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A new class of skew-Cauchy distributions. (English) Zbl 1096.60011
Summary: We discuss here a new class of skew-Cauchy distributions, which is related to {\it A. Azzalini}’s skew-normal distribution [Scand. J. Stat., Theory Appl. 12, 171--178 (1985; Zbl 0581.62014)] denoted by $Z_\lambda\sim \text{SN}(\lambda)$. A random variable $W_\lambda$ is said to have a skew-Cauchy distribution (denoted by SC($\lambda$)) with parameter $\lambda\in R$ if $W_\lambda\overset\text{d}\to= Z_\lambda/|X|$, where $Z_\lambda\sim \text{SN}(\lambda)$ and $X\sim \text{N}(0,1)$ are independent. We discuss some simple properties of $W_\lambda$, 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 $W_\lambda$ are presented.

60E05General theory of probability distributions
Full Text: DOI
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