PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [Consorci de Biblioteques Universitaries de Catalunya]On: 2 March 2010Access details: Access Details: [subscription number 919083124]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in Statistics - Theory and MethodsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713597238

Goodness-of-Fit Tests for the Skew-Normal Distribution When theParameters Are Estimated from the DataGlòria Mateu-Figueras a; Pedro Puig b; Arthur Pewsey c

a Departament d'Informàtica i Matemàtica Aplicada, Universitat de Girona, Girona, Spain b Unitatd'Estadística i Investigació Operativa, Departament de Matemàtiques, Universitat Autònoma deBarcelona, Bellaterra, Spain c Departamento de Matemáticas, Escuela Politécnica, Universidad deExtremadura, Cáceres, Spain

To cite this Article Mateu-Figueras, Glòria, Puig, Pedro and Pewsey, Arthur(2007) 'Goodness-of-Fit Tests for the Skew-Normal Distribution When the Parameters Are Estimated from the Data', Communications in Statistics - Theory andMethods, 36: 9, 1735 — 1755To link to this Article: DOI: 10.1080/03610920601126217URL: http://dx.doi.org/10.1080/03610920601126217

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

Communications in Statistics—Theory and Methods, 36: 1735–1755, 2007Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920601126217

Inference

Goodness-of-Fit Tests for the Skew-NormalDistributionWhen the Parameters Are

Estimated from the Data

GLÒRIA MATEU-FIGUERAS1, PEDRO PUIG2,AND ARTHUR PEWSEY3

1Departament d’Informàtica i Matemàtica Aplicada,Universitat de Girona, Girona, Spain2Unitat d’Estadística i Investigació Operativa,Departament de Matemàtiques,Universitat Autònoma de Barcelona,Bellaterra, Spain3Departamento de Matemáticas, Escuela Politécnica,Universidad de Extremadura, Cáceres, Spain

In this article, tests are developed which can be used to investigate the goodness-of-fit of the skew-normal distribution in the context most relevant to the dataanalyst, namely that in which the parameter values are unknown and are estimatedfrom the data. We consider five test statistics chosen from the broad Cramér–vonMises and Kolmogorov–Smirnov families, based on measures of disparity betweenthe distribution function of a fitted skew-normal population and the empiricaldistribution function. The sampling distributions of the proposed test statistics areapproximated using Monte Carlo techniques and summarized in easy to use tabularform. We also present results obtained from simulation studies designed to explorethe true size of the tests and their power against various asymmetric alternativedistributions.

Keywords Asymmetry; Cramér–von Mises; Empirical distribution function;Kolmogorov–Smirnov.

Mathematics Subject Classification Primary 62G10; Secondary 62G30.

Received November 9, 2005; Accepted September 19, 2006Address correspondence to Glòria Mateu-Figueras, Departament d’Informàtica i

Matemàtica Aplicada, Universitat de Girona, Campus Montilivi P-IV, E-17071 Girona,Spain; E-mail: gloria.mateu@udg.es

1735

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1736 Mateu-Figueras et al.

1. Introduction

The univariate skew-normal class of distributions is one defined on the real line,with a shape parameter that controls the skewness of the density. Azzalini (1985,1986) studied the class in detail and in Azzalini and Capitanio (1999) and Azzaliniand Dalla Valle (1996), proposed and investigated its multivariate extension. Themodel has certain appealing algebraic properties similar to those of the ubiquitousnormal distribution (see, e.g., Azzalini, 2005), but provides a more appropriatemodel when the assumption of normality is untenable due to skewness in the data.The skew-normal distribution, and its extensions, have been applied in numerousscientific disciplines. In combination, the book length treatment edited by Genton(2004) and the resumé by Azzalini (2005) provide extensive details of developmentsin this line of research as well as illustrative examples of applications to real data.

Certain aspects of goodness-of-fit testing for the skew-normal distribution havebeen considered in the literature. Specifically, Salvan (1986) studied the problem oftesting for normality within the skew-normal class and showed that the coefficientof skewness is the locally most powerful location and scale invariant test statistic.Pewsey (2000) used the same statistic to test for departures from the limiting half-normal distribution. Gupta and Chen (2001) applied empirical distribution function(EDF)-based statistics to test the null hypothesis of a skew-normal distribution forthe situation in which the values taken by the parameters are completely specifiedwithout any reference to the data (henceforth referred to simply as “completelyspecified”). To date, however, there are no references in the literature to thegoodness-of-fit testing problem considered here, namely that of testing the nullhypothesis that a random sample comes from a skew-normal population whoseparameters have been estimated from the data. We note that the unpublished workof Dalla Valle (in press) proposes a modification of the Anderson–Darling statisticfor the goodness-of-fit testing context considered here.

The remainder of the article is structured as follows. Fundamental properties ofthe skew-normal distribution, and point estimation for its parameters, are reviewedin Sec. 2. In Sec. 3, general EDF-based test procedures are briefly reviewed andfive specific EDF-based statistics described. Monte Carlo techniques are used inSec. 4.1 to approximate the sampling distributions of the five test statistics whenthe parameters of the hypothesized skew-normal distribution are estimated fromthe data. A series of tables appearing in the Appendix provide estimates of thequantiles of these sampling distributions most relevant to hypothesis testing. Thesteps involved in carrying out the tests are described in Sec. 4.2. In Secs. 4.3 and 4.4,the results from simulation studies designed to explore the true size and power ofthe tests are presented. Finally, Sec. 5 contains some concluding remarks.

2. Fundamental Properties of, and Point Estimation for,the Skew-Normal Distribution

Under the so-called “direct” parametrization, X is a skew-normal random variablewith location, scale, and skewness (or shape) parameters, � ∈ �, � > 0, and � ∈ �,respectively, if its density is given by

f�x� �� �� �� = 2��

(x − �

�

)

(�x − �

�

)� −� < x < +��

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1737

where ��·� and �·� represent the standard normal density and distributionfunction, respectively. We denote the fact by X ∼ ������ �� ��, the � subscriptindicating the use of the direct parametrization. When the shape parameter, �, iszero, the � ��� �2� density is obtained. As � → ±�, the density tends to that of the(negative) half-normal distribution. In what follows we limit our attention to X with� ≥ 0, as if X ∼ ������ �� ��, −X ∼ ����−�� ��−��.

Results for the moments of X are given in Azzalini (1985). In particular,E�X� = �+ b�, var�X� = �2�1− b22� and the coefficient of skewness of X isgiven by �1 = b3�2b2 − 1�/�1− b22�3/2 ∈ �−0�99527� 0�99527�, where b = √

2/ and = �/

√1+ �2. As � → ±�, �1 → ±0�99527. Thus the skew-normal class is

capable of modelling up to moderate levels of skewness only.To compute the distribution function, F�x� = P�X ≤ x�, we can use the general

expression F�x� = ���x − ��/��− 2T��x − ��/�� �� provided by Azzalini (1985),where T�·� denotes Owen’s function (Owen, 1956). An R-based routine to evaluateOwen’s function is available at http://www.r-project.org. Gupta and Chen(2001) tabulate the distribution function of the ����0� 1� �� distribution for a rangeof �-values.

Maximum likelihood methods can be employed to estimate the parametersof the distribution. Since analytic expressions do not exist for these estimators,numerical methods must be used to compute them. As Azzalini and Capitanio(1999) and Pewsey (2000) point out, when � = 0, there is no unique solution tothe likelihood equations. Moreover, the shape of the log-likelihood function canbe irregular and consequently numerical optimization procedures can be slow toconverge. The former problem can be circumvented by basing maximum likelihoodestimation on the so-called “centered” parametrization of the distribution, proposedby Azzalini (1985). Given X ∼ ������ �� ��, the random variable

Y = � + �

(X − E�X�√

var�X�

)

is said to be a skew-normal variable with centred parameters �� �, and �1. Weuse the notation Y ∼ �N���� �� �1�, the � subscript highlighting that the centeredparameterization is being used. For this reparametrization, � = E�Y�� �2 = var�Y�,and �1 is the coefficient of skewness of both X and Y . It has also been found thatthe shape of the log-likelihood surface under this parametrization is generally betterbehaved than that under the direct parametrization (see Azzalini and Capitanio,1999; Pewsey, 2000, for detailed discussion). Having obtained estimates of thecentered parameters, those for the direct parameters can be calculated using:

� = � − c��1/31 � � = �

(1+ c2�

2/31

)1/2� (1)

� = c�1/31

(b2 + c2�b2 − 1��2/31

)−1/2� (2)

where c = �2/�4− ��1/3 and, as before, b = √2/ . A further inconvenience under

either parametrization is that the estimate of the skewness parameter (either � or �1)can occur on the boundary of the parameter space, corresponding to fitting a half-normal distribution. The frequency of this phenomenon is known to diminish withincreasing sample size (see Pewsey, 2000, for the results from a simulation study).

Azzalini and Capitanio (1999) give a web address from which the sn libraryof routines, designed to address the major computational aspects associated with

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1738 Mateu-Figueras et al.

the skew-normal distribution, can be obtained. The library includes routines forrandom variate generation, evaluation of the distribution and density functions, andparameter estimation. The library is freely available in S-plus, R, and Matlab code.

3. General EDF-Based Goodness-of-Fit Tests

In this section we discuss general EDF-based goodness-of-fit statistics designedto test the null hypothesis H0 � “the random sample x1� x2� � � � � xn comes from apopulation with distribution function F�·�.” EDF-based test statistics measure thedifference between the distribution function, F�·�, stated in the null hypothesis andthe EDF, a step function denoted by Fn�·�, calculated from the sample as

Fn�x� =

0� x < x�1��

i/n� x�i� ≤ x < x�i+1��

1� x�n� < x�

where x�1� < x�2� < · · · < x�n� denote the ordered data.To compare the two distribution functions, several statistics can be used that

Stephens (1986) divides into two families. The Cramér–von Mises family containsthe Cramér–von Mises statistic, W 2, Watson’s U 2 statistic, and the Anderson–Darling statistic, A2, defined as:

W 2 = n∫ �

−��Fn�x�− F�x��2dF�x��

U 2 = n∫ �

−�

[Fn�x�− F�x�−

∫ �

−��Fn�t�− F�t��dF�t�

]2

dF�x��

A2 = n∫ �

−��Fn�x�− F�x��2�F�x��1− F�x���−1dF�x��

The Kolmogorov–Smirnov family contains the statistics D+� D−, theKolmogorov–Smirnov statistic, D, and the Kuiper statistic, V , defined as:

D+ = supx

�Fn�x�− F�x��� D− = supx

�F�x�− Fn�x���

D = max�D+� D−�� V = D+ +D−�

Stephens (1986) provides the following simple formulae for calculating thesestatistics:

W 2 =n∑

i=1

(p�i� −

2i− 12n

)2

+ 112n

� (3)

U 2 = W 2 − n

(p̄− 1

2

)2

� (4)

A2 = −n− 1n

n∑i=1

�2i− 1��log p�i� + log�1− p�n+1−i���� (5)

D+ = maxi

(i

n− p�i�

)� (6)

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1739

D− = maxi

(p�i� −

i− 1n

)� (7)

D = max�D+� D−�� (8)

V = D+ +D−� (9)

where p�i� = F�x�i�� and p̄ = ∑ni=1 p�i�/n. Large values of a given statistic indicate

significant differences between the empirical and hypothesized distribution functionsand thus that we should reject the null hypothesis.

In general, when the parameter values of the hypothesized distribution arecompletely specified, the sampling distribution of any of these EDF statistics isknown exactly, and tables of percentage points are available (see Stephens, 1986,Table 4.2). However, when the values taken by the parameters of the distributionare unknown and have to be estimated from the sample, the sampling distributionof any EDF statistic depends on the distribution being tested, sample size, truevalues of the unknown parameters, and method used to estimate the parameters. Inthese circumstances, one is forced to use Monte Carlo techniques to approximatethe sampling distribution of a test statistic. We would note, however, that a generaltheory is available which can be used to obtain the asymptotic distributions of thestatistics W 2� U 2, and A2. See Puig and Stephens (2000, 2001) for a consideration ofthe situation in which the parent population is either Laplace or hyperbolic.

When the unknown parameters are location or scale parameters, and theyare estimated using location and scale equivariant estimators (as are maximumlikelihood estimators), the sampling distributions of the EDF statistics do notdepend on the true values of those parameters. In the case of the skew-normaldistribution, �� �, and � are location, scale, and shape parameters, respectively. Themaximum likelihood estimators �̂� �̂, and �̂ have the property that the distributionsof ��̂− ��/�̂ and �̂ depend only on the sample size n and on the true value of �(see Eastman and Bain, 1973). Consequently, when the parameters are estimated, thesampling distributions of the EDF statistics considered here will also only dependon the true value of � and the sample size, n. Monte Carlo techniques can be usedto estimate those sampling distributions which can then be summarized in tablescontaining their quantiles for specified combinations of n and �.

4. EDF-Based Tests for Skew-Normality When the ParametersAre Estimated

4.1. Sampling Distributions

In this section we describe the Monte Carlo techniques used to estimate thequantiles of the test statistics W 2, U 2, A2, D, and V when the hypothesizeddistribution is skew-normal with parameter values estimated from the data using themaximum likelihood method.

For each specified combination of n and �, we generated 10,000 samples ofsize n from the ����0� 1� �� distribution. The values of � = 0 and � = 1 wereused for simplicity because of the sampling distributions of the statistics beinginvariant to changes in the location and scale parameters, as explained in Sec. 3. Foreach simulated sample, we estimated the parameters of an assumed skew-normaldistribution using the maximum likelihood method, so obtaining the values of �̂� �̂,and �̂. To compute these estimates we used the sn.mle routine from the sn library

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1740 Mateu-Figueras et al.

referred to in Sec. 2. This routine employs gradient based methods to maximizethe log-likelihood function expressed in terms of the centered parameterization.As suggested by Pewsey (2000), we used this routine in combination with a grid ofstarting values in an attempt to ensure that the true global maximum was identified.

Having obtained the maximum likelihood estimates, the values of p�i� =F�x�i�� �̂� �̂� �̂�, i = 1� 2� � � � � n, were then computed using

p�i� =

(x�i� − �̂

�̂

)− 2T

(x�i� − �̂

�̂� �̂

)

and the value of each statistic calculated using Eqs. (3)–(9). The 10,000 valuesof each statistic were then ordered so as to obtain estimates of the quantiles,Qp, for p = 0�5� 0�75� 0�85� 0�9� 0�95� 0�975, and 0.99. Tables 3–7 of the Appendixpresent the quantiles obtained for � = 0� 1� 2� 3� 5� 7� 10� 20 and samples sizes ofn = 20� 30� 50�50�200� 300� 500. For n > 500, the quantiles of the test statistics werefound to be almost identical to those for n = 500. We also note that the distributionsof all five statistics are invariant to changes in the sign of �, and that the differencesbetween any two skew-normal densities with � ≥ 20 are very slight. The tablescorresponding to the Kolmogorov–Smirnov and Kuiper statistics (Tables 6 and 7)contain the quantiles of

√nD and

√nV instead of the quantiles of D and V . This

is common practice, introduced by Lilliefors (1967), as the D and V statistics tendto 0 when n → +�.

4.2. Realization of the Tests

Given a random sample of n data values, the steps necessary to carry out a giventest can be summarized as follows:

• Estimate the centred parameters using the maximum likelihood method andthen calculate those for the direct parameters, �̂� �̂, and �̂, using expressions(1) and (2).

• Obtain the p�i�-values.• Calculate the value of the chosen test statistic, T , using the appropriateformula(e) from Eqs. (3)–(9).

• For a given significance level, �, identify the quantile Q1−� of the chosentest statistic from the appropriate table (Tables 3–7) corresponding to � =�̂ and n. If either � = �̂ or the value of n do not appear in the table werecommend the use of linear interpolation between the nearest relevant valuesin the table. If neither � = �̂ nor the value of n appear in the table, bivariateinterpolation should be used. For sample sizes greater than 500 or �̂-values inexcess of 20, the critical values for n = 500 or � = 20, as appropriate, shouldbe used. If �̂ < 0, use the tables with � = −�̂.

• If T is greater than Q1−�, the null hypothesis is rejected at the � significancelevel.

A routine written in R has been developed by us to perform the steps involvedin carrying out the tests. The routine makes use of the sn package for R, as well asfiles containing the quantiles in Tables 3–7. The routine and the supporting files arefreely available from the leading author.

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1741

4.3. True Size of the Tests

When performing any one of the tests, the true value of the skewness parameter,�, is unknown and estimated using �̂. Also, linear or bivariate interpolationmust generally be applied when using the tables of quantiles. The use of theseapproximations will result in the true size of a given test generally differing from thenominal significance level. In this section we present the results from a simulationstudy designed to investigate the true size of the tests.

In our study, for all possible combinations of n = 20, 25, 50, 100, 125, 200,400, 500 and � = 0, 1, 2, 3, 5, 7, 10, 20, we simulated 10,000 samples of size nfrom the ����0� 1� �� distribution. We then applied all five tests to each sample,using nominal significance levels of 10%, 5%, 2.5%, and 1%. Finally, for each testand nominal significance level combination, we recorded the number of occasionson which the null hypothesis was rejected. In what follows we present the resultsfor the nominal significance level of 5%, these being broadly representative of thefindings for the other levels. Of course, for this nominal level, if the true and nominalsignificance levels were equal, we would expect the null hypothesis to be rejected for500 out of the 10,000 samples.

We note that when n = 20� 50� 100� 200, or 500 it is only generally necessary toapply linear interpolation (when, as will generally be the case, the value of �̂ does notcoincide with a �-value in Tables 3–7). However, when n = 25� 125, or 400, bivariateinterpolation must generally be applied.

The results obtained from the study are presented in Table 1. From aconsideration of the content of this table it is evident that, for small-sized samples(n ≤ 50), all five tests are generally conservative. Sample sizes of around 125 arerequired before the true size of the tests roughly matches the nominal significancelevel. In general, for n ≤ 125, the test based on the V statistic maintains the nominalsignificance level best, and the tests are most conservative when the population ismoderately skew �� ≈ 5�. Undoubtedly, these features are consequences of the effectsof sample size and the true value of � on the sampling properties of the ML estimatesof the parameters, in particular those of �̂: see the relevant results in Sec. 4 of Pewsey(2000). Even for a sample size as small as 25, the effect of bivariate interpolation onthe size of the tests is only marginal when compared with that of linear interpolation.

4.4. Power of the Tests

In this section we consider the results of a power study designed to explore the powerof the five EDF-based tests against various asymmetric alternative distributions. Asthe alternative distributions considered in any such power study cannot be fullyexhaustive, the results presented here can only be considered as partially indicative ofthe performance of the tests.

In our study we used certain cases of the log-normal (���� ��), Weibull(W(�� �)), gamma (G(�� �)), exponential (EXP(�)), and chi-squared (�2� ) distributionsas asymmetric alternatives. The actual distributions involved were ��0� 1�, ��0� 0�3�,W(1,2), G(4,1), G(5,1), EXP(1), �22, and �24. Note that, in fact, all of the last fiveof these belong to the gamma family of distributions. In Fig. 1 we represent thedensities of four of the distributions together with matched skew-normal densities.The ��0� 1� and �22 distributions, represented in plots (a) and (b), have coefficientsof skewness in excess of the upper limit for the skew-normal class of distributions,namely 0.99527, corresponding to a half-normal distribution. These two densities

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1742 Mateu-Figueras et al.

Table 1Empirical size �×10� 000� of the five EDF tests for a nominal significance levelof � = 0�05 for �N�0� 1� �� samples of size n. The standard error of any entry is,

at most, approximately 24

�

Statistic n 0 1 2 3 5 7 10 20

A2 20 171 180 110 109 93 163 262 55325 212 206 140 95 88 161 257 52350 353 296 246 235 227 231 262 528100 483 427 413 356 304 327 324 482125 484 453 408 387 369 364 380 562200 487 461 433 431 442 404 446 554400 512 504 540 455 451 486 498 552500 549 516 470 452 487 476 488 507

W 2 20 169 191 175 145 144 198 302 56725 166 204 182 147 123 185 301 54150 299 266 234 210 233 255 296 535100 454 418 393 339 312 348 372 462125 505 458 403 362 395 393 422 570200 487 440 393 432 448 407 462 536400 512 504 510 456 466 511 502 577500 531 516 467 430 480 483 500 509

U 2 20 271 288 280 254 272 350 380 54425 294 332 303 245 232 318 396 53050 446 398 372 332 368 348 397 545100 494 448 447 405 367 396 430 478125 523 495 465 419 442 441 455 525200 496 460 422 449 477 423 492 499400 515 511 530 456 458 543 494 545500 532 523 488 448 508 503 512 526

D 20 160 189 183 180 210 289 373 60325 155 191 209 173 179 266 366 55650 305 277 258 261 334 335 365 579100 432 417 383 362 359 369 444 474125 466 478 408 378 437 438 461 528200 486 424 374 440 446 451 491 540400 506 450 467 494 496 505 532 538500 489 509 452 435 512 490 507 522

V 20 327 338 325 307 314 395 372 57525 352 379 360 289 281 327 402 52250 481 433 394 388 396 363 403 553100 508 464 445 412 370 393 447 472125 521 511 469 419 434 445 454 503200 480 455 451 479 466 459 485 497400 476 483 518 474 494 532 494 549500 523 514 487 430 493 474 503 554

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1743

Figure 1. Densities of four of the alternative distributions (——) used in the power studytogether with their matched skew-normal densities (– – –); (a) ��0� 1� and ����0� 1�25���;(b) �22 and ����0� 1�51���; (c) W(1,2) and ����0�36� 0�70� 2�78�; and (d) G(5,1) and����2�14� 3�63� 6�11�.

have been matched by half-normal densities with a lower threshold of � = 0 andmean values equal to those of the ��0� 1� and �22 distributions, respectively. Theskew-normal densities in plots (c) and (d) have been matched so as to have thesame first three moments as their respective W(1,2) and G(1,5) densities. Alternativecriteria could have been employed to match the densities, but those used by usprovide the reader with an appreciation of how closely the original four densitiescan be approximated by skew-normal densities. A consideration of these plots, andsimilarly generated plots for the other four alternative distributions used in thepower study, identified the greatest disparities between the pairs of densities as beingthose for the ��0� 1�, EXP(1), and �22 distributions. We would therefore expect thetests to be most powerful against these three alternatives. The other five distributionscan be better approximated by their matched skew-normal distributions and so wewould expect the tests to have much lower power against them.

The values of n considered in the study were the same as those used inthe simulation study described in Sec. 4.3. For each sample size and alternativedistribution combination, we generated 10,000 samples and performed each of thefive tests using a nominal significance level of 5%. Our results for the number oftimes the null hypothesis of an underlying skew-normal distribution was rejected arepresented in Table 2.

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1744 Mateu-Figueras et al.

Table 2Empirical power �×10� 000� of the five EDF tests against various alternativedistributions for a nominal significance level of � = 0�05. The standard error

of any entry is at most 50

Alternative distribution

Statistic n ��0� 1� ��0� 0�3� W�1� 2� G�4� 1� G�5� 1� EXP�1� �22 �24

A2 20 6128 128 97 198 157 3666 3858 55225 7004 199 137 204 188 4457 4694 63450 9247 373 197 295 278 7461 7702 999100 9951 615 713 475 417 9518 9592 1377125 9990 705 927 590 497 9787 9823 1456200 10000 969 1746 746 676 9990 9990 2343400 10000 1466 3808 1182 962 10000 10000 4351500 10000 1817 4682 1420 1239 10000 10000 5168

W 2 20 6342 213 158 262 213 3581 3751 66425 7203 266 160 269 229 4369 4600 76350 9286 352 195 279 251 7219 7488 1064100 9953 560 697 462 410 9401 9502 1393125 9990 606 937 574 467 9734 9781 1473200 10000 778 1634 658 606 9988 9989 2167400 10000 1173 3602 943 832 10000 10000 3978500 10000 1426 4389 1138 1056 10000 10000 4711

U 2 20 5083 308 268 295 308 2551 2552 57525 5962 363 282 367 310 2969 3065 59150 8632 483 297 390 396 5333 5469 828100 9866 625 671 561 505 8114 8360 1302125 9968 661 879 677 548 8866 9041 1493200 9999 808 1468 752 668 9823 9848 2250400 10000 1262 3191 1201 1003 9998 9996 4338500 10000 1473 3875 1462 1231 10000 10000 5156

D 20 6094 229 159 280 231 3327 3463 68525 6863 281 169 282 256 3944 4133 73050 9148 397 297 339 320 6661 6996 1065100 9926 539 696 494 456 9046 9220 1307125 9980 590 904 556 508 9493 9604 1370200 10000 692 1403 631 593 9960 9970 1955400 10000 1061 2906 868 733 10000 10000 3479500 10000 1231 3509 953 889 10000 10000 4084

V 20 5140 335 312 331 348 2560 2578 58225 5907 395 348 390 355 2966 3042 58950 8548 514 358 428 444 5252 5433 744100 9830 621 610 579 532 8084 8334 1121125 9957 720 762 690 583 8876 9004 1354200 9998 781 1209 776 717 9817 9854 1990400 10000 1184 2584 1210 995 10000 9996 3840500 10000 1375 3189 1471 1184 10000 10000 4553

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1745

As expected, the tests are most powerful against the ��0� 1�, EXP(1), and �22alternatives identified above as having densities most dissimilar to those of theirmatched skew-normal distributions. The tests based on the A2 and W 2 statistics arethe most powerful against these alternative distributions as well as against the �24distribution.

Against the ��0� 0�3�, W(1,2), G(4,1), and G(5,1) distributions the performanceof all five tests is poor. Indeed, for sample sizes of 50 or less their power againstthese four alternatives is often considerably lower than the nominal significance level.Under these same circumstances, the test based on the V statistic at least has powerapproaching the nominal significance level. For larger sample sizes, the test based onthe A2 statistic is the most powerful of the five tests against the ��0� 0�3� alternative,although its power is only 18.17% for a sample size as large as 500. This test alsohas highest power against the W(1,2) alternative. Against the G(4,1) and G(5,1)alternatives the test based on the V statistic generally performs best. However, themaximum power achieved against these two alternatives is a miserly 14.71% and thisfor a considerable n = 500. Despite these observations, the rather dismal performanceof the tests against these four alternatives needs to be put into perspective. As hasbeen stated above, these distributions can be approximated by reasonably closelyfitting skew-normal distributions and hence we would not expect the tests to be ableto discriminate well between them and the skew-normal distribution.

5. Concluding Remarks

In this article, detailed consideration has been given to five EDF-based goodness-of-fit tests; three from the Cramér–von Mises family (W 2� U 2, and A2) and two from theKolmogorov–Smirnov family (D and V ). In addition to describing how to performthe tests, we have provided tables of estimated quantiles for use with the tests andpresented Monte Carlo results for their true size and power.

The results in Table 1 indicate that, in general, all five tests are conservativeand that, generally, the one based on the V statistic holds the nominal significancelevel best.

Of the five statistics considered by us, it is known that tests based on A2 aregenerally the most powerful when there are large differences between the tails of thenull and alternative distributions (see Stephens, 1986, p. 167). Our results in Table 2suggest that the tests based onA2 andW 2 are themost powerful against the alternativesincluded in our study with markedly different distributional forms from those inthe skew-normal class. As was to be expected, against the alternatives which can bewell approximated by some member of the skew-normal class, the power of all fivetests is low. It would be unjustified to say anything more definitive about the generalperformance of the tests, given the inevitably limited scope of our power study.

As we have noted in the Introduction, Dalla Valle (in press) has proposed agoodness-of-fit test for skew-normality based on a modification of the Anderson–Darling statistic. She only considers a nominal significance level of 5%, and exploresthree different variants of the test employing criteria which, without going intotheir detail, we will refer to simply as “minimum”, “mean” and “maximum”. Acomparison of our Monte Carlo results with hers suggests that the modifiedAnderson–Darling test based on the “minimum” criterion generally appears tomaintain the nominal significance level better than our test based on the V statistic.It is more difficult to compare the power of the various tests as just three of the

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1746 Mateu-Figueras et al.

alternative distributions included in Dalla Valle’s power study were considered inours. Nevertheless, against the ��0� 1� distribution, all five of our tests appear tohave higher power than all three variants of Dalla Valle’s modified Anderson–Darling test. However, against the G(4,1) and G(5,1) alternatives, Dalla Valle’s testemploying the “minimum” criterion appears to be the more powerful.

Finally, our R-based software for performing the tests, referred to at the endof Sec. 4.2, makes use of the quantiles in Tables 3–7 together with interpolation toestablish whether skew-normality should be rejected or not for a specified nominalsignificance level. It would be possible to develop software to compute MonteCarlo p-values for the tests, calculated directly from the values of the test statisticsobtained for (a large number of) simulated samples from the same skew-normaldistribution as that fitted to the original data. However, we have shied away fromso doing primarily because of the increase in the execution time which would resultfrom employing such an approach.

Appendix

Table 3Quantile Qp of A2 estimated via 10,000 simulations

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

0 20 0.2856 0.4054 0.4905 0.5591 0.6823 0.8810 2.094030 0.2750 0.3750 0.4524 0.5144 0.6231 0.7451 0.984750 0.2750 0.3675 0.4355 0.4921 0.5964 0.7039 0.8808100 0.2747 0.3665 0.4339 0.4842 0.5745 0.6640 0.8201150 0.2762 0.3704 0.4355 0.4896 0.5717 0.6656 0.7818200 0.2794 0.3767 0.4440 0.4953 0.5826 0.6724 0.7803300 0.2821 0.3800 0.4452 0.4996 0.5879 0.6823 0.7963500 0.2809 0.3788 0.4460 0.4996 0.5801 0.6622 0.7657

1 20 0.2867 0.4044 0.4929 0.5661 0.6951 0.8797 1.577130 0.2774 0.3812 0.4553 0.5212 0.6318 0.7589 1.005950 0.2717 0.3684 0.4368 0.4919 0.6001 0.7250 0.9227100 0.2758 0.3711 0.4363 0.4930 0.5809 0.6746 0.8211150 0.2789 0.3726 0.4367 0.4881 0.5794 0.6657 0.7805200 0.2804 0.3796 0.4470 0.4984 0.5833 0.6760 0.7875300 0.2812 0.3781 0.4492 0.5030 0.5908 0.6706 0.7875500 0.2794 0.3753 0.4424 0.4893 0.5772 0.6713 0.8120

2 20 0.2982 0.4186 0.5087 0.5793 0.7212 0.8891 1.223130 0.2820 0.3896 0.4714 0.5401 0.6518 0.7727 0.929550 0.2793 0.3806 0.4560 0.5171 0.6266 0.7315 0.8938100 0.2772 0.3748 0.4421 0.4940 0.5829 0.6768 0.8279150 0.2763 0.3770 0.4425 0.4945 0.5878 0.6833 0.8134200 0.2747 0.3701 0.4402 0.4945 0.5881 0.6843 0.8230300 0.2734 0.3685 0.4288 0.4768 0.5573 0.6561 0.7726500 0.2790 0.3773 0.4426 0.4969 0.5844 0.6713 0.8057

(continued)

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1747

Table 3Continued

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

3 20 0.3162 0.4555 0.5625 0.6445 0.8228 1.0379 1.394130 0.2950 0.4179 0.5059 0.5812 0.7121 0.8540 1.060750 0.2897 0.4063 0.4914 0.5591 0.6853 0.8203 0.9967100 0.2842 0.3908 0.4712 0.5380 0.6580 0.8017 1.0201150 0.2853 0.3885 0.4618 0.5278 0.6268 0.7487 0.9108200 0.2866 0.3882 0.4625 0.5193 0.6198 0.7302 0.8747300 0.2832 0.3852 0.4586 0.5207 0.6311 0.7392 0.8775500 0.2856 0.3876 0.4617 0.5226 0.6294 0.7254 0.8660

5 20 0.3648 0.5520 0.6897 0.8097 1.0434 1.3263 1.771030 0.3271 0.4772 0.5872 0.6844 0.8566 1.0522 1.327350 0.3123 0.4458 0.5453 0.6312 0.7781 0.9219 1.1503100 0.3103 0.4416 0.5519 0.6423 0.7957 0.9644 1.2031150 0.3117 0.4408 0.5424 0.6218 0.7719 0.9234 1.1384200 0.3101 0.4390 0.5335 0.6082 0.7498 0.8891 1.1028300 0.3065 0.4303 0.5178 0.5921 0.7341 0.8740 1.0589500 0.3062 0.4304 0.5150 0.5920 0.7121 0.8609 1.0761

7 20 0.4085 0.6373 0.8058 0.9539 1.2590 1.5981 2.146130 0.3671 0.5416 0.6855 0.8012 1.0129 1.2819 1.652050 0.3435 0.5006 0.6174 0.7170 0.9052 1.1002 1.3855100 0.3336 0.4843 0.5990 0.6900 0.8650 1.0791 1.3244150 0.3327 0.4767 0.5908 0.6792 0.8537 1.0409 1.3044200 0.3351 0.4796 0.5900 0.6866 0.8553 1.0417 1.2726300 0.3321 0.4779 0.5913 0.6789 0.8459 1.0301 1.2472500 0.3343 0.4799 0.5887 0.6749 0.8264 0.9683 1.1980

10 20 0.4856 0.7567 0.9844 1.1596 1.5254 1.9456 2.595430 0.4228 0.6534 0.8405 1.0029 1.3077 1.6240 2.096350 0.3824 0.5707 0.7260 0.8469 1.0786 1.3520 1.7093100 0.3687 0.5442 0.6696 0.7790 0.9761 1.1954 1.5684150 0.3631 0.5338 0.6719 0.7831 0.9554 1.1593 1.4517200 0.3614 0.5289 0.6593 0.7727 0.9705 1.1993 1.5404300 0.3596 0.5296 0.6590 0.7679 0.9598 1.1701 1.4386500 0.3605 0.5294 0.6520 0.7525 0.9549 1.1421 1.4294

20 20 0.6187 0.9791 1.2734 1.5149 1.9398 2.3472 3.146530 0.5468 0.8556 1.0997 1.3256 1.7519 2.1812 2.864950 0.4795 0.7475 0.9606 1.1473 1.5106 1.9292 2.4599100 0.4404 0.6857 0.8701 1.0281 1.2958 1.6020 2.1487150 0.4208 0.6387 0.8153 0.9640 1.2152 1.5001 1.8582200 0.4189 0.6273 0.7902 0.9256 1.1798 1.4286 1.7817300 0.4149 0.6271 0.7894 0.9296 1.1703 1.4344 1.7789500 0.4080 0.6139 0.7822 0.9198 1.1654 1.4064 1.7560

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1748 Mateu-Figueras et al.

Table 4Quantile Qp of W 2 estimated via 10,000 simulations

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

0 20 0.0423 0.0617 0.0755 0.0856 0.1014 0.1180 0.143830 0.0415 0.0594 0.0733 0.0847 0.1036 0.1224 0.142050 0.0411 0.0581 0.0711 0.0814 0.1004 0.1227 0.1552100 0.0409 0.0577 0.0695 0.0789 0.0956 0.1126 0.1390150 0.0408 0.0579 0.0703 0.0796 0.0968 0.1124 0.1354200 0.0414 0.0586 0.0710 0.0809 0.0974 0.1142 0.1328300 0.0422 0.0593 0.0717 0.0817 0.0978 0.1169 0.1376500 0.0419 0.0589 0.0718 0.0814 0.0973 0.1131 0.1331

1 20 0.0426 0.0617 0.0750 0.0858 0.1027 0.1209 0.149830 0.0416 0.0601 0.0745 0.0863 0.1059 0.1248 0.149150 0.0409 0.0580 0.0713 0.0823 0.1020 0.1244 0.1585100 0.0412 0.0583 0.0699 0.0801 0.0978 0.1141 0.1406150 0.0414 0.0582 0.0700 0.0788 0.0964 0.1127 0.1353200 0.0418 0.0596 0.0724 0.0821 0.0976 0.1142 0.1376300 0.0417 0.0592 0.0721 0.0821 0.0987 0.1147 0.1370500 0.0416 0.0584 0.0705 0.0801 0.0961 0.1150 0.1402

2 20 0.0439 0.0635 0.0773 0.0882 0.1078 0.1314 0.167030 0.0426 0.0619 0.0766 0.0897 0.1109 0.1323 0.158850 0.0419 0.0612 0.0755 0.0875 0.1095 0.1336 0.1694100 0.0415 0.0590 0.0711 0.0826 0.0989 0.1178 0.1473150 0.0413 0.0590 0.0713 0.0812 0.0981 0.1159 0.1450200 0.0407 0.0578 0.0706 0.0811 0.0990 0.1178 0.1426300 0.0406 0.0572 0.0689 0.0776 0.0931 0.1114 0.1321500 0.0411 0.0590 0.0707 0.0806 0.0974 0.1127 0.1361

3 20 0.0460 0.0677 0.0842 0.0974 0.1205 0.1482 0.198930 0.0450 0.0661 0.0822 0.0960 0.1217 0.1467 0.177950 0.0445 0.0664 0.0829 0.0983 0.1233 0.1552 0.1928100 0.0436 0.0636 0.0790 0.0929 0.1176 0.1497 0.1962150 0.0436 0.0628 0.0771 0.0892 0.1111 0.1368 0.1705200 0.0435 0.0628 0.0771 0.0883 0.1089 0.1314 0.1591300 0.0428 0.0616 0.0767 0.0879 0.1078 0.1299 0.1588500 0.0432 0.0623 0.0761 0.0884 0.1083 0.1287 0.1551

5 20 0.0511 0.0807 0.1041 0.1222 0.1605 0.1979 0.265830 0.0498 0.0762 0.0970 0.1149 0.1459 0.1799 0.234850 0.0491 0.0744 0.0957 0.1126 0.1459 0.1764 0.2214100 0.0495 0.0758 0.0977 0.1171 0.1501 0.1883 0.2427150 0.0499 0.0763 0.0966 0.1135 0.1438 0.1790 0.2256200 0.0496 0.0752 0.0949 0.1116 0.1417 0.1718 0.2186300 0.0487 0.0730 0.0915 0.1081 0.1377 0.1673 0.2107500 0.0489 0.0732 0.0915 0.1063 0.1338 0.1656 0.2105

(continued)

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1749

Table 4Continued

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

7 20 0.0575 0.0926 0.1214 0.1452 0.1940 0.2493 0.331530 0.0550 0.0863 0.1113 0.1335 0.1753 0.2239 0.283350 0.0549 0.0857 0.1108 0.1310 0.1711 0.2123 0.2715100 0.0547 0.0853 0.1094 0.1301 0.1667 0.2112 0.2667150 0.0548 0.0847 0.1084 0.1274 0.1644 0.2066 0.2596200 0.0552 0.0856 0.1080 0.1282 0.1661 0.2046 0.2588300 0.0545 0.0854 0.1084 0.1279 0.1642 0.2036 0.2509500 0.0552 0.0845 0.1081 0.1269 0.1599 0.1935 0.2457

10 20 0.0665 0.1100 0.1453 0.1759 0.2356 0.3089 0.423330 0.0629 0.1016 0.1343 0.1652 0.2187 0.2761 0.361350 0.0612 0.0975 0.1269 0.1522 0.2018 0.2522 0.3212100 0.0613 0.0970 0.1235 0.1464 0.1894 0.2380 0.3122150 0.0611 0.0964 0.1247 0.1479 0.1854 0.2296 0.2986200 0.0605 0.0964 0.1225 0.1459 0.1891 0.2438 0.3114300 0.0606 0.0952 0.1230 0.1478 0.1874 0.2319 0.2932500 0.0609 0.0965 0.1222 0.1458 0.1876 0.2315 0.2922

20 20 0.0824 0.1425 0.1899 0.2323 0.3065 0.3816 0.527130 0.0794 0.1314 0.1761 0.2140 0.2913 0.3740 0.485450 0.0749 0.1249 0.1662 0.2015 0.2703 0.3414 0.4357100 0.0729 0.1199 0.1575 0.1895 0.2461 0.3090 0.4113150 0.0699 0.1137 0.1501 0.1815 0.2328 0.2945 0.3742200 0.0698 0.1124 0.1462 0.1751 0.2290 0.2805 0.3563300 0.0699 0.1135 0.1467 0.1744 0.2230 0.2837 0.3528500 0.0689 0.1108 0.1457 0.1750 0.2257 0.2757 0.3516

Table 5Quantile Qp of U 2 estimated via 10,000 simulations

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

0 20 0.0404 0.0568 0.0684 0.0778 0.0936 0.1091 0.131430 0.0402 0.0562 0.0678 0.0768 0.0920 0.1056 0.126350 0.0405 0.0568 0.0688 0.0779 0.0931 0.1086 0.1290100 0.0406 0.0572 0.0686 0.0778 0.0936 0.1097 0.1357150 0.0405 0.0575 0.0698 0.0790 0.0959 0.1115 0.1326200 0.0412 0.0581 0.0705 0.0802 0.0969 0.1128 0.1314300 0.0420 0.0590 0.0713 0.0813 0.0971 0.1163 0.1369500 0.0417 0.0586 0.0713 0.0809 0.0966 0.1120 0.1326

(continued)

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1750 Mateu-Figueras et al.

Table 5Continued

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

1 20 0.0407 0.0568 0.0678 0.0775 0.0938 0.1113 0.134830 0.0402 0.0572 0.0688 0.0777 0.0938 0.1091 0.130050 0.0402 0.0566 0.0688 0.0778 0.0940 0.1104 0.1313100 0.0409 0.0577 0.0691 0.0791 0.0963 0.1120 0.1346150 0.0412 0.0577 0.0694 0.0782 0.0954 0.1116 0.1329200 0.0415 0.0592 0.0719 0.0815 0.0969 0.1133 0.1356300 0.0414 0.0588 0.0716 0.0816 0.0979 0.1139 0.1368500 0.0414 0.0581 0.0702 0.0797 0.0954 0.1143 0.1392

2 20 0.0415 0.0579 0.0694 0.0780 0.0937 0.1120 0.137830 0.0407 0.0576 0.0693 0.0786 0.0947 0.1114 0.133550 0.0405 0.0580 0.0701 0.0807 0.0966 0.1127 0.1363100 0.0406 0.0574 0.0689 0.0784 0.0931 0.1091 0.1313150 0.0405 0.0574 0.0694 0.0789 0.0940 0.1114 0.1353200 0.0399 0.0567 0.0689 0.0787 0.0961 0.1130 0.1357300 0.0399 0.0561 0.0672 0.0757 0.0906 0.1082 0.1285500 0.0405 0.0577 0.0693 0.0788 0.0948 0.1099 0.1323

3 20 0.0427 0.0597 0.0722 0.0821 0.1005 0.1195 0.148630 0.0418 0.0592 0.0715 0.0820 0.0996 0.1170 0.141450 0.0419 0.0601 0.0733 0.0837 0.1019 0.1187 0.1451100 0.0413 0.0590 0.0723 0.0828 0.1014 0.1213 0.1502150 0.0414 0.0586 0.0709 0.0816 0.0985 0.1165 0.1400200 0.0414 0.0583 0.0712 0.0802 0.0985 0.1156 0.1412300 0.0408 0.0580 0.0705 0.0800 0.0976 0.1161 0.1377500 0.0413 0.0585 0.0707 0.0812 0.0992 0.1159 0.1393

5 20 0.0455 0.0675 0.0840 0.0960 0.1192 0.1425 0.173530 0.0445 0.0643 0.0791 0.0903 0.1116 0.1322 0.164250 0.0439 0.0635 0.0773 0.0891 0.1093 0.1299 0.16126100 0.0444 0.0646 0.0803 0.0942 0.1162 0.1356 0.1633150 0.0447 0.0654 0.0798 0.0916 0.1120 0.1329 0.1606200 0.0442 0.0642 0.0786 0.0902 0.1121 0.1328 0.1607300 0.0434 0.0631 0.0774 0.0884 0.1093 0.1322 0.1605500 0.0436 0.0634 0.0772 0.0881 0.1072 0.1274 0.1576

7 20 0.0492 0.0731 0.0915 0.1051 0.1317 0.1543 0.188530 0.0470 0.0692 0.0857 0.1000 0.1226 0.1460 0.177550 0.0468 0.0690 0.0849 0.0985 0.1227 0.1444 0.1764100 0.0467 0.0683 0.0852 0.0975 0.1207 0.1499 0.1766150 0.0469 0.0687 0.0842 0.0972 0.1198 0.1446 0.1744200 0.0477 0.0693 0.0849 0.0976 0.1220 0.1446 0.1758300 0.0469 0.0697 0.0858 0.0999 0.1210 0.1444 0.1771500 0.0473 0.0698 0.0853 0.0974 0.1183 0.1416 0.1694

(continued)

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1751

Table 5Continued

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

10 20 0.0528 0.0798 0.0979 0.1142 0.1406 0.1706 0.207930 0.0512 0.0762 0.0944 0.1102 0.1375 0.1621 0.195550 0.0499 0.0743 0.0922 0.1073 0.1344 0.1582 0.1918100 0.0507 0.0751 0.0915 0.1048 0.1282 0.1550 0.1910150 0.0503 0.0741 0.0916 0.1058 0.1293 0.1534 0.1892200 0.0500 0.0741 0.0921 0.1068 0.1313 0.1576 0.1937300 0.0499 0.0740 0.0929 0.1070 0.1326 0.1601 0.1956500 0.0500 0.0747 0.0922 0.1060 0.1312 0.1564 0.1909

20 20 0.0598 0.0902 0.1116 0.1302 0.1574 0.1864 0.223030 0.0580 0.0876 0.1086 0.1253 0.1571 0.1851 0.222850 0.0561 0.0844 0.1056 0.1221 0.1518 0.1803 0.2190100 0.0563 0.0848 0.1038 0.1202 0.1485 0.1786 0.2104150 0.0550 0.0817 0.1012 0.1166 0.1467 0.1741 0.2159200 0.0546 0.0810 0.1018 0.1172 0.1456 0.1752 0.2085300 0.0551 0.0821 0.1014 0.1171 0.1451 0.1694 0.2113500 0.0547 0.0814 0.1005 0.1178 0.1446 0.1724 0.2101

Table 6Quantile Qp of

√nD estimated via 10,000 simulations

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

0 20 0.5392 0.6354 0.6944 0.7358 0.7952 0.8609 0.927930 0.5407 0.6373 0.6930 0.7356 0.8013 0.8658 0.939950 0.5441 0.6371 0.6975 0.7410 0.8076 0.8666 0.9540100 0.5492 0.6379 0.6928 0.7320 0.7944 0.8565 0.9336150 0.5548 0.6460 0.6996 0.7375 0.7968 0.8521 0.9257200 0.5575 0.6471 0.7036 0.7426 0.7972 0.8498 0.9103300 0.5658 0.6558 0.7114 0.7476 0.8049 0.8658 0.9308500 0.5629 0.6531 0.7092 0.7491 0.8096 0.8570 0.9151

1 20 0.5405 0.6349 0.6936 0.7329 0.7929 0.8551 0.929730 0.5414 0.6408 0.7006 0.7452 0.8117 0.8680 0.946950 0.5449 0.6359 0.6961 0.7397 0.8017 0.8691 0.9572100 0.5503 0.6421 0.6984 0.7355 0.7986 0.8512 0.9220150 0.5544 0.6469 0.6979 0.7375 0.7967 0.8472 0.9323200 0.5593 0.6520 0.7060 0.7474 0.8086 0.8627 0.9239300 0.5626 0.6558 0.7097 0.7469 0.8089 0.8669 0.9363500 0.5631 0.6514 0.7070 0.7474 0.8075 0.8619 0.9378

(continued)

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1752 Mateu-Figueras et al.

Table 6Continued

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

2 20 0.5445 0.6427 0.7007 0.7407 0.8129 0.8796 0.966430 0.5501 0.6514 0.7120 0.7547 0.8268 0.8982 0.988850 0.5534 0.6538 0.7158 0.7607 0.8389 0.9078 0.9866100 0.5560 0.6498 0.7073 0.7503 0.8159 0.8821 0.9570150 0.5567 0.6487 0.7043 0.7451 0.8138 0.8688 0.9648200 0.5568 0.6477 0.7052 0.7497 0.8183 0.8802 0.9573300 0.5563 0.6492 0.7048 0.7443 0.8056 0.8658 0.9354500 0.5635 0.6571 0.7111 0.7501 0.8124 0.8735 0.9413

3 20 0.5613 0.6668 0.7296 0.7767 0.8571 0.9273 1.026930 0.5615 0.6663 0.7349 0.7845 0.8595 0.9329 1.032650 0.5696 0.6782 0.7491 0.7969 0.8835 0.9615 1.0525100 0.5705 0.6762 0.7435 0.7965 0.8843 0.9677 1.0803150 0.5721 0.6735 0.7351 0.7870 0.8646 0.9303 1.0263200 0.5753 0.6773 0.7408 0.7859 0.8682 0.9369 1.0184300 0.5750 0.6742 0.7369 0.7841 0.8594 0.9226 1.0118500 0.5774 0.6808 0.7460 0.7967 0.8650 0.9334 1.0142

5 20 0.5888 0.7138 0.7932 0.8460 0.9440 1.0341 1.145230 0.5866 0.7075 0.7855 0.8388 0.9260 1.0079 1.132150 0.5941 0.7135 0.7932 0.8494 0.9343 1.0121 1.1098100 0.6053 0.7294 0.8076 0.8689 0.9684 1.0519 1.1537150 0.6128 0.7353 0.8112 0.8659 0.9511 1.0336 1.1478200 0.6099 0.7279 0.8050 0.8656 0.9593 1.0376 1.1390300 0.6099 0.7267 0.8036 0.8612 0.9433 1.0304 1.1277500 0.6120 0.7287 0.8039 0.8555 0.9430 1.0233 1.1195

7 20 0.6122 0.7470 0.8368 0.8991 1.0042 1.0960 1.220830 0.6145 0.7413 0.8258 0.8878 0.9859 1.0816 1.189250 0.6213 0.7514 0.8302 0.8960 0.9944 1.0810 1.1899100 0.6269 0.7599 0.8394 0.8982 0.9978 1.0900 1.2092150 0.6326 0.7649 0.8441 0.9023 0.9976 1.0855 1.1887200 0.6378 0.7681 0.8503 0.9060 0.9951 1.0761 1.1855300 0.6378 0.7680 0.8496 0.9094 1.0053 1.0794 1.2006500 0.6408 0.7710 0.8520 0.9081 0.9982 1.0802 1.1737

10 20 0.6504 0.7960 0.8881 0.9555 1.0735 1.1767 1.308330 0.6446 0.7854 0.8799 0.9484 1.0584 1.1543 1.270750 0.6474 0.7862 0.8719 0.9376 1.0396 1.1361 1.2451100 0.6582 0.7946 0.8787 0.9386 1.0324 1.1313 1.2474150 0.6615 0.7949 0.8800 0.9388 1.0369 1.1292 1.2336200 0.6541 0.7957 0.8787 0.9434 1.0437 1.1438 1.2588300 0.6598 0.7999 0.8842 0.9472 1.0438 1.1370 1.2619500 0.6642 0.8056 0.8880 0.9501 1.0489 1.1334 1.2541

(continued)

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1753

Table 6Continued

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

20 20 0.6976 0.8612 0.9672 1.0410 1.1436 1.2595 1.383230 0.6986 0.8570 0.9550 1.0340 1.1535 1.2635 1.393750 0.6911 0.8441 0.9452 1.0196 1.1240 1.2395 1.3587100 0.6952 0.8483 0.9409 1.0055 1.1127 1.2237 1.3486150 0.6875 0.8362 0.9301 0.9989 1.1080 1.2016 1.3059200 0.6905 0.8341 0.9286 0.9967 1.0973 1.1973 1.3029300 0.6951 0.8441 0.9319 0.9930 1.0978 1.1927 1.3077500 0.6885 0.8401 0.9301 0.9958 1.0960 1.1884 1.3065

Table 7Quantile Qp of

√nV estimated via 10,000 simulations

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

0 20 0.9578 1.0947 1.1797 1.2441 1.3405 1.4280 1.533030 0.9682 1.1067 1.1919 1.2487 1.3369 1.4283 1.545750 0.9827 1.1316 1.2182 1.2757 1.3771 1.4628 1.5668100 1.0032 1.1455 1.2337 1.2967 1.3948 1.4853 1.6196150 1.0087 1.1616 1.2461 1.3085 1.4066 1.4896 1.6034200 1.0162 1.1646 1.2538 1.3164 1.4166 1.4978 1.6017300 1.0271 1.1781 1.2659 1.3307 1.4292 1.5258 1.6253500 1.0246 1.1734 1.2671 1.3330 1.4312 1.5139 1.6118

1 20 0.9570 1.0961 1.1784 1.2406 1.3372 1.4317 1.547930 0.9704 1.1150 1.1982 1.2565 1.3568 1.4501 1.564650 0.9826 1.1248 1.2146 1.2757 1.3665 1.4585 1.5661100 1.0027 1.1480 1.2382 1.3037 1.4017 1.4882 1.5960150 1.0113 1.1595 1.2453 1.3062 1.4068 1.4923 1.6010200 1.0179 1.1717 1.2612 1.3245 1.4228 1.5071 1.6208300 1.0217 1.1738 1.2687 1.3302 1.4300 1.5265 1.6345500 1.0270 1.1721 1.2650 1.3291 1.4268 1.5265 1.6523

2 20 0.9613 1.1023 1.1823 1.2413 1.3336 1.4261 1.543730 0.9754 1.1162 1.1999 1.2572 1.3476 1.4383 1.560050 0.9856 1.1346 1.2232 1.2850 1.3799 1.4626 1.5662100 1.0001 1.1457 1.2321 1.2957 1.3879 1.4684 1.5700150 1.0050 1.1544 1.2414 1.2999 1.3972 1.4817 1.5933200 1.0062 1.1511 1.2409 1.3058 1.4030 1.5056 1.6323300 1.0062 1.1518 1.2369 1.2994 1.3942 1.4884 1.6088500 1.0189 1.1698 1.2574 1.3175 1.4099 1.5004 1.6147

(continued)

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

1754 Mateu-Figueras et al.

Table 7Continued

p

� n 0.5 0.75 0.85 0.9 0.95 0.975 0.99

3 20 0.9713 1.1160 1.2010 1.2658 1.3609 1.4435 1.563530 0.9755 1.1229 1.2132 1.2696 1.3767 1.4657 1.586850 0.9929 1.1466 1.2350 1.2952 1.3979 1.4860 1.6034100 1.0015 1.1548 1.2389 1.3070 1.4050 1.4970 1.6288150 1.0081 1.1538 1.2406 1.3014 1.3939 1.4902 1.6042200 1.0158 1.1570 1.2436 1.3055 1.4106 1.4994 1.6231300 1.0127 1.1596 1.2453 1.3109 1.4110 1.4944 1.6067500 1.0197 1.1700 1.2561 1.3190 1.4220 1.5034 1.6042

5 20 0.9953 1.1556 1.2532 1.3237 1.4364 1.5163 1.634530 0.9948 1.1493 1.2464 1.3120 1.4106 1.5021 1.624350 1.0073 1.1610 1.2506 1.3126 1.4157 1.5118 1.6264100 1.0271 1.1811 1.2751 1.3458 1.4560 1.5609 1.6716150 1.0325 1.1909 1.2809 1.3476 1.4488 1.5348 1.6447200 1.0307 1.1859 1.2744 1.3442 1.4443 1.5419 1.6582300 1.0305 1.1847 1.2781 1.3447 1.4417 1.5304 1.6426500 1.0318 1.1899 1.2809 1.3447 1.4385 1.5250 1.6417

7 20 1.0195 1.1880 1.2874 1.3637 1.4753 1.5683 1.685330 1.0172 1.1811 1.2799 1.3489 1.4578 1.5527 1.666450 1.0310 1.1892 1.2865 1.3572 1.4682 1.5693 1.6791100 1.0391 1.2016 1.2985 1.3665 1.4791 1.5821 1.7047150 1.0452 1.2037 1.3052 1.3710 1.4736 1.5836 1.7073200 1.0532 1.2132 1.3081 1.3758 1.4767 1.5788 1.6925300 1.0550 1.2209 1.3166 1.3832 1.4906 1.5952 1.7160500 1.0627 1.2203 1.3160 1.3803 1.4768 1.5686 1.6927

10 20 1.0493 1.2199 1.3216 1.3973 1.5153 1.6124 1.749830 1.0476 1.2169 1.3210 1.3953 1.5159 1.6249 1.739750 1.0483 1.2214 1.3214 1.3943 1.5030 1.6034 1.7272100 1.0690 1.2368 1.3325 1.4066 1.5119 1.6154 1.7205150 1.0726 1.2370 1.3379 1.4123 1.5196 1.6134 1.7213200 1.0699 1.2388 1.3432 1.4177 1.5306 1.6304 1.7569300 1.0775 1.2427 1.3438 1.4171 1.5400 1.6392 1.7659500 1.0808 1.2487 1.3488 1.4186 1.5358 1.6358 1.7512

20 20 1.0890 1.2735 1.3785 1.4561 1.5707 1.6843 1.808730 1.0939 1.2780 1.3861 1.4608 1.5892 1.6803 1.823650 1.0959 1.2713 1.3819 1.4556 1.5797 1.6831 1.8281100 1.1065 1.2860 1.3909 1.4679 1.5832 1.6963 1.8283150 1.1041 1.2778 1.3830 1.4580 1.5782 1.6856 1.8044200 1.1025 1.2827 1.3890 1.4646 1.5813 1.6872 1.7981300 1.1151 1.2896 1.3936 1.4683 1.5788 1.6952 1.8267500 1.1171 1.2919 1.3918 1.4649 1.5892 1.6993 1.8169

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010

Goodness-of-Fit Tests for Skew-Normality 1755

Acknowledgments

This research has been supported by the Dirección General de Enseñanza Superior(DGES) of the Spanish Ministry for Education and Culture through the projectBFM2003-05640/MATE.

References

Azzalini, A. (1985). A class of distribution which includes the normal ones. Scand. J. Statist.12:171–178.

Azzalini, A. (1986). Further results on a class of distributions which includes the normalones. Statistica 46:199–208.

Azzalini, A. (2005). The skew-normal distribution and related multivariate families. Scand.J. Statist. 32:159–188.

Azzalini, A., Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika83:715–726.

Azzalini, A., Capitanio, A. (1999). Statistical applications of the multivariate skew-normaldistribution. J. Roy. Statist. Soc. Ser. B 61:579–602.

Dalla Valle, A., A test for the hypothesis of skew normality in a population. J. Statist.Comput. Simul. In press.

Eastman, J., Bain, L. J. (1973). A property of maximum likelihood estimators in the presenceof location-scale nuisance parameters. Commun. Statist. 2:23–28.

Genton, M. G., ed. (2004). Skew-elliptical Distributions and Their Applications: A JourneyBeyond Normality. London: Chapman & Hall/CRC.

Gupta, A. K., Chen, T. (2001). Goodness-of-fit tests for the skew-normal distribution.Commun. Statist. Simul. Comput. 30:907–930.

Lilliefors, H. W. (1967). On the Kolmogorov–Smirnov test for normality with mean andvariance unknown. J. Amer. Statist. Assoc. 62:399–402.

Owen, D. B. (1956). Tables for computing bivariate normal probabilities. Ann. Math. Statist.27:1075–1090.

Pewsey, A. (2000). Problems of inference for Azzalini’s skew-normal distribution. J. Appl.Statist. 27:859–870.

Puig, P., Stephens, M. A. (2000). Test of fit for the Laplace distribution. Technometrics42:417–424.

Puig, P., Stephens, M. A. (2001). Goodness-of-fit for the hyperbolic distribution. Canad. J.Statist. 29:1–12.

Salvan, A. (1986). Locally most powerful invariant tests of normality (in Italian). Atti dellaXXXIII Riunione Scientifica della Società Italiana di Statistica 2:173–179.

Stephens, M. A. (1986). Tests based on EDF statistics. In: D’Agostino, R. B., Stephens, M.A., eds. Goodness-of-Fit Techniques. New York: Marcel Dekker.

Downloaded By: [Consorci de Biblioteques Universitaries de Catalunya] At: 12:07 2 March 2010