A GENERALIZED SKEW LOGISTIC DISTRIBUTION

Transcription

1 REVSTAT Statistica Journa Voume 11, Number 3, November 013, A GENERALIZED SKEW LOGISTIC DISTRIBUTION Authors: A. Asgharzadeh Department of Statistics, University of Mazandaran Babosar, Iran a.asgharzadeh@umz.ac.ir L. Esmaeii Department of Statistics, University of Mazandaran Babosar, Iran.esmaiy@stu.umz.ac.ir S. Nadarajah Schoo of Mathematics, University of Manchester Manchester M13 9PL, UK mbbsssn@manchester.ac.uk S.H. Shih Department of Mathematics and Statistics, American University of Sharjah Sharjah, UAE sshih@aus.edu Received: Juy 01 Revised: March 013 Accepted: March 013 Abstract: In this paper, we introduce a generaized skew ogistic distribution that contains the usua skew ogistic distribution as a specia case. Severa mathematica properties of the distribution are discussed ike the cumuative distribution function and moments. Furthermore, estimation using the method of maximum ikeihood and the Fisher information matrix are investigated. Two rea data appications iustrate the performance of the distribution. Key-Words: estimation; ogistic distribution; moments. AMS Subject Cassification: 6E15.

2 318 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih

3 A Generaized Skew Logistic Distribution INTRODUCTION Azzaini [] introduced the skew norma distribution specified by the probabiity density function pdf: 1.1 f SN x; λ = φxφλx, < x <, where λ R is the skewness parameter, φx is the standard norma pdf, and Φx is the standard norma cumuative distribution function cdf. Athough, Azzaini introduced the skew version 1.1 for the norma distribution, this idea can be appied to any symmetric pdf. Aong the same ine, the skew ogistic distribution with the skewness parameter λ can be proposed as foows. Consider the standard ogistic distribution specified by the cdf Hx = exp x, < x <, and the pdf hx = exp x 1 + exp x, < x <. Using the idea of Azzaini [], the pdf of the usua skew ogistic distribution is given by 1. f SL x; λ = hxhλx = exp x 1 + exp x 1 + exp λx for < x < and λ R. The properties of this distribution have been studied extensivey in the iterature. See, for exampe, Nadarajah [1] and Gupta and Kundu [9]. The skew ogistic distribution in 1. has aso received appications; for exampe, Koesser and Kumar [11] iustrate an appication with respect to an adaptive test for the two-sampe scae probem based on U-statistics. Because of the increasing popuarity of 1., one woud ike to have generaizations that are more fexibe. The aim of this paper is to construct a new generaization of 1. using the type III generaized ogistic distribution instead of the standard ogistic distribution. We study mathematica properties of this new generaization and discuss rea data appications.

4 30 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih [10] The type III generaized ogistic distribution has the pdf see Johnson et a. g α x = 1 Bα, α exp αx 1 + exp x α for < x < and α > 0. This distribution is symmetric for every α. When α = 1, the above pdf reduces to the standard ogistic pdf. This distribution has the cdf where y = 1 + exp x 1 and Here, Γa = 0 G α x = B yα, α Bα, α, Bα, α = { Γα } Γα t a 1 exp tdt, B x a, b =. x 0 t a 1 1 t b 1 dt are the gamma function and the incompete beta function, respectivey. Now, we define the new skew ogistic distribution as foows. If a random variabe X has the foowing pdf 1.3 fx; α, λ = g α xg α λx, < x <, α > 0, λ R, then we say that X has a genera skew ogistic GSL distribution. We write X GSLα, λ. From 1.3, some basic properties of GSLα, λ can be noted as foows: i When α = 1, 1.3 reduces to the usua skew ogistic pdf; ii When λ = 0, 1.3 reduces to the type III generaized ogistic pdf; iii If X GSLα, λ, then X GSLα, λ; iv fx; α, λ + fx; α, λ = g α x for a x R; v fx; α, λ g α xi{x 0} as λ + and fx; α, λ g α x I{x 0} as λ for a α; vi fx; α, λ 0 as x ± for a α > 0 and λ R. Numerica investigations show that 1.3 has a singe mode. The mode is at x 0, where x 0 is the root of λ α g α x G α x 1 exp x 1 + exp x = 0.

5 A Generaized Skew Logistic Distribution 31 Figures 1 and iustrate possibe shapes of the pdf 1.3 for α = and seected vaues of λ. Figure 1: Pots of GSLα,λ pdf for α = and λ > 0. Figure : Pots of GSLα,λ pdf for α = and λ < 0.

6 3 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih Simuation from 1.3 is straight-forward by using the foowing representation due to Azzaini [3]: X = S U U, where, conditionay on U = u, S U = +1 with probabiity G α λu and S U = 1 with probabiity 1 G α λu; X = S U U, where, conditionay on U = u, S U = +1 with probabiity G α λ u and S U = 1 with probabiity 1 G α λ u. Both these representations have physica meanings as expained in Azzaini [3]. In the seque, we sha use the foowing functions: τ 1 b, q = q j b, τ a, b, q, λ = q + j j=0 q j a λb + q + j j=0 and, the Gauss hypergeometric function defined by F 1 a, b; c; x = k=0 a k b k c k x k k!, where z k = zz + 1 z + k 1 denotes the ascending factoria. Throughout the rest of this paper uness otherwise stated, we sha assume that λ > 0 since the corresponding resuts for λ < 0 can be obtained using the fact that X has the pdf g α xg α λx. Some resuts of this paper require certain series representations of the genera skew ogistic pdf 1.3, which we derive now. Using the Tayor series expansion for [1 + exp λx] 1, we can obtain G α λx = 1 Bα, α 1 Bα, α α 1 α + i 1i exp jλx i j, x > 0, α 1 α + i 1i exp λx + j i j, x < 0. Substituting this into 1.3, a doube series representation for the genera skew

7 A Generaized Skew Logistic Distribution 33 ogistic pdf can be obtained as 1.4 fx; α, λ = Bα, α 1+ exp x α 1i exp xλj + α Bα, α 1+ exp x α i 1i exp xλi + λα + λj α α 1 α+i i, x > 0, α 1 α+i j j, x < 0. By expanding the terms in the denominators of 1.4, one can aso obtain the tripe series representation 1.5 fx; α, λ = α 1 α+i α Bα, α i j k k=0 1i exp xλj + k + α Bα, α, x > 0, α 1 α+i α i j k k=0 1 i exp x λ + j + k + α, x < 0.. CUMULATIVE DISTRIBUTION FUNCTION Using the doube and tripe series representations in 1.4 and 1.5, we derive some formuas for the cdf corresponding to 1.3. First, we use the doube series representation in 1.4. If x > 0, then the cdf Fx can be written as.1 x Fx = F0 + Bα, α 0 exp tλj + α dt = α 1 α + i 1 i i j 1 + exp t α

8 34 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih..3.4 α 1 i = F0 + Bα, α x exp tλj + α α dt exp t α 1 = F0 + 1 exp x = F0 + Bα, α z λj+α z Bα, α α dz i α + i j α + i j 1 i 1 i α 1 α + i 1 i i j Ix, where Ix = 1 exp x = I 1 I. z λj+α 1 α dz = 1 + z 1 z λj+α 1 0 α dz 1 + z exp x z λj+α z α dz By equation in Gradshteyn and Ryzhik [8], the integras I 1 and I can be cacuated as.5 I 1 = 1 0 z λj+α z α dz = 1 λj + α F 1 α, α + λj; α + λj + 1; 1, and.6 exp x z λj+α 1 I = z = exp α + λjy α + λj α dz F 1 α, α + λj; α + λj + 1; exp x. Combining.5 and.6 and substituting into.4, the cdf Fx for x > 0 becomes α 1 α + i 1 i i j Fx = F0 + Bα, α { 1 λj + α F 1 α, α + λj; α + λj + 1; 1 exp α + λjx } F 1 α, α + λj, α + λj + 1; exp x. α + λj

9 A Generaized Skew Logistic Distribution 35 Repeating the above argument with x = 0 yieds the form for F0 as F0 = = = = 0 Bα, α ftdt α 1 α + i 1 i i j t λ + j + α 0 exp Bα, α Bα, α F expt α dt α 1 α + i 1 i i j α 1 α + i 1 i 1 i j α, λ + j + α; λ + j + α + 1; 1 λ + j + α 0 z λi+α+j+α z α dz If x < 0, then simiar arguments by using equation in Gradshteyn and Ryzhik [8] yieds α 1 α + i x 1 i i j Fx = ftdt = Bα, α I, where I = = x exp t λj + α + i + α expx z λi+α+j+α 1 α = 1 + expt z α dz exp x λ + j + α λ + j + α F 1 α, λ + j + α; λ + j + α + 1, expx, and so the resut α 1 α + i 1 i i j Fx = Bα, α exp x λ + j + α λ + j + α F 1 α, λ + j + α; λ + j + α + 1, expx..

10 36 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih Using the tripe series representation, 1.5, the cdf Fx can be cacuated as Fx = 1 Bα, α Bα, α k=0 α 1 α + i α i j k λj + k + α 1 i exp xλj + k + α, x > 0, α 1 α + i α 1 i i j k λ + j + k + α k=0 exp x λ + j + k + α, x < MOMENTS Many of the interesting characteristics of the genera skew ogistic distribution can be studied through its moments. Let X GSLα, λ. In this section, we derive the nth moment of X. It is easy to show that if X foows GSLα, λ then Y = X has the foded form of the type III generaized ogistic distribution specified by the pdf gy; α, λ = Bα, α exp αy 1 + exp y α for y > 0. Thus, the even order moments of X are obtained as E X n = = = Bα, α Bα, α n! Bα, α 0 i=0 x n exp αx 1 + exp x α dx = Bα, α 1 α 1 n 1 n z i i=0 0 z α+i 1 dz α i n! n+1 = α + i Bα, α τ 1n + 1, α, 0 n 1 n z α 1 z 1 + z α dz where the penutimate step foows by using equation in Gradshteyn and Ryzhik [8].

11 A Generaized Skew Logistic Distribution 37 If n is odd then, using the tripe series representation, 1.5, one obtains α 1 α + i α E X n 1 i i j k = B α, α k=0 { 0 x n exp x λ + j + k + α dx + x n exp x λj + k + α } dx 0 α 1 α + i α n! 1 i i j k = B α, α { k=0 } 1 λj + k + α n+1 1 n+1 λ + j + k + α α 1 α + i n! 1 i i j = B n + 1, α, λ, α, α where n + 1, α, λ = τ n + 1, j, α, λ τ n + 1, + j, α, λ. and Using these, the first four moments of X can be obtained as α 1 α + i 1 i i j EX = B, α, λ, α, α EX = EX 3 = EX 4 = 4 Bα, α τ 13, α, α 1 1 i B α, α 48 Bα, α τ 15, α. α + i j 1 i 4, α, λ, Using the above moments, we can cacuate the four measures EX, V arx, SkewnessX and KurtosisX. Figures 3 to 6 iustrate the behavior of the four measures for λ = 10,...,10 and α = 1,, 5. From these figures, we see that: i EX increases with increasing λ; ii EX decreases with increasing α; iii V arx decreases with increasing λ ;

12 38 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih iv V arx decreases with increasing α; v SkewnessX increases with increasing λ; vi SkewnessX decreases with increasing α; vii KurtosisX initiay decreases before increasing with increasing λ ; viii KurtosisX decreases with increasing α. Figure 3: Pot of EX. Figure 4: Pot of V ariancex.

13 A Generaized Skew Logistic Distribution 39 Figure 5: Pot of SkewnessX. Figure 6: Pot of KurtosisX.

14 330 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih 4. ESTIMATION Let us first consider a version of 1.3 with the ocation parameter µ R and scae parameter σ > 0, i.e., fx; µ, σ, α, λ = [ ] x µ x µ 4.1 σ g α G α λ σ σ for < x <, α > 0 and λ R. In this section, we consider estimation of the parameters µ, σ, α and λ and provide expressions for the Fisher information matrix. The og-ikeihood for a random sampe x 1,..., x n from 4.1 is: 4. 1 exp x 1+exp x = nlµ, σ, α, λ = n nσ + n ng α y i + i=1 n ζ 0 λy i, where y i = x i µ σ and ζ 0 x = n{g α x}. We aso define the derivative ζ m x = d m ζ 0 x/dx m, m = 1,, 3,... and note that ζ 1 x = g α x/g α x. A subsequent derivatives can be expressed as functions of ζ 1 x; in particuar, ζ x = α ζ 1 x ζ 1 x. By differentiating 4. with respect to µ, σ, α and λ, and equating the derivatives to zero, the maximum ikeihood estimators are the simutaneous soutions of n exp y i n 4.3 α 1 + exp y i + λ ζ 1 λy i = n α, i=1 i=1 i=1 4.4 n n y i 1 exp yi n + λ y i ζ 1 λy i = α 1 + exp y i i=1 i=1, n n y i + n { 1+ exp y i } n n { G α λy i } 4.5 = n Ψα Ψα α i=1 i=1 i=1 and n 4.6 y i ζ 1 λy i = 0, i=1 where Ψx = nγx/dx is the digamma function. In 4.5, we have n { G α λy } α = Ψα Ψα λy λy tg α tdt G α λy n 1 + exp t g α tdt. G α λy

15 A Generaized Skew Logistic Distribution 331 The maximum ikeihood estimators µ, σ, λ, α of µ, σ, λ, α are consistent estimators, and n µ µ, σ σ, λ λ, α α is asymptoticay norma with zero means and variance covariance matrix I 1, where E E E E µ µ σ µ λ µ α I = 1 E E σ µ E E σ σ λ σ α n. E E E E E λ µ α µ E λ σ α σ E λ α λ λ α E α Now, we compute the Fisher information matrix based on the ikeihood equations. These enabe, for exampe, construction of confidence intervas based on pivota quantities using the imiting norma distribution. For simpicity, et us consider interva estimation of µ, σ, λ when α is known. In this case, the eements of the Fisher information matrix can be written as E = αn µ σ I 1 + nλ σ I, E = n σ σ + nα 4nα EZ σ σ I 3 + nα σ I 4 + nλ σ a λ, E = na λ, λ E = n µ λ σ I 5 nαλ σ I 6 nλ σ a 1λ, E = nαλ σ λ σ I 6 nλ σ a 1λ, E = nα µ σ σ + nα σ I 1 nα σ I 7 nλ σ I 5 nλ α σ I 6 nλ σ a 1λ, E = 4n Ψ1, α Ψ1, α ni 8 4nI 9 4nI 10 α ni nI 1 + 4nI 13, E = n α µ σ I 17 nλ σ I 18 + nλ σ I 19 + nλ σ I 0, E = n α σ σ I 1 nλ Ψα Ψα I5 nλ σ σ a 1λ nλ σ I 3 + nλ σ I + nλ σ I 16, E = ni 14 ni 15 ni 16, α λ

16 33 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih where I 1 = E exp Z, I = E ζ 1 + exp Z 1λZ Z exp Z, I 3 = E, 1 + exp Z Z exp Z I 4 = E { }, I 5 = E ζ 1 λz, 1 + exp Z Z 1 exp λz ζ1 λz Z exp Z I 6 = E, I 7 = E { } 1 + exp λz, 1 + exp Z b λz I 8 = E, I 9 = E G α λz b I 11 = E 1 λz G, I 1 = E αλz c11 λz, I 10 = E G α λz c 01 λz G αλz I 14 = E Zζ 1 λz n 1 + exp λz,, I 13 = E c0 λz, G α λz b1 λzc 01 λz G αλz I 15 = E Zζ 1 λz b 1λZ, G α λz I 16 = E Zζ 1 λz c 01λZ 1 exp Z, I 17 = E, G α λz 1 + exp Z I 18 = E ζ 1 λz n 1 + exp λz, I 19 = E ζ 1 λz b 1λZ, G α λz I 0 = E ζ 1 λz c 01λZ, I 1 = E Z exp Z 1, G α λz exp Z + 1 I = E Zζ 1 λz b 1 λz, I 3 = E Zζ 1 λz n 1 + exp Z,, where Z = X µ/σ, a kh λ = E λ { Z k ζ h 1 λz }, b k x = x t k g α tdt, Ψn, x = dn d n x Ψx and c kh x = x t k n h{ 1 + exp t } g α tdt. Note that a k1 λ = 0 when k is odd, and that a kh λ 0 when both k and h are even. Aso EhZζ 1 λz = 0 when hx is an odd function and EhZζ 1 λz 0 when hx is a even function. In genera, these expectations wi have to be computed numericay. However, cosed-form expressions are possibe in some particuar cases.

17 A Generaized Skew Logistic Distribution REAL DATA APPLICATIONS In this section, we fit the genera skew ogistic GSLµ, σ, λ, α distribution to two rea data sets. We compare the fits with those of the usua ogistic distribution Lµ, σ, the type III generaized ogistic distribution GLµ, σ, α, the skew ogistic distribution SLµ, σ, λ, Azzaini s [] skew norma distribution SNµ, σ, λ, and Azzaini and Capitanio s [5] skew t distribution STµ, σ, λ, α. The parameter λ in the skew norma and skew t distributions is the skewness parameter. The parameter α in the skew t distribution is the degree of freedom parameter. As with Azzaini s [] skew norma distribution, Azzaini and Capitanio s [5] skew t distribution has been studied by many authors. Two most recent papers are Areano-Vae and Azzaini [1] and Azzaini and Areano-Vae [4]. Exampe 1. The first data set represents the strength data originay reported in Badar and Priest [6]. It represents the strength measured in GPA for singe carbon fibers and impregnated 1000-carbon fiber tows. Singe fibers were tested under tension at gauge ength of 10mm. This data have been anayzed previousy by Raqab and Kundu [13] and Gupta and Kundu [9]. The data are as foows: We fitted a six distributions to the above data by the method of maximum ikeihood. The GSL distribution was fitted by soving Tabe 1 presents the parameter estimates, the og ikeihoods LL, the Komogorov Smirnov K-S statistics and respective p-vaues. Tabe presents the chi-squared statistics with observed and expected frequencies. Note that the ast two coumns of Tabes 1 and appear identica. This can be expained by the we-known fact that the ST distribution reduces to the SN distribution as α approaches infinity. Note aso that the α for the GSL distribution is very arge. Some eementary cacuations show that and g λ x G λ x I{x 0} 1 4 α I{x = 0} Bα, α

18 334 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih as α. So, the pdf of the GSL distribution in 1.3 reduces to as α. fx; α, λ 1 α I{x = 0} I{λx 0} Bα, α Tabe 1: MLEs, og-ikeihoods, Komogorov Smirnov statistics and corresponding p-vaues for Exampe 1. Distribution Lµ, σ SLµ, σ, λ GLµ, σ, α GSLµ, σ, λ, α SNµ, σ, λ STµ, σ, λ, α µ σ λ α Log-ikeihood KSS p-vaue Tabe : Observed and expected frequencies and chi-squared statistics for Exampe 1. Intervas Observed Lµ, σ SLµ, σ, λ GLµ, σ, α GSLµ, σ, λ, α SNµ, σ, λ STµ, σ, λ, α < > χ =0.883 χ = χ = χ =0.68 χ =0.684 χ =0.684 From Tabes 1 and, we see that the GSL distribution provides a better fit for the data than the other five distributions. The GSL distribution takes the smaest chi-squared statistic, the smaest K-S statistic, and the argest p-vaue. The SN and ST distributions take the second smaest chi-squared statistic, the second smaest K-S statistic, and the second argest p-vaue. The argest ogikeihood of is shared by the GSL, SN and ST distributions. Because of this, one can argue that the SN distribution is a competitor to the GSL distribution or perhaps that the SN distribution is a better choice than the GSL distribution since the former has one ess parameter. Figure 7 pots the fitted pdfs on top of the empirica histogram of the data. Figure 8 pots the fitted cdfs on top of the empirica cdf of the data. Both these figures support concusions based on Tabes 1 and. In both these figures, the

19 A Generaized Skew Logistic Distribution 335 fitted pdfs for the GSL, SN and ST distributions appear amost indistinguishabe. Both figures suggest that the GSL distribution captures the tais of the data better than most other distributions. Fitted PDFs Logistic Skew ogistic Gen ogistic GSL Skew norma Skew t Figure 7: Histogram of the first data set and the fitted pdfs. Fitted CDFs Logistic Skew ogistic Gen ogistic GSL Skew norma Skew t Figure 8: Empirica cdf of the first data set and the fitted cdfs.

20 336 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih Exampe. Here, we anayze the ean body mass of Austraian athetes. The data given in Cook and Weisberg [7] are as foows: We fitted a six distributions to the above data by the method of maximum ikeihood. Tabe 3 presents the parameter estimates, the og ikeihoods, the Komogorov Smirnov statistics and respective p-vaues. The corresponding chi-squared statistics with observed and expected frequencies are reported in Tabe 4. Tabe 3: MLEs, og-ikeihoods, Komogorov Smirnov statistics and corresponding p-vaues for Exampe. Distribution Lµ, σ SLµ, σ, λ GLµ, σ, α GSLµ, σ, λ, α SNµ, σ, λ STµ, σ, λ, α µ σ λ α Log-ikeihood KSS p-vaue Tabe 4: Observed and expected frequencies and chi-squared statistics for Exampe. Intervas Observed Lµ, σ SLµ, σ, λ GLµ, σ, α GSLµ, σ, λ, α SNµ, σ, λ STµ, σ, λ, α < > χ = χ = χ = χ = χ =3.157 χ =1.74 From Tabes 3 and 4, we can see that the GSL distribution takes the argest og ikeihood, the smaest chi-squared statistic, the smaest K-S statistic, and the argest p-vaue. The GL distribution takes the second argest og ikeihood,

21 A Generaized Skew Logistic Distribution 337 the second smaest chi-squared statistic, the second smaest K-S statistic, and the second argest p-vaue. The SN distribution takes the smaest og ikeihood, the argest chi-squared statistic, the argest K-S statistic, and the smaest p-vaue. Figures 9 and 10 pot the fitted pdfs and fitted cdfs, respectivey. Both these figures support concusions based on Tabes 3 and 4. Both figures suggest that the GSL distribution captures the midde part of the data better than most other distributions. Fitted PDFs Logistic Skew ogistic Gen ogistic GSL Skew norma Skew t Figure 9: Histogram of the second data set and the fitted pdfs. Fitted CDFs Logistic Skew ogistic Gen ogistic GSL Skew norma Skew t Figure 10: Empirica cdf of the second data set and the fitted cdfs.

22 338 A. Asgharzadeh, L. Esmaeii, S. Nadarajah and S.H. Shih ACKNOWLEDGMENTS The authors woud ike to thank the Editor and the referee for carefu reading and for their comments which greaty improved the paper. REFERENCES [1] Areano-Vae, R.B. and Azzaini, A The centred parameterization and reated quantities of the skew-t distribution, Journa of Mutivariate Anaysis, 113, [] Azzaini, A A cass of distributions which incudes the norma ones, Scandinavian Journa of Statistics, 1, [3] Azzaini, A Further resuts on a cass of distributions which incudes the norma ones, Statistica, 46, [4] Azzaini, A. and Areano-Vae, R.B Maximum penaized ikeihood estimation for skew-norma and skew-t distributions, Journa of Statistica Panning and Inference, 143, [5] Azzaini, A. and Capitanio, A Distributions generated by perturbation of symmetry with emphasis on a mutivariate skew t distribution, Journa of the Roya Statistica Society, B, 65, [6] Badar, M.G. and Priest, A.M Statistica aspects of fiber and bunde strength in hybrid composites. In Progress in Science and Engineering Composites T. Hayashi, K. Kawata and S. Umekawa, Eds., ICCM-IV, Tokyo, pp [7] Cook, R.D. and Weisberg, S Regression Graphics, John Wiey and Sons, New York. [8] Gradshteyn, I.S. and Ryzhik, I.M Tabe of Integras, Series and Products, sixth edition, Academic Press, San Diego. [9] Gupta, R.D. and Kundu, D Generaized ogistic distributions, Journa of Appied Statistica Sciences, 18, [10] Johnson, N.L.; Kotz, S. and Baakrishnan, N Continuous Univariate Distributions, voume, Wiey, New York. [11] Koesser, W. and Kumar, N An adaptive test for the two-sampe scae probem based on U-statistics, Communications in Statistics Simuation and Computation, 39, [1] Nadarajah, S The skew ogistic distribution, AStA Advances in Statistica Anaysis, 93, [13] Raqab, M.Z. and Kundu, D Comparison of different estimators of PY < X for a scaed Burr type X distribution, Communications in Statistics Simuation and Computation, 34,