A new class of skew-Cauchy distributions

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 Zλ∼SN(λ)Zλ∼SN(λ). A random variable WλWλ is said to have a skew-Cauchy distribution (denoted by SC(λ)SC(λ)) with parameter λ∈Rλ∈R if Wλ=dZλ/|X|, where Zλ∼SN(λ)Zλ∼SN(λ) and X∼N(0,1)X∼N(0,1) are independent. In this paper, we discuss some simple properties of WλWλ, 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 WλWλ are presented.