A NOTE ON GENERALIZED ALPHA-SKEW-NORMAL DISTRIBUTION. A.H. Handam Department of Mathematics Al Al-Bayt University P.O. Box , Al Mafraq, JORDAN

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1 International Journal of Pure and Applied Mathematics Volume 74 No , ISSN: printed version url: PA ijpam.eu A NOTE ON GENERALIZED ALPHA-SKEW-NORMAL DISTRIBUTION A.H. Handam Department of Mathematics Al Al-Bayt University P.O. Box , Al Mafraq, JORDAN Abstract: The alpha-skew-normal distributions is suggested by Elal-Olivero in [4]. In this paper, we modified this distribution to a generalized alpha-skewnormal distribution. Some properties of GASN are investigated. AMS Subject Classification: 60E05, 44A60 Key Words: alpha-skew-normal distribution, generalized alpha-skew-normal distribution, moment generating function 1. Introduction Azzalini [1] defined the skew-normal distribution for a random variable Z with the parameter λ be: gz = 2φzΦλz < z <, λ R, where φ and Φ are the standard normal density and distribution function, respectively. The term skew normal SN refers to a parametric class of probability distributions that extend the normal distribution by an additional shape parameter that regulates the skewness, allowing for a continuous variation from normality to non-normality. Henze [5] gave a stochastic representation for this distribution, obtaining explicitly its odd moments. Also Azzalini [2] discussed stochastic representations in a more general context. On the other hand, there has been a number of works exploring bimodality arising from skew distributions, see [7], [3], [8]. Received: November 3, 2011 c 2012 Academic Publications, Ltd. url:

2 492 A.H. Handam In[4], Elal-Olivero introduced a new family called alpha-skew-normalasn and it is denoted by {ASNα : α R} where α represents the asymmetric parameter with effect of uni-bimodality. This family of distributions is flexible enough to support both unimodal and bimodal shape. In this paper, we introduce a new class of skew normal distributions, which is a generaliztion of alpha-skew-normal distribution. This new class is called generalized alphaskew-normal GASN and is denoted by {GASNα : α R} where α represents the asymmetric parameter with effect of uni-bimodality, so that GASN0 corresponds to the normal distribution. 2. Generalized Alpha-Skew-Normal Distribution In this section, we present the definition and some simple properties of GASNα. Definition 2.1. see [4] If a random variable X has density function, fx α αx2 1 2α 2 φx, x R, where α R, then we say that X is a alpha-skew-normal random variable with parameter α. We denote this as X ASNα. Definition 2.2. We say that a random variable X has the generalized alpha-skew-normal distribution if its density is given by fx α αx 1 φx, x R, where α R, n N {0} and = 2 n this by X GASN α. α l 2j 1. We denote It is easy to show that the GASN satisfies conditions of the pdf. If X GASN α, the following properties are satisfied: 1 GASN 0 = N0,1. 2 If α ±, then X d 3 X GASN α. 1 n 2j 1 x φx. 4 if n, then GASN α = ASN α. Proposition 2.3. Let Z be the standard normal distribution, then

3 A NOTE ON GENERALIZED E Z 2k = k 2j 1 and E X 2k 1 = 0, k,2,3,... Theorem 2.4. If X GASN α, then for k = 0,1,2,..., we have E X 2k 2 k 2j 1 kl α 2j 1 and E X 2k1 kl 1 α 1 2j 1. 1 Proof. E X 2k x 2k 2 x 2k 1 αx 1 φxdx x 2k 2 = 2E Z 2k k 2 2j 1 on the other hand E X 2k 1 2 m=1 αx α E Z 2k x 2k 1 2 α kl αx m φxdx m αx 1 φxdx 1 1 α 1 E Z 2k 1 2j 1 by Proposition 2.3, x 2k 1 1 αx 1 φxdx αx αx 1 φxdx 1

4 494 A.H. Handam = 2E Z 2k 1 α E Z 2k 1 1 α 1 kl 1 1 α 1 E Z 2k 2 2j 1 by Proposition 2.3. Proposition 2.5. [6]. If W is a normal distribution with mean µ and variance σ 2, then E [ W k] = σ k k1/2 i=1 k!µ 2i 1 2i 1![k1/2 i]!2 k1/2 i σ 2i 1, k,3,5,... σ k k/2 i=0 k!µ 2i 2i!k/2 i!2 k/2 i σ 2i, k = 2,4,6,... Theorem 2.6. If M X t is the moment generating function of X GASN α, then [ M X t = et2 /2 2 α P,t ] α 1 Q 1,t 1 where and Qk,t = Pk,t = k1/2 i=1 k/2 i=0 k!t 2i 2i!k/2 i!2 k/2 i k!t 2i 1 2i 1![k 1/2 i]!2 k1/2 i. Proof. M X t = E e tx e tx 2 e tx 1 αx 1 φxdx αx αx 1 φxdx 1

5 A NOTE ON GENERALIZED = et2 /2 2 = e t2 /2 2 αx α E Y αx 1 1 e 1 2 x t2 dx 1 2π 1 α 1 E Y 1 [ = et2 /2 2 α P,t where Y N t,1 ] α 1 Q 1,t 1 by Proposition 2.5, where and Qk,t = Pk,t = k1/2 i=1 k/2 i=0 k!t 2i 2i!k/2 i!2 k/2 i k!t 2i 1 2i 1![k 1/2 i]!2 k1/2 i. References [1] A. Azzalini, Class of distributions which includes the normal ones, Scandinavian Journal of Statistics, , [2] A. Azzalini, Further results on a class of distributions which includes the normal ones, Statistica, , [3] A. Azzalini, A. Capitonio, Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution, Journal of the Royal Statistical Society, Series B, , [4] D. Elal-Olivero, Alpha-skew-normal distribution, Proyecciones, , [5] N. Henze, A probabilistic representation of the skew-normal distribution, Scandinavian Journal of Statistics, ,

6 496 A.H. Handam [6] K. Krishnamoorthy, Handbook of Statistical Distributions with Applications, Chapman & Hall/CRC Press [7] Y.Ma, M.G. Genton, Flexible class of skew-symmetric distribution, Scandinavian Journal of Statistics, , [8] H.S. Salinas, R.B. Arellano-Valle, H.W. Gómez, The extended skewexponencial power distribution and its derivations, Communications in Statistics-Theory and Methods, ,