A doubly skewed normal distribution

Abstract

We consider a distribution obtained by combining two well-known mechanisms for generating skewed distributions. In this manner we arrive at a flexible model which subsumes and extends several skew distributions which have been discussed in the literature. One approach to the problem of generating skewed distributions was first popularized by Azzalini [A class of distributions which includes the normal ones. Scand J Stat. 1985;12:171–178]. The single constraint skew normal distribution that was studied by Azzalini is of the form fSN(x)=2φ(x)Φ(αx), where φ and Φ denote, respectively, the standard normal density and distribution function and α∈ℝ is a skewing parameter. Multiple constraint variations of this distribution have also been considered. The second skewing approach that we will consider was proposed by Mudholkar and Hutson [The epsilon-skew-normal distribution for analyzing near-normal data. J Statist Plann Inference. 2000;83:291–309] and called an epsilon-skew-normal distribution. The combination of an Azzalini mechanism with that of Mudholkar and Hutson is investigated in this paper with special focus on the distributions obtained using the standard normal as the base distribution. The resulting flexible model includes both unimodal and bimodal cases and can be expected to fit a wider variety of data configurations than either of the models involving a single skewing mechanism. Distributional and inferential properties of the doubly skewed model are discussed and the model is used to obtain improved fits to two well-known data sets.