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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 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:

60E05 Probability distributions: general theory

Citations:

Zbl 0581.62014
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References:

[1] Arnold, B. C.; Beaver, R. J., The skew-Cauchy distribution, Statist. Probab. Lett., 49, 285-290 (2000) · Zbl 0969.62037
[2] Arnold, B. C.; Beaver, R. J., The skew multivariate models related to hidden truncation and/or selective reporting, Test, 11, 7-54 (2002) · Zbl 1033.62013
[3] Azzalini, A., A class of distributions which includes the normal ones, Scand. J. Statist., 12, 171-178 (1985) · Zbl 0581.62014
[4] Azzalini, A., Further results on a class of distributions which includes the normal ones, Statistica, 46, 199-208 (1986) · Zbl 0606.62013
[5] Azzalini, A.; Dalla Valle, A., The multivariate skew-normal distribution, Biometrika, 74, 715-729 (1996) · Zbl 0885.62062
[6] Fang, K. T.; Kotz, S.; Ng, K. W., Symmetric Multivariate and Related Distributions (1990), Chapman & Hall: Chapman & Hall London · Zbl 0699.62048
[7] Genton, M. G.; Loperfido, N. M.R., Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math., 57, 389 (2005) · Zbl 1083.62043
[8] Henze, N. A., A probabilistic representation of the skew-normal distribution, Scand. J. Statist., 13, 271-275 (1986) · Zbl 0648.62016
[9] Johnson, N.L., Kotz, S., Balakrishnan, N., 1994. Continuous Univariate Distributions, second ed., vol. 1. Wiley, New York.; Johnson, N.L., Kotz, S., Balakrishnan, N., 1994. Continuous Univariate Distributions, second ed., vol. 1. Wiley, New York. · Zbl 0811.62001
[10] Loperfido, N. M.R., Quadratic forms of skew-normal random vectors, Statist. Probab. Lett., 54, 381-387 (2001) · Zbl 1002.62039
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