Some Observations on a Simple Means of Generating Skew Distributions

Abstract

During the last decade, a substantial part of Barry Arnold’s research effort has been directed towards developing models capable of describing the forms of asymmetry manifested by real data. One general and seemingly elegant means of constructing skew distributions is provided by a lemma presented in (1985). The now widely known skew-normal distribution is just one special case belonging to the family of distributions generated using the construction implicit in that lemma. In this paper, a simple alternative proof of the lemma is given, and reflections are made upon how the construction arising from it has been employed in the literature. The densities of various special cases are presented, which highlight both the flexibility and limitations of the construction. Likelihood-based inference for the parameters of the location-scale extensions of classes arising from the construction is also considered. General results are given for the solutions to the score equations and for the observed information matrix. For the special case of the skew-normal distribution, it is shown that, for one of the solutions to the score equations, the observed information matrix is always singular.