183

.870

.970

.384

.073

.681

.181

.889

.788

.571

.754

.340

.780

.984

.413

.291

.183

.858

.285

.581

.185

.050

.687

.985

.777

.337

.382

.186

.981

.774

.291

.694

.586

.5

10G

EP

(0.4

,-1.

7,0)

73.2

85.7

82.9

82.0

85.8

85.2

89.7

90.1

91.9

88.6

46.1

18.0

9.0

89.6

90.7

21.7

87.9

82.4

72.3

91.7

87.3

90.3

63.8

91.8

90.8

84.8

50.0

88.1

91.6

86.0

65.6

88.4

91.6

90.8

10G

EP

(0.5

,-1,

0)83

.191

.987

.086

.892

.088

.492

.592

.995

.794

.670

.841

.323

.592

.493

.918

.595

.191

.276

.794

.591

.893

.872

.195

.394

.189

.852

.392

.394

.991

.962

.495

.397

.093

.8

10G

EP

(0.7

,0,0

)96

.999

.094

.294

.999

.093

.796

.597

.099

.299

.396

.889

.077

.596

.598

.129

.599

.698

.985

.098

.398

.098

.688

.999

.198

.797

.457

.898

.298

.798

.249

.299

.799

.899

.1

10G

EP

(1.4

,5.7

,0)

92.2

96.8

90.9

91.3

96.9

91.4

94.9

95.3

98.1

98.0

89.9

74.2

58.1

94.8

96.5

21.3

98.6

96.8

80.8

96.7

95.7

96.9

81.1

97.8

97.1

94.4

54.9

96.0

97.2

96.0

56.4

98.7

99.3

97.4

10G

EP

(0,6

,12.

2)88

.895

.087

.687

.995

.188

.592

.993

.397

.096

.786

.068

.652

.892

.794

.818

.497

.795

.076

.895

.293

.595

.275

.396

.595

.591

.851

.594

.095

.894

.060

.497

.998

.795

.9

10G

EP

(0,1

7,-1

2.6)

89.0

95.3

88.1

88.5

95.4

88.9

93.1

93.6

97.3

97.0

86.4

69.1

53.1

93.0

95.0

18.8

97.7

95.3

77.5

95.5

93.8

95.4

76.1

96.7

95.7

92.2

52.0

94.2

96.1

94.3

60.1

98.0

98.8

96.1

20G

EP

(0.3

,-1.

3,0)

97.3

99.1

97.1

97.4

99.1

97.1

98.5

98.7

99.5

99.5

93.3

71.7

46.8

98.5

99.1

31.3

99.5

99.0

91.8

99.1

98.8

99.2

93.1

99.4

99.2

98.4

66.9

98.9

99.2

98.9

36.4

99.6

99.7

99.2

20G

EP

(0.5

,0,0

)99

.910

099

.699

.710

099

.299

.799

.810

010

099

.999

.296

.699

.799

.955

.010

010

097

.199

.999

.910

099

.310

010

099

.974

.410

099

.999

.914

.310

010

010

0

20G

EP

(0.8

,2.3

,0)

99.4

99.8

98.8

99.0

99.8

98.6

99.3

99.4

99.9

99.9

98.8

93.0

81.7

99.3

99.7

38.2

99.9

99.8

95.0

99.7

99.7

99.8

97.4

99.9

99.8

99.6

71.2

99.7

99.8

99.7

24.5

100

100

99.8

20G

EP

(0,1

4,-1

7.6)

96.8

98.9

95.9

96.3

98.9

95.9

97.8

98.0

99.3

99.3

95.0

82.1

66.0

97.7

98.6

26.3

99.5

98.9

89.2

98.7

98.4

98.8

90.9

99.2

98.9

97.8

63.8

98.5

98.9

98.6

40.3

99.6

99.8

99.1

20G

EP

(0,6

,2.6

)95

.098

.294

.294

.598

.294

.396

.897

.199

.098

.992

.778

.161

.796

.797

.923

.699

.298

.286

.498

.197

.598

.287

.498

.898

.396

.660

.597

.798

.497

.746

.099

.399

.698

.5

50G

EP

(0.3

,-0.

7,0)

100

100

99.9

99.9

100

99.8

99.9

99.9

100

100

100

99.2

95.8

99.9

100

60.5

100

100

99.0

100

100

100

99.8

100

100

100

80.6

100

100

100

6.1

100

100

100

50G

EP

(0.4

,0,0

)10

010

010

010

010

099

.910

010

010

010

010

010

099

.810

010

078

.410

010

099

.710

010

010

010

010

010

010

084

.510

010

010

01.

610

010

010

0

50G

EP

(0.6

,1.2

,0)

100

100

100

100

100

99.9

100

100

100

100

100

99.8

98.3

100

100

68.7

100

100

99.5

100

100

100

99.9

100

100

100

83.8

100

100

100

2.8

100

100

100

50G

EP

(0,5

.1,2

.1)

96.9

99.0

96.3

96.6

99.0

96.3

98.0

98.2

99.4

99.4

95.1

82.4

66.3

98.0

98.8

26.6

99.6

99.0

90.1

98.9

98.5

99.0

91.7

99.3

99.0

98.0

64.7

98.7

99.1

98.7

37.7

99.6

99.8

99.2

50G

EP

(0,8

,-7.

2)98

.899

.698

.298

.499

.698

.099

.099

.199

.899

.897

.788

.674

.199

.099

.532

.399

.999

.694

.199

.599

.499

.695

.899

.799

.699

.170

.299

.499

.699

.527

.299

.999

.999

.7

50G

EP

(0,1

0,-1

3.8)

99.5

99.9

99.1

99.3

99.9

98.9

99.5

99.6

99.9

99.9

99.0

93.2

80.9

99.5

99.8

38.6

100

99.9

96.3

99.8

99.8

99.8

98.0

99.9

99.9

99.7

74.0

99.8

99.8

99.8

19.2

100

100

99.9

50G

EP

(0,1

0.8,

-17.

3)99

.810

099

.699

.710

099

.599

.899

.810

010

099

.696

.587

.799

.899

.946

.510

010

097

.899

.999

.999

.999

.110

010

099

.977

.099

.999

.999

.912

.910

010

010

0

Mea

n

100

62.4

68.1

59.4

58.5

68.1

54.5

51.1

57.3

43.6

63.5

60.0

56.5

52.9

50.6

58.3

36.7

69.0

64.4

36.0

62.5

65.3

64.2

56.3

49.3

63.9

64.8

35.4

65.2

61.7

60.6

24.2

67.3

49.2

73.5

Ran

k

100

143.

518

193.

524

2621

3012

1722

2527

2031

29

3213

610

2328

118

337

1516

345

291

Dev

.to

the

Bes

t-

Mea

n

100

16.6

10.9

19.5

20.5

10.9

24.4

27.9

21.7

35.4

15.5

19.0

22.4

26.1

28.4

20.6

42.3

10.0

14.6

43.0

16.4

13.7

14.8

22.6

29.7

15.1

14.2

43.6

13.8

17.2

18.4

54.8

11.7

29.8

5.5

Ran

k

100

143.

518

193.

524

2621

3012

1722

2527

2031

29

3213

610

2328

118

337

1516

345

291

Dev

.to

the

Bes

t-

Max

100

74.5

44.3

62.6

62.4

44.5

99.2

89.0

79.1

99.5

73.3

77.4

88.9

91.9

89.6

76.2

83.6

64.2

59.5

93.9

71.6

46.7

50.8

66.0

99.8

52.8

47.7

87.5

45.7

82.7

72.3

100

65.1

97.8

40.0

Ran

k

100

182

1110

331

2621

3217

2025

2827

1923

129

2915

57

1433

86

244

2216

3413

301

35

Page 36: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le6:

Pow

erag

ainst

GE

Pdis

trib

uti

onfo

r5%

leve

lte

sts.M

=10

0,00

0.n

=20

0(b

egin

nin

g)

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

1.1

GE

P(1

,-40

.5,0

)10

010

010

010

010

00.

010

010

010

010

010

010

010

010

010

095

.810

010

099

.610

010

010

010

010

010

010

010

010

010

010

00.

599

.910

010

0

1.1

GE

P(4

,-18

.1,0

)10

010

010

010

010

00.

010

010

010

010

010

010

010

010

010

098

.110

010

010

010

010

010

010

010

010

010

010

010

010

010

00.

399

.910

010

0

1.1

GE

P(6

,-14

.6,0

)10

010

010

010

010

00.

010

010

010

010

010

010

010

010

010

099

.110

010

010

010

010

010

010

010

010

010

010

010

010

010

00.

310

010

010

0

1.1

GE

P(8

,-12

.9,0

)10

010

010

010

010

00.

010

010

010

010

010

010

010

010

010

099

.610

010

010

010

010

010

010

010

010

010

010

010

010

010

00.

210

010

010

0

1.1

GE

P(1

5,-1

0.9,

0)10

010

010

010

010

00.

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

00.

210

010

010

0

1.1

GE

P(3

0,-1

0,0)

100

100

100

100

100

0.0

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

0.2

100

100

100

1.1

GE

P(6

0,-9

.8,0

)10

010

010

010

010

00.

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

00.

210

010

010

0

1.5

GE

P(1

,-8.

3,0)

100

100

100

100

100

43.0

100

100

100

100

100

100

100

100

100

95.0

100

100

99.9

100

100

100

100

0.1

100

100

71.1

100

100

100

0.0

100

69.9

100

1.5

GE

P(2

,-5.

7,0)

100

100

100

100

100

45.0

100

100

100

100

100

100

100

100

100

94.2

100

100

100

100

100

100

100

0.4

100

100

86.3

100

100

100

0.0

100

89.0

100

1.5

GE

P(4

,-3.

7,0)

100

100

100

100

100

46.8

100

100

100

100

100

100

100

100

100

94.3

100

100

100

100

100

100

100

31.9

100

100

98.2

100

100

100

0.0

100

99.6

100

1.5

GE

P(6

,-2.

9,0)

100

100

100

100

100

47.8

100

100

100

100

100

100

100

100

100

96.8

100

100

100

100

100

100

100

59.7

100

100

99.9

100

100

100

0.0

100

100

100

1.5

GE

P(8

,-2.

5,0)

100

100

100

100

100

48.0

100

100

100

100

100

100

100

100

100

97.8

100

100

100

100

100

100

100

70.6

100

100

100

100

100

100

0.0

100

100

100

1.5

GE

P(1

5,-2

.1,0

)10

010

010

010

010

048

.410

010

010

010

010

010

010

010

010

098

.610

010

010

010

010

010

010

080

.810

010

010

010

010

010

00.

010

010

010

0

1.5

GE

P(3

0,-1

.9,0

)10

010

010

010

010

048

.810

010

010

010

010

010

099

.910

010

098

.710

010

010

010

010

010

010

083

.210

010

010

010

010

010

00.

010

010

010

0

1.5

GE

P(6

0,-1

.8,0

)10

010

010

010

010

048

.510

010

010

010

010

010

099

.910

010

098

.610

010

010

010

010

010

010

083

.910

010

010

010

010

010

00.

010

010

010

0

1.84

GE

P(0

.6,-

8.4,

0)10

010

010

010

010

098

.997

.598

.499

.310

010

010

010

097

.210

094

.510

010

083

.510

010

010

010

00.

110

010

032

.510

099

.110

00.

099

.910

.610

0

1.84

GE

P(0

.8,-

6,0)

100

100

100

100

100

99.3

98.0

98.8

99.4

100

100

100

100

97.9

100

94.2

100

100

83.6

100

100

100

100

1.1

100

100

34.1

100

99.4

100

0.0

100

11.4

100

1.84

GE

P(1

,-5,

0)10

010

010

010

010

099

.698

.699

.299

.610

010

010

010

098

.410

092

.810

010

083

.910

010

010

010

07.

110

010

036

.910

099

.610

00.

010

012

.510

0

1.84

GE

P(1

.5,-

3.9,

0)10

010

010

010

010

099

.999

.499

.799

.810

010

010

010

099

.310

081

.510

010

082

.310

010

010

010

058

.910

010

044

.810

099

.910

00.

010

017

.310

0

1.84

GE

P(2

,-3.

2,0)

100

100

100

100

100

100

99.7

99.9

99.9

100

100

100

100

99.7

100

71.7

100

100

80.0

100

100

100

100

89.8

100

100

52.9

100

100

100

0.0

100

23.8

100

1.84

GE

P(3

,-2.

1,0)

99.9

100

100

100

100

100

99.9

100

99.7

100

100

99.9

99.6

99.9

100

65.4

100

100

76.7

100

100

100

100

99.0

100

100

66.9

100

100

100

0.0

100

39.0

100

1.84

GE

P(4

,-1.

4,0)

99.4

100

100

100

100

100

100

100

99.5

100

99.9

98.8

95.5

100

100

63.6

100

100

74.2

100

100

100

100

99.7

100

100

78.6

100

100

100

0.0

100

55.6

100

1.84

GE

P(6

,-0.

7,0)

97.4

100

100

100

100

100

100

100

98.8

100

98.7

90.3

75.8

100

100

60.8

100

99.8

71.0

100

100

100

100

99.9

99.9

100

91.8

100

100

100

0.0

100

80.5

100

1.84

GE

P(8

,-0.

4,0)

95.5

100

100

100

100

100

100

100

98.2

100

96.7

79.8

58.7

100

100

58.2

100

99.5

69.4

100

100

100

100

100

100

100

97.0

100

100

100

0.0

100

92.7

100

1.84

GE

P(1

5.2,

0,0)

91.7

99.9

100

100

99.9

100

100

100

97.0

100

90.3

59.2

35.7

100

100

50.8

99.9

98.7

66.5

100

100

100

100

100

100

100

99.9

100

100

99.9

0.0

99.9

99.6

99.9

2G

EP

(0.5

,-8.

9,0)

100

100

100

100

100

95.5

89.0

92.6

95.6

100

100

100

100

88.2

99.5

94.4

100

100

65.5

99.5

100

100

100

0.7

100

100

23.7

100

93.9

100

0.0

99.9

3.8

100

2G

EP

(0.8

,-5.

1,0)

100

100

100

100

100

96.5

89.9

93.5

95.8

100

100

100

100

89.0

99.3

93.0

100

100

63.0

99.4

100

100

100

6.8

100

100

24.6

100

95.3

100

0.0

99.9

4.0

100

2G

EP

(1,-

4.2,

0)10

010

099

.999

.910

097

.390

.894

.396

.010

010

010

010

089

.999

.188

.410

010

060

.799

.410

010

010

023

.310

010

026

.010

096

.510

00.

099

.94.

410

0

2G

EP

(1.5

,-3.

2,0)

100

100

99.4

99.1

100

98.6

92.6

95.9

95.3

100

100

100

100

91.7

98.6

68.0

100

100

53.4

99.5

100

100

99.9

76.2

100

100

32.0

100

98.4

100

0.0

100

5.9

100

2G

EP

(2,-

2.5,

0)99

.710

098

.798

.210

099

.393

.796

.793

.410

010

099

.999

.892

.898

.355

.110

010

047

.899

.599

.899

.599

.893

.499

.499

.937

.899

.899

.299

.90.

010

07.

910

0

2G

EP

(3,-

1.4,

0)94

.999

.598

.598

.199

.699

.795

.297

.887

.399

.898

.694

.788

.694

.397

.844

.199

.999

.440

.699

.599

.297

.699

.799

.096

.999

.549

.699

.199

.899

.50.

099

.914

.199

.5

2G

EP

(4,-

0.7,

0)86

.698

.498

.998

.898

.599

.895

.798

.281

.199

.593

.378

.863

.494

.997

.339

.999

.697

.336

.399

.699

.096

.499

.899

.695

.499

.460

.698

.799

.998

.90.

099

.622

.498

.4

2G

EP

(6,0

,0)

72.8

96.7

99.6

99.6

96.9

99.9

96.1

98.5

71.5

98.9

76.4

47.0

29.1

95.2

96.7

33.4

98.2

90.9

31.4

99.6

99.3

96.5

99.9

99.8

95.3

99.7

77.6

99.0

100

97.6

0.0

98.2

42.1

97.0

2G

EP

(8,0

.3,0

)64

.895

.899

.999

.996

.199

.996

.298

.565

.698

.263

.030

.716

.695

.396

.228

.996

.685

.529

.099

.699

.697

.210

099

.896

.299

.887

.799

.310

096

.70.

096

.559

.996

.5

2G

EP

(15,

0.7,

0)55

.195

.710

010

095

.899

.996

.198

.558

.397

.144

.015

.77.

895

.395

.522

.192

.876

.526

.699

.499

.998

.810

099

.898

.210

097

.999

.810

095

.00.

092

.988

.996

.2

2G

EP

(0,8

65,8

73)

94.3

98.4

92.5

92.6

98.5

93.6

96.6

96.8

99.2

99.1

94.1

82.0

67.5

96.4

97.9

24.8

99.6

98.7

79.6

98.3

97.6

98.3

84.1

99.1

98.4

96.5

49.3

97.7

98.6

97.8

88.4

99.6

99.8

98.8

2.25

GE

P(0

.5,-

7.2,

0)10

010

099

.699

.610

080

.863

.272

.177

.699

.710

010

010

061

.491

.693

.610

010

038

.493

.510

010

010

03.

410

010

016

.310

073

.799

.80.

299

.50.

810

0

2.25

GE

P(0

.7,-

4.7,

0)10

010

098

.598

.410

080

.260

.770

.474

.599

.610

010

010

058

.788

.691

.410

010

034

.191

.910

010

099

.99.

110

010

015

.910

074

.099

.60.

199

.50.

710

0

2.25

GE

P(0

.8,-

3.8,

0)10

010

096

.495

.710

080

.158

.368

.971

.399

.410

010

010

056

.285

.386

.110

010

030

.590

.610

010

099

.618

.999

.910

016

.010

074

.699

.20.

199

.50.

710

0

2.25

GE

P(1

,-3.

3,0)

100

100

92.6

91.4

100

80.3

56.6

67.8

67.3

99.1

100

100

100

54.3

81.9

76.2

100

100

26.8

89.0

99.8

99.6

98.7

32.9

99.6

99.8

16.0

99.8

76.0

98.5

0.0

99.6

0.7

100

2.25

GE

P(1

.5,-

2.4,

0)96

.898

.977

.875

.999

.081

.051

.264

.450

.296

.699

.199

.098

.548

.570

.743

.799

.798

.617

.584

.493

.488

.892

.067

.987

.394

.318

.292

.979

.893

.20.

099

.10.

899

.1

2.25

GE

P(2

,-1.

7,0)

81.6

91.8

69.7

68.7

92.2

81.6

47.5

61.8

35.8

92.2

92.0

87.8

82.6

44.5

62.6

29.6

98.2

91.8

12.6

80.7

82.3

71.4

86.2

80.5

68.5

85.4

20.6

80.7

82.1

85.5

0.0

97.4

1.0

91.8

2.25

GE

P(3

,-0.

5,0)

48.1

70.7

66.5

66.9

71.6

82.4

43.1

58.6

18.9

81.3

60.4

43.6

32.4

39.7

52.5

19.3

87.8

68.4

7.9

76.2

69.3

51.3

82.7

87.7

47.3

74.9

26.3

66.0

85.4

71.1

0.0

87.3

1.6

69.8

2.25

GE

P(3

.7,0

,0)

35.4

61.0

68.5

69.3

61.9

82.9

41.4

57.6

13.6

74.5

42.2

25.1

16.6

37.9

48.6

16.1

78.9

55.3

6.6

74.7

66.6

46.4

83.9

88.6

42.1

73.4

30.5

62.7

87.3

64.0

0.0

78.8

2.1

61.5

2.25

GE

P(4

,0.2

,0)

30.5

56.9

70.0

71.3

57.8

83.1

40.4

56.9

11.8

71.1

34.9

19.0

12.2

37.0

46.8

14.7

74.1

49.1

6.1

74.0

66.2

44.8

85.2

88.5

40.3

73.5

32.6

61.9

88.1

60.6

0.0

74.1

2.4

58.5

2.25

GE

P(8

,1.2

,0)

13.7

43.1

86.7

88.4

43.7

83.4

36.4

53.6

4.6

50.4

8.2

4.4

5.0

32.9

37.1

7.8

43.3

22.5

3.8

70.0

76.2

48.6

95.7

85.3

42.8

83.9

56.3

70.5

92.8

42.6

0.0

44.0

9.7

53.4

2.25

GE

P(0

,785

,782

)94

.598

.592

.792

.898

.593

.996

.796

.999

.199

.194

.282

.067

.596

.697

.925

.099

.698

.779

.898

.397

.698

.384

.299

.198

.496

.649

.697

.898

.597

.988

.299

.699

.898

.9

2.5

GE

P(0

.5,-

6.1,

0)10

010

094

.093

.810

058

.036

.646

.848

.996

.110

010

010

034

.669

.892

.299

.910

022

.675

.710

010

099

.25.

510

010

013

.910

046

.896

.31.

098

.20.

210

0

2.5

GE

P(0

.7,-

3.9,

0)10

010

084

.483

.810

054

.030

.941

.740

.693

.310

010

010

029

.060

.386

.999

.899

.918

.068

.899

.799

.696

.510

.399

.599

.712

.699

.744

.992

.10.

797

.90.

110

0

2.5

GE

P(0

.8,-

3.2,

0)99

.910

072

.571

.510

051

.126

.737

.733

.189

.999

.810

010

024

.851

.975

.599

.599

.614

.662

.797

.896

.890

.417

.296

.497

.712

.097

.843

.986

.60.

597

.40.

199

.9

2.5

GE

P(1

,-2.

7,0)

98.8

99.3

58.7

58.0

99.3

47.7

22.4

33.3

24.8

85.1

99.0

99.7

99.8

20.7

42.8

58.6

98.8

97.7

11.3

56.1

90.4

86.8

80.5

25.4

85.6

90.5

11.2

90.2

42.6

78.6

0.3

96.4

0.1

99.6

2.5

GE

P(1

.5,-

1.8,

0)66

.273

.432

.834

.074

.140

.213

.924

.08.

864

.378

.278

.576

.012

.423

.924

.287

.371

.25.

939

.851

.040

.052

.740

.637

.454

.410

.349

.340

.351

.90.

184

.80.

177

.2

2.5

GE

P(2

,-1,

0)34

.743

.724

.627

.244

.736

.010

.019

.13.

747

.245

.940

.134

.58.

815

.614

.166

.042

.93.

931

.231

.620

.342

.044

.118

.336

.210

.729

.439

.135

.40.

064

.90.

141

.9

2.5

GE

P(2

.8,0

,0)

14.1

22.2

21.4

24.5

22.8

32.0

6.7

14.7

1.2

29.6

17.3

11.8

9.1

5.7

9.5

8.9

38.3

20.5

2.5

23.9

21.3

10.6

37.7

43.2

9.0

26.4

12.0

18.8

38.4

22.0

0.0

38.4

0.1

20.1

2.5

GE

P(3

,0.2

,0)

11.6

18.8

21.1

24.4

19.2

31.0

6.1

13.7

1.0

26.2

13.3

8.7

6.8

5.1

8.4

8.0

32.9

16.9

2.2

22.6

19.8

9.4

37.6

42.4

7.9

25.2

12.3

17.3

38.4

19.3

0.0

33.1

0.2

17.7

2.5

GE

P(4

,0.9

,0)

6.5

12.1

22.9

26.4

12.4

27.9

4.3

11.0

0.3

16.0

5.3

4.7

5.1

3.6

5.5

5.8

16.9

9.0

1.6

18.1

18.1

7.1

41.2

37.8

5.7

24.1

14.5

15.0

38.0

12.6

0.0

17.2

0.3

15.4

2.5

GE

P(8

,1.9

,0)

6.4

12.1

39.7

43.4

12.3

24.1

2.6

7.4

0.1

6.0

7.5

14.7

17.7

2.1

2.9

4.1

4.8

4.5

0.9

12.9

28.8

9.5

66.5

27.9

7.4

38.2

24.3

22.7

41.8

6.5

0.0

5.0

1.8

28.7

2.5

GE

P(0

,730

,739

)94

.498

.492

.692

.698

.593

.796

.796

.999

.199

.194

.382

.267

.796

.697

.924

.899

.698

.679

.698

.397

.698

.384

.099

.198

.396

.549

.397

.798

.697

.888

.299

.699

.898

.8

2.75

GE

P(0

.5,-

5.3,

0)10

010

077

.878

.210

038

.321

.729

.425

.782

.199

.910

010

020

.447

.689

.998

.899

.716

.253

.299

.699

.593

.16.

899

.499

.513

.599

.626

.481

.22.

894

.40.

110

0

2.75

GE

P(0

.6,-

4.1,

0)10

010

069

.369

.410

035

.119

.226

.521

.577

.999

.810

010

018

.041

.886

.598

.399

.414

.448

.198

.598

.187

.77.

897

.998

.212

.998

.524

.675

.42.

693

.50.

099

.9

2.75

GE

P(0

.7,-

3.3,

0)99

.999

.958

.558

.699

.931

.916

.623

.417

.072

.299

.310

010

015

.435

.579

.797

.398

.412

.542

.095

.093

.878

.98.

893

.394

.412

.195

.222

.967

.62.

491

.80.

099

.9

2.75

GE

P(0

.8,-

2.8,

0)98

.999

.147

.147

.199

.228

.514

.020

.312

.965

.397

.699

.699

.812

.929

.368

.395

.295

.310

.836

.186

.383

.767

.010

.882

.685

.611

.186

.521

.158

.52.

189

.40.

099

.6

2.75

GE

P(1

,-2.

3,0)

83.8

84.9

28.8

29.3

85.5

22.6

9.7

15.1

6.5

49.7

85.7

92.8

94.3

8.9

19.1

40.4

84.5

75.8

8.0

25.2

56.3

51.1

42.8

14.7

49.5

56.5

9.5

56.1

18.4

39.8

1.6

79.2

0.0

94.4

2.75

GE

P(1

.5,-

1.4,

0)26

.927

.611

.913

.428

.313

.64.

68.

12.

022

.933

.435

.334

.14.

27.

512

.541

.724

.94.

711

.716

.612

.318

.916

.211

.418

.37.

316

.013

.616

.30.

940

.60.

131

.4

2.75

GE

P(2

,-0.

5,0)

9.9

10.8

7.3

9.4

11.1

9.2

2.6

5.2

1.2

11.8

11.5

10.4

9.3

2.4

3.6

7.2

18.2

10.2

3.3

6.4

8.1

4.9

13.4

13.5

4.4

9.7

6.2

7.3

10.9

8.9

0.6

18.2

0.3

8.8

2.75

GE

P(2

.3,0

,0)

6.4

7.3

6.2

8.4

7.5

7.6

2.0

4.2

1.0

8.5

6.8

6.1

5.6

1.7

2.4

5.8

11.5

7.1

2.8

4.8

6.4

3.5

12.5

12.0

3.1

7.9

5.9

5.5

9.9

6.9

0.5

11.6

0.6

5.6

2.75

GE

P(3

,0.8

,0)

4.7

5.0

5.3

7.5

5.0

5.4

1.1

2.7

0.9

4.7

4.6

5.6

6.0

1.0

1.2

4.6

5.0

4.7

2.1

2.9

4.9

2.2

12.6

8.9

1.9

6.5

5.6

4.1

8.2

4.7

0.4

5.1

2.0

5.6

2.75

GE

P(4

,1.6

,0)

6.8

6.4

5.7

7.7

6.5

3.9

0.7

1.9

0.7

3.3

8.7

12.5

13.8

0.6

0.7

4.3

3.8

4.4

1.7

1.8

5.6

2.2

16.0

6.3

1.9

7.5

5.7

4.4

7.2

4.0

0.4

3.9

5.0

11.7

2.75

GE

P(0

,690

,706

)94

.498

.392

.692

.798

.493

.796

.796

.999

.199

.194

.182

.267

.596

.597

.924

.899

.698

.779

.698

.397

.698

.284

.199

.098

.396

.549

.597

.798

.597

.888

.299

.699

.898

.8

3G

EP

(0.5

,-4.

6,0)

100

100

59.9

60.8

100

26.9

18.4

22.4

14.5

59.3

99.1

100

100

17.6

34.8

86.6

93.9

98.0

15.7

36.6

96.6

96.1

76.9

9.0

95.7

95.9

14.8

96.8

15.2

56.7

6.6

85.3

0.0

99.9

3G

EP

(0.6

,-3.

6,0)

99.7

99.8

50.0

50.6

99.8

24.2

16.6

20.0

11.8

52.2

97.5

99.8

100

15.8

30.2

81.0

91.0

95.7

14.2

31.7

90.9

89.9

65.6

8.8

89.2

89.7

14.1

91.3

14.0

47.8

6.5

82.0

0.0

99.8

3G

EP

(0.7

,-2.

9,0)

97.9

98.0

39.9

39.9

98.2

21.3

15.0

17.8

9.5

43.6

93.3

98.9

99.6

14.3

25.7

70.2

85.4

89.4

12.8

26.7

78.8

77.2

51.0

8.9

76.3

76.9

13.1

79.3

12.5

38.0

6.3

76.4

0.0

99.4

3G

EP

(0.8

,-2.

5,0)

89.6

89.5

30.7

30.4

90.1

18.6

13.3

15.7

7.5

35.4

84.3

94.9

97.4

12.7

21.2

54.6

76.1

75.4

11.8

22.0

60.8

59.0

37.0

9.2

58.0

58.9

11.9

61.4

11.1

29.0

6.2

68.1

0.0

97.8

3G

EP

(1,-

1.9,

0)51

.147

.918

.517

.548

.813

.610

.711

.95.

620

.251

.565

.770

.810

.214

.925

.747

.436

.59.

414

.728

.627

.817

.78.

727

.427

.610

.129

.18.

815

.25.

743

.60.

172

.9

3G

EP

(1.5

,-1,

0)9.

48.

68.

37.

78.

77.

67.

07.

25.

07.

29.

811

.111

.36.

77.

97.

710

.37.

36.

57.

48.

48.

46.

56.

48.

48.

17.

08.

46.

06.

25.

110

.21.

411

.6

3G

EP

(2,0

,0)

5.0

5.0

5.2

5.2

5.0

5.1

5.2

5.1

5.2

5.1

4.9

5.0

4.8

5.0

5.2

5.0

4.9

5.1

5.2

5.0

5.2

5.1

5.1

4.9

5.2

5.0

5.1

5.1

4.9

5.0

5.1

5.0

4.9

4.8

36

Page 37: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le6:

Pow

erag

ainst

GE

Pdis

trib

uti

onfo

r5%

leve

lte

sts.M

=10

0,00

0.n

=20

0(c

onti

nuin

g)

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

3G

EP

(3,1

.4,0

)9.

99.

03.

03.

69.

22.

53.

02.

75.

67.

414

.115

.815

.32.

72.

94.

59.

97.

23.

62.

95.

24.

36.

54.

44.

25.

03.

14.

74.

05.

75.

110

.116

.611

.5

3G

EP

(0,6

65,6

61)

94.5

98.4

92.8

92.8

98.5

93.9

96.7

96.9

99.1

99.1

94.3

82.0

67.7

96.6

97.9

24.8

99.6

98.7

79.7

98.3

97.7

98.3

84.1

99.1

98.4

96.6

49.5

97.8

98.6

97.9

88.2

99.6

99.8

98.9

3.5

GE

P(0

.5,-

3.8,

0)97

.597

.945

.444

.698

.027

.328

.328

.116

.623

.484

.597

.899

.427

.734

.578

.263

.782

.620

.932

.878

.779

.441

.419

.379

.175

.219

.479

.717

.022

.520

.452

.60.

599

.5

3.5

GE

P(0

.6,-

2.8,

0)90

.090

.339

.037

.090

.826

.128

.027

.516

.617

.770

.992

.397

.227

.532

.567

.651

.068

.020

.030

.964

.166

.129

.018

.865

.959

.918

.565

.517

.117

.021

.442

.20.

698

.6

3.5

GE

P(0

.7,-

2.3,

0)70

.069

.033

.530

.070

.025

.428

.127

.117

.212

.950

.877

.087

.127

.530

.749

.434

.844

.519

.229

.747

.951

.318

.918

.451

.543

.317

.549

.417

.712

.922

.929

.61.

095

.0

3.5

GE

P(0

.8,-

1.9,

0)42

.340

.228

.623

.941

.224

.227

.726

.518

.310

.129

.550

.662

.127

.229

.030

.319

.823

.018

.228

.734

.539

.111

.918

.239

.630

.216

.436

.218

.810

.624

.417

.92.

280

.0

3.5

GE

P(1

,-1.

4,0)

12.6

14.5

22.7

17.4

14.7

23.0

27.5

26.1

22.0

9.7

8.4

13.1

16.5

27.0

27.0

11.0

8.4

8.6

16.5

28.1

22.8

27.6

6.5

19.0

28.3

19.1

14.2

24.2

21.2

10.4

27.7

8.7

8.1

37.3

3.5

GE

P(1

.6,0

,0)

13.2

18.3

17.3

12.6

18.6

21.0

27.0

25.6

32.1

23.4

14.8

11.1

9.0

26.4

26.2

5.4

26.7

17.3

14.5

29.2

20.5

24.9

5.8

26.9

25.6

16.6

10.3

21.4

27.8

17.7

35.8

27.2

37.3

24.3

3.5

GE

P(2

,0.8

,0)

21.4

26.9

16.1

12.1

27.4

20.2

27.0

25.5

36.7

32.4

27.3

22.1

18.1

26.2

27.2

6.1

39.1

26.0

14.0

30.3

23.4

27.5

7.4

31.7

28.2

19.0

8.5

24.1

30.9

23.1

39.7

39.8

51.5

31.1

3.5

GE

P(3

,2.3

,0)

42.0

48.9

14.6

12.8

49.7

17.5

25.2

23.9

45.7

51.7

57.0

51.1

43.6

24.4

30.3

9.9

62.7

46.2

13.4

33.0

33.2

35.8

14.5

42.2

36.5

28.1

5.0

33.2

36.2

36.2

47.2

63.5

74.3

52.7

3.5

GE

P(0

,620

,609

)94

.598

.492

.992

.898

.593

.996

.897

.099

.299

.194

.382

.367

.896

.797

.925

.099

.598

.779

.998

.397

.798

.384

.299

.198

.496

.649

.497

.898

.697

.988

.199

.699

.898

.9

4G

EP

(0.4

,-4.

5,0)

94.2

95.6

52.0

50.0

96.0

38.8

42.8

41.7

28.4

15.1

67.5

93.5

98.3

42.3

45.5

77.8

39.9

72.0

28.9

45.1

76.1

78.7

35.6

34.5

78.7

71.6

25.8

77.4

31.0

19.2

36.1

32.2

2.5

99.3

4G

EP

(0.5

,-3.

2,0)

83.3

86.0

48.8

45.4

86.7

39.7

44.5

43.2

30.6

13.5

50.3

83.1

93.2

43.9

45.7

68.1

28.4

57.0

28.9

46.0

66.0

70.0

27.6

35.7

70.3

60.6

25.1

67.7

33.5

18.0

39.2

23.8

3.6

98.4

4G

EP

(0.6

,-2.

3,0)

62.4

66.1

45.5

40.7

67.3

40.6

45.9

44.4

33.4

13.3

30.9

61.9

78.4

45.3

45.7

52.8

18.3

38.0

28.7

47.2

55.5

61.1

20.9

37.2

61.6

50.0

24.3

57.4

36.6

17.8

42.8

16.8

6.0

95.6

4G

EP

(0.7

,-1.

9,0)

35.9

42.1

42.4

36.4

43.0

41.3

47.4

46.0

37.3

15.4

15.0

33.2

47.4

46.8

46.3

32.2

12.8

21.4

28.2

48.6

47.0

53.3

15.9

38.9

54.1

41.4

23.7

48.9

40.1

19.3

46.7

13.5

11.2

83.8

4G

EP

(0.8

,-1.

5,0)

19.8

29.3

40.3

33.3

29.7

42.4

49.0

47.7

42.2

20.4

7.4

13.0

19.2

48.4

47.3

16.8

15.9

15.9

28.0

50.6

42.1

48.8

13.4

41.8

49.8

36.5

22.9

44.0

44.1

22.6

51.1

17.1

20.9

63.2

4G

EP

(1,-

1.1,

0)15

.628

.238

.731

.428

.443

.451

.249

.950

.130

.98.

45.

96.

050

.549

.38.

629

.921

.828

.153

.640

.947

.712

.647

.648

.735

.121

.742

.750

.029

.157

.330

.739

.548

.3

4G

EP

(1.4

,0,0

)34

.748

.539

.233

.549

.146

.155

.654

.665

.257

.336

.624

.718

.454

.856

.07.

864

.048

.829

.260

.549

.255

.417

.362

.056

.442

.518

.850

.561

.246

.868

.864

.774

.256

.3

4G

EP

(2,1

.5,0

)55

.267

.941

.938

.268

.648

.059

.058

.475

.174

.364

.151

.641

.558

.163

.011

.481

.768

.631

.266

.860

.465

.426

.172

.766

.353

.716

.361

.369

.061

.676

.982

.388

.672

.1

4G

EP

(3,3

.1,0

)74

.384

.647

.046

.285

.050

.363

.263

.184

.387

.784

.975

.464

.562

.371

.618

.492

.784

.234

.474

.573

.677

.040

.783

.477

.667

.913

.274

.077

.276

.485

.093

.396

.286

.4

4G

EP

(0,5

80,5

84)

94.4

98.4

92.7

92.8

98.5

93.8

96.8

96.9

99.1

99.1

94.4

82.2

67.7

96.6

98.0

25.1

99.6

98.7

80.1

98.3

97.7

98.3

84.4

99.1

98.4

96.7

49.9

97.8

98.6

97.9

88.0

99.6

99.8

98.9

4.5

GE

P(0

.4,-

4,0)

81.6

87.0

58.5

55.1

87.8

51.8

57.4

56.0

43.6

17.5

38.6

76.7

91.0

56.9

57.0

70.9

21.3

56.4

37.2

58.8

72.4

76.6

32.8

50.7

77.0

67.2

30.9

74.0

48.5

25.5

52.6

19.9

9.1

98.7

4.5

GE

P(0

.5,-

2.7,

0)64

.372

.956

.952

.274

.053

.559

.658

.447

.620

.522

.654

.874

.959

.158

.358

.617

.242

.837

.661

.166

.271

.628

.353

.672

.160

.730

.767

.953

.028

.057

.018

.213

.997

.1

4.5

GE

P(0

.6,-

2,0)

43.2

55.8

55.6

49.8

56.9

55.7

62.3

61.2

53.1

26.4

11.3

28.6

45.7

61.7

60.5

41.0

19.2

31.5

38.2

64.0

61.1

67.3

24.7

57.0

68.0

55.3

30.8

63.0

57.7

32.2

62.1

21.4

22.8

91.5

4.5

GE

P(0

.7,-

1.5,

0)28

.745

.755

.048

.646

.358

.365

.564

.560

.236

.57.

210

.416

.764

.963

.521

.530

.428

.539

.067

.658

.665

.023

.261

.865

.952

.630

.060

.463

.539

.267

.632

.238

.677

.9

4.5

GE

P(0

.8,-

1.2,

0)27

.646

.555

.348

.746

.960

.368

.267

.567

.549

.111

.46.

26.

367

.666

.612

.247

.437

.740

.271

.059

.365

.823

.767

.266

.653

.229

.561

.068

.747

.472

.748

.657

.568

.0

4.5

GE

P(1

,-0.

8,0)

37.9

56.9

56.9

51.1

57.4

63.2

71.5

71.0

76.2

65.1

29.0

14.9

9.5

70.9

71.2

8.9

68.8

55.4

42.2

75.3

63.9

69.8

27.5

74.8

70.7

57.8

28.8

65.3

75.1

58.9

78.4

69.5

77.5

68.6

4.5

GE

P(1

.3,0

,0)

58.7

74.9

61.1

57.3

75.5

67.2

76.3

76.1

85.6

81.8

60.2

42.3

31.0

75.6

78.2

11.2

86.7

76.2

45.5

81.5

73.9

78.4

36.7

84.2

79.2

68.0

28.1

74.8

82.7

73.2

85.2

87.1

91.8

80.3

4.5

GE

P(2

,2.1

,0)

79.9

90.0

68.2

67.1

90.4

72.1

81.5

81.7

92.8

93.1

85.5

72.9

60.2

80.9

85.9

18.5

96.0

90.7

51.1

88.0

85.2

88.0

52.9

92.2

88.4

80.8

26.2

85.7

89.4

86.4

91.4

96.3

98.0

91.9

4.5

GE

P(3

,3.9

,0)

90.0

96.0

75.3

76.0

96.2

76.5

86.0

86.4

96.4

97.3

94.4

86.2

75.0

85.5

91.4

26.0

98.7

96.3

56.0

92.5

92.1

93.7

67.0

96.3

94.0

89.2

24.3

92.4

93.6

93.3

95.3

98.8

99.4

96.7

4.5

GE

P(0

,550

,557

)94

.498

.392

.792

.798

.493

.996

.896

.999

.199

.194

.382

.267

.796

.697

.925

.099

.698

.779

.798

.397

.698

.384

.299

.198

.496

.549

.897

.798

.697

.888

.299

.699

.898

.8

5G

EP

(0.5

,-2.

4,0)

53.1

68.8

65.5

60.8

69.7

65.6

72.0

71.2

63.5

36.3

10.6

29.4

49.6

71.5

70.1

50.5

26.9

42.9

46.2

73.8

71.7

76.9

34.9

68.9

77.6

66.3

36.3

73.3

69.2

43.4

69.9

30.1

32.4

95.9

5G

EP

(0.6

,-1.

7,0)

41.5

61.2

66.0

60.8

61.9

69.0

75.4

74.8

70.8

48.4

7.9

11.6

20.4

75.0

73.8

32.0

40.4

41.5

48.3

77.4

70.8

76.1

34.1

73.8

76.8

65.3

36.6

72.3

74.7

51.9

75.1

43.0

49.0

89.2

5G

EP

(0.7

,-1.

3,0)

40.2

61.8

67.0

61.8

62.2

71.9

78.7

78.3

78.3

62.9

14.6

6.6

6.7

78.2

77.7

16.8

60.5

51.5

50.5

81.1

71.9

77.3

35.5

79.2

78.1

66.4

36.3

73.4

80.0

61.7

80.0

61.8

69.6

80.9

5G

EP

(0.8

,-1,

0)49

.169

.469

.264

.869

.974

.881

.681

.584

.975

.832

.214

.18.

281

.281

.711

.577

.666

.553

.284

.875

.980

.739

.984

.781

.470

.636

.377

.184

.871

.384

.078

.284

.780

.3

5G

EP

(1,-

0.6,

0)65

.281

.573

.270

.282

.078

.285

.285

.290

.987

.159

.637

.124

.684

.786

.412

.190

.382

.356

.788

.882

.586

.248

.590

.386

.777

.936

.483

.389

.481

.587

.590

.794

.286

.6

5G

EP

(1.1

,0,0

)77

.489

.576

.875

.289

.881

.087

.787

.994

.393

.177

.759

.044

.587

.389

.915

.595

.790

.460

.091

.787

.890

.557

.093

.890

.984

.036

.288

.492

.588

.190

.195

.997

.792

.1

5G

EP

(2,2

.7,0

)91

.597

.084

.784

.797

.286

.392

.292

.497

.998

.293

.983

.771

.191

.995

.024

.799

.197

.467

.195

.894

.896

.173

.997

.896

.392

.736

.095

.196

.495

.593

.799

.299

.697

.7

5G

EP

(3,4

.6,0

)95

.898

.989

.490

.098

.989

.794

.795

.099

.199

.397

.590

.880

.394

.497

.331

.299

.799

.071

.997

.797

.598

.283

.199

.098

.396

.335

.597

.698

.197

.995

.999

.899

.999

.0

5G

EP

(0,5

00,6

00)

94.5

98.5

92.7

92.8

98.5

93.8

96.8

96.9

99.2

99.1

94.4

82.4

68.0

96.6

97.9

24.9

99.6

98.7

79.9

98.3

97.7

98.3

84.5

99.1

98.4

96.6

49.7

97.8

98.6

97.9

88.2

99.6

99.8

98.9

6G

EP

(0.4

,-2.

9,0)

62.7

80.7

78.5

75.3

81.5

79.8

84.9

84.6

79.9

59.0

8.8

19.4

38.0

84.5

83.4

54.0

48.8

59.5

59.8

86.5

83.8

87.5

51.1

84.5

87.9

79.7

45.0

84.9

84.6

64.8

78.8

52.5

56.5

97.5

6G

EP

(0.5

,-1.

9,0)

59.7

79.5

80.6

77.5

80.2

83.3

88.0

87.8

85.9

71.8

11.8

8.4

15.3

87.7

87.1

39.4

65.0

64.5

63.1

89.6

85.2

88.5

53.6

88.5

89.0

81.3

46.2

86.1

88.7

73.9

82.3

67.4

72.8

95.5

6G

EP

(0.6

,-1.

3,0)

64.1

82.6

82.8

80.2

82.9

86.2

90.8

90.7

91.4

84.1

28.5

9.1

6.3

90.5

90.6

23.7

82.6

75.8

66.7

92.6

87.5

90.5

58.0

92.4

90.9

84.0

46.7

88.3

92.4

82.8

85.1

83.5

87.9

92.5

6G

EP

(0.8

,-0.

8,0)

81.4

92.7

87.5

86.2

93.0

90.3

94.2

94.3

96.9

95.5

71.5

42.9

26.0

94.0

95.1

16.7

96.7

92.8

73.1

96.1

93.2

95.0

70.1

96.9

95.3

90.9

47.9

93.7

96.4

92.9

87.6

96.9

98.2

95.1

6G

EP

(1,0

,0)

94.4

98.4

92.6

92.6

98.5

93.7

96.7

96.8

99.1

99.1

94.1

81.9

67.6

96.6

97.9

24.7

99.6

98.7

79.4

98.3

97.7

98.3

83.8

99.1

98.4

96.6

49.1

97.8

98.5

97.9

88.0

99.6

99.8

98.9

6G

EP

(2,3

.9,0

)98

.199

.696

.196

.399

.696

.298

.398

.499

.799

.898

.391

.981

.298

.299

.233

.499

.999

.785

.199

.399

.299

.491

.899

.799

.598

.751

.699

.399

.499

.388

.899

.910

099

.7

6G

EP

(3,6

.2,0

)98

.799

.897

.397

.699

.897

.398

.998

.999

.899

.998

.893

.983

.898

.899

.436

.899

.999

.887

.899

.699

.599

.694

.399

.899

.799

.253

.199

.599

.699

.689

.110

010

099

.8

6G

EP

(0,1

40,1

29.1

)94

.898

.693

.393

.398

.694

.397

.097

.299

.399

.294

.582

.768

.496

.998

.125

.299

.698

.880

.898

.597

.998

.585

.399

.298

.596

.950

.598

.098

.798

.187

.699

.799

.899

.0

10G

EP

(0.4

,-1.

7,0)

95.0

98.7

97.7

97.5

98.7

98.2

99.1

99.1

99.4

99.0

72.8

27.2

10.4

99.0

99.2

42.4

98.8

98.0

91.2

99.4

98.9

99.2

92.0

99.6

99.3

98.4

67.0

99.0

99.5

98.7

74.0

98.9

99.4

99.5

10G

EP

(0.5

,-1,

0)98

.599

.798

.998

.999

.799

.199

.699

.799

.999

.994

.267

.739

.499

.699

.836

.999

.999

.794

.799

.899

.799

.896

.499

.999

.899

.569

.999

.799

.999

.769

.499

.999

.999

.8

10G

EP

(0.7

,0,0

)10

010

099

.999

.910

099

.899

.910

010

010

010

099

.497

.299

.910

055

.710

010

098

.410

010

010

099

.810

010

010

074

.610

010

010

051

.310

010

010

0

10G

EP

(1.4

,5.7

,0)

99.8

100

99.5

99.6

100

99.6

99.8

99.8

100

100

99.5

95.7

86.5

99.8

99.9

41.6

100

100

96.8

100

99.9

100

98.8

100

100

99.9

71.9

100

100

100

61.3

100

100

100

10G

EP

(0,6

,12.

2)99

.399

.999

.199

.199

.999

.199

.699

.699

.999

.999

.093

.181

.999

.699

.836

.410

099

.995

.099

.899

.899

.997

.699

.999

.999

.768

.499

.899

.999

.867

.210

010

099

.9

10G

EP

(0,1

7,-1

2.6)

99.5

99.9

99.1

99.2

99.9

99.2

99.7

99.7

100

100

99.0

93.4

82.2

99.7

99.9

37.1

100

99.9

95.1

99.9

99.8

99.9

97.8

100

99.9

99.7

69.0

99.8

99.9

99.9

66.5

100

100

99.9

20G

EP

(0.3

,-1.

3,0)

100

100

100

100

100

100

100

100

100

100

99.8

94.7

75.8

100

100

60.3

100

100

99.6

100

100

100

99.9

100

100

100

83.6

100

100

100

31.3

100

100

100

20G

EP

(0.5

,0,0

)10

010

010

010

010

010

010

010

010

010

010

010

010

010

010

084

.210

010

010

010

010

010

010

010

010

010

089

.210

010

010

06.

810

010

010

0

20G

EP

(0.8

,2.3

,0)

100

100

100

100

100

100

100

100

100

100

100

99.8

98.4

100

100

67.7

100

100

99.9

100

100

100

100

100

100

100

86.9

100

100

100

17.0

100

100

100

20G

EP

(0,1

4,-1

7.6)

100

100

99.9

100

100

99.9

100

100

100

100

99.9

98.2

91.9

100

100

50.2

100

100

99.2

100

100

100

99.8

100

100

100

81.2

100

100

100

36.7

100

100

100

20G

EP

(0,6

,2.6

)99

.910

099

.899

.910

099

.899

.910

010

010

099

.897

.189

.399

.910

045

.710

010

098

.510

010

010

099

.510

010

010

077

.710

010

010

045

.310

010

010

0

50G

EP

(0.3

,-0.

7,0)

100

100

100

100

100

100

100

100

100

100

100

100

99.9

100

100

89.0

100

100

100

100

100

100

100

100

100

100

93.4

100

100

100

1.4

100

100

100

50G

EP

(0.4

,0,0

)10

010

010

010

010

010

010

010

010

010

010

010

010

010

010

096

.710

010

010

010

010

010

010

010

010

010

095

.210

010

010

00.

110

010

010

0

50G

EP

(0.6

,1.2

,0)

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

93.1

100

100

100

100

100

100

100

100

100

100

95.1

100

100

100

0.3

100

100

100

50G

EP

(0,5

.1,2

.1)

100

100

99.9

100

100

99.9

100

100

100

100

99.9

98.4

92.3

100

100

51.1

100

100

99.3

100

100

100

99.8

100

100

100

81.7

100

100

100

33.3

100

100

100

50G

EP

(0,8

,-7.

2)10

010

010

010

010

010

010

010

010

010

010

099

.495

.910

010

059

.510

010

099

.810

010

010

010

010

010

010

086

.210

010

010

020

.010

010

010

0

50G

EP

(0,1

0,-1

3.8)

100

100

100

100

100

100

100

100

100

100

100

99.8

98.2

100

100

68.5

100

100

99.9

100

100

100

100

100

100

100

89.3

100

100

100

10.9

100

100

100

50G

EP

(0,1

0.8,

-17.

3)10

010

010

010

010

010

010

010

010

010

010

010

099

.310

010

077

.110

010

010

010

010

010

010

010

010

010

091

.610

010

010

05.

510

010

010

0

Mea

n

200

76.0

81.1

73.3

72.5

81.3

63.4

68.0

70.1

66.7

74.2

72.2

69.7

66.8

67.4

71.9

48.9

78.4

77.0

52.6

74.9

79.6

79.2

70.2

62.8

79.0

78.7

44.5

79.6

72.8

72.5

29.8

77.8

51.2

85.6

Ran

k

200

123

1517

.52

2824

2227

1419

2326

2520

329

1130

134.

56

2129

78

334.

516

17.5

3410

311

Dev

.to

the

Bes

t-

Mea

n

200

13.2

8.1

15.9

16.7

7.9

25.8

21.2

19.1

22.5

15.0

17.0

19.4

22.4

21.7

17.3

40.3

10.7

12.2

36.5

14.2

9.6

10.0

18.9

26.4

10.2

10.5

44.7

9.5

16.4

16.7

59.4

11.4

37.9

3.6

Ran

k

200

123

1517

.52

2824

2227

1419

2326

2520

329

1130

135

621

297

833

416

17.5

3410

311

Dev

.to

the

Bes

t-

Max

200

82.0

54.4

67.1

67.4

54.2

100

85.8

82.1

91.1

84.9

88.7

91.3

92.2

86.9

76.6

87.9

79.9

73.2

91.9

75.8

47.1

57.0

76.1

99.9

59.1

51.7

88.7

45.6

87.1

82.1

100

78.9

100

42.3

Ran

k

200

176

910

533

2118

.527

2025

.528

3022

1424

1611

2912

37

1331

84

25.5

223

18.5

3315

331

37

Page 38: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le7:

Pow

erag

ainst

sym

met

ric

dis

trib

uti

onfo

r5%

leve

lte

sts.M

=1,

000,

000.n

=20

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

3N

(0,1

)5.

05.

05.

15.

05.

05.

04.

95.

15.

05.

05.

05.

05.

04.

95.

04.

95.

15.

05.

05.

05.

05.

05.

05.

15.

05.

05.

05.

05.

15.

05.

15.

05.

15.

1

1.5

Bet

a(0.

5,0.

5)31

.861

.577

.766

.561

.248

.00.

647

.80.

470

.327

.611

.56.

50.

627

.545

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38

Page 39: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le8:

Pow

erag

ainst

sym

met

ric

dis

trib

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9

39

Page 40: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

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Pow

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9.1

4.9

12.9

59.7

8.6

11.0

9.4

Ran

k

100

2616

813

156

2418

3011

2729

3225

19.5

334.

523

313

4.5

1712

2219

.52

289

121

347

1410

Dev

.to

the

Bes

t-

Max

100

82.3

52.8

59.9

65.2

52.3

76.1

86.5

75.9

99.7

54.2

85.6

91.5

92.3

86.9

72.6

88.2

45.4

70.5

94.8

73.8

38.6

57.3

44.9

98.5

61.1

37.1

75.7

43.4

73.3

61.4

100

45.4

73.8

54.8

Ran

k

100

248

1215

723

2622

339

2529

3027

1728

5.5

1631

19.5

211

432

131

213

1814

345.

519

.510

40

Page 41: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

Tab

le10

:P

ower

agai

nst

sym

met

ric

dis

trib

uti

onfo

r5%

leve

lte

sts.M

=1,

000,

000.n

=20

0

β2

Alt

ern

ativ

eK−S

AD∗

ZC

ZA

PS

K2

JB

DH

RJB

TLmom

T(1

)Lmom

T(2

)Lmom

T(3

)Lmom

BM

3−

4BM

3−

6TMC−LR

Tw

TMC−LR−Tw

TS,5

TK,5

WWSFWRG

Da

rCS

QBCMR

β2 3

TEP

I nRsJ

SRn

3N

(0,1

)5.

05.

05.

05.

05.

05.

05.

05.

05.

05.

05.

05.

05.

05.

05.

04.

95.

04.

95.

05.

05.

05.

15.

05.

05.

05.

05.

05.

05.

04.

95.

05.

05.

05.

0

1.5

Bet

a(0.

5,0.

5)10

010

010

010

010

049

.210

010

010

010

010

099

.795

.210

010

099

.710

010

099

.910

010

010

010

078

.710

010

010

010

010

010

00.

010

010

010

0

1.51

LoC

onN

(0.5

,5)

100

100

100

100

100

54.7

100

100

100

100

100

100

100

100

100

91.3

100

100

100

100

100

100

100

6.5

100

100

89.7

100

100

100

0.0

100

92.4

100

1.63

JS

B(0

,0.5

)10

010

010

010

010

097

.710

010

010

010

099

.995

.278

.810

010

093

.410

010

098

.010

010

010

010

099

.610

010

010

010

010

010

00.

010

010

010

0

1.72

LoC

onN

(0.5

,4)

100

100

100

100

100

99.8

100

100

100

100

100

100

100

100

100

77.5

100

100

94.6

100

100

100

100

89.3

100

100

71.8

100

100

100

0.0

100

55.1

100

1.75

Tu

key(1

.5)

97.9

100

100

100

100

100

100

100

99.6

100

96.3

71.3

45.1

100

100

67.4

100

99.8

85.1

100

100

100

100

100

100

100

100

100

100

100

0.0

100

100

100

1.8

Tu

key(2

)94

.610

010

010

010

010

010

010

098

.610

091

.960

.536

.110

010

055

.299

.999

.275

.610

010

010

010

010

010

010

010

010

010

010

00.

099

.910

010

0

1.87

JS

B(0

,0.7

)88

.099

.910

010

099

.910

099

.910

094

.399

.986

.253

.632

.199

.999

.944

.299

.797

.759

.910

010

010

010

010

010

010

099

.910

010

099

.90.

099

.799

.899

.9

2.04

LoC

onN

(0.5

,3)

94.7

99.3

96.3

95.6

99.3

98.9

89.3

94.8

79.6

99.7

98.6

95.0

88.8

89.3

95.3

44.2

99.9

98.9

35.2

98.8

98.4

96.0

99.0

97.8

95.0

98.9

39.9

98.1

99.0

99.1

0.0

99.8

8.1

99.2

2.06

Tu

key(3

)35

.688

.810

010

088

.899

.688

.495

.136

.290

.924

.58.

75.

788

.487

.411

.485

.060

.815

.797

.799

.997

.810

099

.196

.710

099

.699

.710

086

.60.

084

.796

.489

.0

2.1

Tu

key(0

.5)

43.9

84.4

99.5

99.7

84.8

99.2

84.0

92.7

38.0

91.8

42.7

20.5

12.5

84.1

85.1

16.6

90.3

70.5

15.3

97.3

97.9

89.2

99.9

99.1

85.9

99.1

89.6

96.8

99.9

87.5

0.0

90.0

51.4

86.7

2.14

Bet

a(2,

2)33

.371

.197

.297

.971

.696

.968

.082

.522

.382

.532

.315

.710

.168

.170

.513

.282

.557

.89.

592

.492

.474

.899

.697

.369

.596

.077

.889

.599

.275

.80.

082

.126

.774

.6

2.5

LoC

onN

(0.5

,2)

17.8

27.1

22.0

25.3

27.5

33.7

7.3

16.2

1.6

34.7

23.1

15.8

11.9

7.4

11.3

10.6

44.8

24.9

2.8

27.3

23.5

13.1

37.9

44.3

11.2

29.1

11.6

21.3

39.2

25.8

0.0

44.2

0.1

25.1

2.9

Tu

key(5

)77

.688

.575

.669

.988

.90.

30.

10.

29.

540

.994

.796

.895

.50.

25.

233

.769

.957

.61.

43.

388

.573

.297

.715

.869

.991

.216

.984

.52.

630

.97.

170

.580

.796

.9

3T

uke

y(0

.1)

4.9

4.9

3.5

4.2

4.8

3.7

3.4

3.6

3.7

4.7

5.1

5.1

5.0

3.4

3.1

4.9

4.6

4.9

4.2

3.3

4.1

3.7

5.0

4.2

3.6

4.2

3.7

4.0

4.2

4.8

3.9

4.6

4.8

3.8

3.4

JS

U(0

,1,0

,0.3

)7.

710

.516

.311

.610

.418

.422

.421

.223

.212

.87.

16.

05.

622

.521

.85.

014

.29.

913

.323

.515

.920

.04.

718

.120

.513

.311

.117

.119

.511

.025

.714

.321

.018

.4

3.91

GE

P(1

,-1.

1,0)

14.3

25.6

35.4

28.2

25.5

40.3

47.4

46.4

46.7

28.6

8.3

5.7

5.6

47.4

45.8

8.0

28.4

20.2

25.8

50.3

37.2

44.2

10.9

44.1

45.0

32.0

20.0

39.2

46.3

26.8

54.0

28.8

37.7

44.3

4S

tud

ent-

t(10

)14

.824

.535

.127

.824

.539

.545

.944

.847

.629

.511

.17.

66.

446

.044

.75.

434

.324

.026

.448

.435

.842

.411

.442

.443

.131

.220

.837

.744

.126

.250

.434

.444

.140

.3

4.2

Log

isti

c(0,

1)24

.339

.445

.138

.139

.651

.459

.058

.362

.946

.818

.811

.08.

259

.158

.06.

352

.538

.833

.362

.749

.256

.217

.159

.557

.043

.524

.551

.160

.641

.566

.552

.762

.854

.4

4.51

JS

U(0

,1,0

,0.5

)26

.442

.750

.943

.842

.856

.763

.863

.266

.949

.518

.210

.57.

963

.962

.76.

155

.842

.538

.267

.154

.160

.921

.263

.561

.648

.628

.456

.064

.745

.168

.055

.965

.559

.1

5.4

Tu

key(1

0)10

010

010

010

010

099

.710

010

010

010

010

010

010

010

010

010

010

010

099

.910

010

010

010

010

010

010

011

.010

010

010

00.

810

010

010

0

6L

apla

ce(0

,1)

94.4

98.4

92.5

92.5

98.4

93.7

96.6

96.8

99.1

99.1

94.1

81.8

67.6

96.6

97.8

25.1

99.6

98.6

79.6

98.3

97.5

98.2

83.8

99.1

98.3

96.5

49.3

97.7

98.5

97.8

88.3

99.6

99.8

98.9

10.4

Tu

key(2

0)10

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

010

099

.510

010

010

00.

010

010

010

0

36.1

9JS

U(0

,1,0

,1)

99.3

99.9

99.7

99.7

99.9

99.8

99.9

99.9

100

99.9

95.4

71.0

45.1

99.9

99.9

28.5

100

99.9

98.1

99.9

99.9

99.9

98.7

100

99.9

99.8

80.0

99.9

100

99.9

45.6

100

100

99.9

75.9

8JS

U(0

,1,0

,1.1

)99

.910

099

.999

.910

099

.910

010

010

010

098

.884

.659

.410

010

038

.310

010

099

.410

010

010

099

.710

010

010

085

.310

010

010

029

.210

010

010

0

Inf

Stu

den

t-t(

4)75

.389

.290

.088

.289

.392

.494

.894

.996

.191

.747

.322

.213

.594

.894

.89.

694

.089

.379

.596

.092

.594

.572

.995

.994

.790

.658

.393

.195

.890

.472

.294

.196

.294

.1

Inf

Stu

den

t-t(

2)99

.910

010

010

010

010

010

010

010

010

098

.176

.847

.910

010

032

.710

010

099

.810

010

010

099

.910

010

010

088

.810

010

010

015

.410

010

010

0

Inf

Stu

den

t-t(

1)10

010

010

010

010

010

010

010

010

010

010

010

099

.310

010

091

.510

010

010

010

010

010

010

010

010

010

095

.710

010

010

00.

010

010

010

0

Mea

n

200

66.1

75.0

77.3

76.0

75.0

72.5

74.1

75.6

69.0

75.3

64.2

54.3

46.0

74.1

74.6

40.2

77.2

71.4

57.0

77.5

78.3

77.5

73.7

73.5

77.0

77.8

59.9

78.2

77.8

73.4

19.0

77.2

69.6

78.2

Ran

k

200

2715

.58

1215

.523

18.5

1326

1428

3132

18.5

1733

9.5

2430

6.5

16.

520

2111

4.5

292.

54.

522

349.

525

2.5

Dev

.to

the

Bes

t-

Mea

n

200

16.5

7.6

5.3

6.6

7.6

10.1

8.5

7.0

13.6

7.3

18.4

28.3

36.6

8.5

8.0

42.4

5.4

11.2

25.6

5.0

4.3

5.1

8.9

9.1

5.6

4.8

22.7

4.4

4.8

9.2

63.6

5.4

13.0

4.4

Ran

k

200

2715

.58

1215

.523

18.5

1326

1428

3132

18.5

1733

9.5

2430

61

720

2111

4.5

292.

54.

522

349.

525

2.5

Dev

.to

the

Bes

t-

Max

200

66.3

28.5

22.8

28.4

28.5

97.4

97.6

97.5

88.2

56.8

75.5

91.3

94.3

97.5

92.5

88.6

27.8

41.8

96.3

94.4

21.3

31.7

49.4

93.5

33.6

23.0

89.0

23.5

95.1

66.8

100

27.2

91.8

25.0

Ran

k

200

169.

52

89.

530

3331

.519

1518

2226

31.5

2420

713

2927

111

1425

123

214

2817

346

235

41

Page 42: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

−2 −1 0 1 2

0.0

0.5

1.0

1.5

GEP densities with a kurtosis of 1.1

X

Den

sitie

s

−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

GEP densities with a kurtosis of 1.5

X

Den

sitie

s

−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

GEP densities with a kurtosis of 1.84

X

Den

sitie

s

−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

GEP densities with a kurtosis of 2

X

Den

sitie

s

−3 −2 −1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

GEP densities with a kurtosis of 2.25

X

Den

sitie

s

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

GEP densities with a kurtosis of 2.5

X

Den

sitie

s

Figure 1: GEP densities with different tails behaviour

42

Page 43: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

GEP densities with a kurtosis of 2.75

X

Den

sitie

s

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

GEP densities with a kurtosis of 3

X

Den

sitie

s

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

GEP densities with a kurtosis of 3.5

X

Den

sitie

s

−4 −2 0 2 4

0.0

0.2

0.4

0.6

GEP densities with a kurtosis of 4

X

Den

sitie

s

−4 −2 0 2 4

0.0

0.2

0.4

0.6

GEP densities with a kurtosis of 4.5

X

Den

sitie

s

−3 −2 −1 0 1 2 3

0.0

0.2

0.4

0.6

GEP densities with a kurtosis of 5

X

Den

sitie

s

Figure 2: GEP densities with different tails behaviour

43

Page 44: Test of normality against generalized exponential power ... · comes from a normal distribution. There are, however, many other tests of normality in the literature, all based on

−3 −2 −1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

GEP densities with a kurtosis of 6

X

Den

sitie

s

−3 −2 −1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

GEP densities with a kurtosis of 10

X

Den

sitie

s

−3 −2 −1 0 1 2 3

0.0

0.5

1.0

1.5

2.0

GEP densities with a kurtosis of 20

X

Den

sitie

s

−3 −2 −1 0 1 2 3

0.0

0.5

1.0

1.5

2.0

2.5

GEP densities with a kurtosis of 50

X

Den

sitie

s

Figure 3: GEP densities with different tails behaviour

44