Some skew-symmetric distributions which include the bimodal ones Dexter O. Cahoy Department of Mathematics and Statistics College of Engineering and Science Louisiana Tech University Ruston, LA 71271
[email protected] Tel: +1 318 257 3529; Fax: +1 318 257 2182 Abstract The class of skew-symmetric distributions has received much attention in recent years. In this article, we introduce two distributions which can capture the skew-symmetric unimodal (e.g., skew-Laplace, skew-normal) and the skewsymmetric bimodal ones systematically. Their natural generalizations of the skew-Laplace and the skew-normal distributions provide greater flexibility in modeling real data distributions. These models also avoid the identifiability problems of using mixtures to fit bimodal data. The stochastic representations that provide the random number generation algorithms are presented. The explicit forms of the central moments indicated that the proposed distributions have wide ranges of the skewness and kurtosis measures. Keywords: skew-symmetric, Gaussian, skew-normal, M -Wright, Mittag-Leffler, skew-Laplace, Airy
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Introduction
Several extensions of the normal distribution to the skew-symmetric family have already been proposed. Some of these extensions are the skew-normal (Azzalini,
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1985), the Balakrishan skew-normal distribution, and its generalizations (Balakrishnan, 2002; Gupta and Gupta, 2004; Jamalizadeh et. al, 2008, 2009; Hasanalipour and Sharafi, 2012) to name a few. In this paper we introduce two families, which include not only the asymmetric unimodal distributions but also the skew-symmetric bimodal ones using an auxiliary function of Wright type. This auxiliary function has earned a few equivalent names and forms in the literature (see Piryatinska et. al, 2005; Meerschaert and Straka, 2013) but for the sake of clarity, we call it the M -Wright function (see Mainardi et. al, 2010). Recall that the one-sided M -Wright function takes the form ∞ X (−x)j , x ∈ R+ , 0 ≤ ν < 1, (1) Mν (x) = j!Γ (−νj + (1 − ν)) j=0 and satisfies the following integral relation (see Beghin and Orsingher, 2010): Z 1 n+1 e−βx xn Mν (x)dx, Eν,νn+1 (−β) = n! R+
(2)
where κ Eη,θ (ρ)
=
∞ X j=0
(κ)j ρj j!Γ(ηj + θ)
η, θ, κ, ρ ∈ C,
