A new class of skew-Cauchy distributions

Abstract

We discuss here a new class of skew-Cauchy distributions, which is related to Azzalini's [1985. A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178] skew-normal distribution denoted by Z λ ∼ SN ( λ ) . A random variable W λ is said to have a skew-Cauchy distribution (denoted by SC ( λ ) ) with parameter λ ∈ R if W λ = d Z λ / | X | , where Z λ ∼ SN ( λ ) and X ∼ N ( 0 , 1 ) are independent. In this paper, we discuss some simple properties of W λ , such as its density, distribution function, quantiles and a measure of skewness. Next, a bivariate Cauchy distribution is introduced using which some representations and important characteristics of W λ are presented.