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 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 $$W_\lambda\overset\text{d}= Z_\lambda/|X|$$, 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.

MSC:

 6e+06 Probability distributions: general theory

Keywords:

skew-normal; bivariate Cauchy

Zbl 0581.62014
Full Text:

References:

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