ALPHA-SKEW-NORMAL DISTRIBUTION

David Elal-Olivero

doi:10.4067/s0716-09172010000300006

Abstract

The main object of this paper is to introduce an alternative form of generate asymmetry in the normal distribution that allows to fit unimodal and bimodal data sets. Basic properties of this new distribution, such as stochastic representation, moments, maximum likelihood and the singularity of the Fisher information matrix are studied. The methodology developed is illustrated with a real application.

Highlights

The univariate skew-normal (SN) distribution has been studied by Azzalini (1985, 1986), Henze (1986), Pewsey (2000), and others, and synthetized in the book edited by Genton (2004)
In the last years, different families of skew-symmetric distributions have been generated, some of which are related with the SN model introduced by Azzalini (1985). Examples of these families are those considered by Arellano-Valle et al (2004) and Gomez et al (2006). Most of those classes include the normal distribution as a particular case and satisfy similar properties as the normal family
Mudholkar and Hutson (2000) proposed an asymmetric normal family of distributions with a different structure of the SN class considered by Azzalini (1985), which is called epsilon-skew-normal (ESN) and is denoted {ESN ( ) : | | < 1} where represents the asymmetry parameter, so that ESN (0) corresponds to the normal distribution

Summary

Introduction

The univariate skew-normal (SN) distribution has been studied by Azzalini (1985, 1986), Henze (1986), Pewsey (2000), and others, and synthetized in the book edited by Genton (2004). In the last years, different families of skew-symmetric distributions have been generated, some of which are related with the SN model introduced by Azzalini (1985) Examples of these families are those considered by Arellano-Valle et al (2004) and Gomez et al (2006). Many data sets arising in practice can be adequately modeled in this way and so the proposal plays a unifying role in this context This new family is called alpha-skew-normal (ASN) and is denoted by {ASN (α) : α ∈ R} where α represents the asymmetric parameter with effect of uni-bimodality, so that ASN (0) corresponds to the normal distribution. Density and accumulative function of the standard normal distribution will be expressed as φ(·) and Φ(·) respectively

Alpha-Skew-Normal Distribution

Stochastic Representation

Maximum likelihood estimation

An illustrative application