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Extended generalized skew-elliptical distributions and their moments. (English) Zbl 1373.60030

Summary: This paper constructs a family of multivariate distributions which extends the class of generalized skew-elliptical (GSE) distributions, introduced by A. Azzalini and A. Capitanio [J. R. Stat. Soc., Ser. B, Stat. Methodol. 65, No. 2, 367–389 (2003; Zbl 1065.62094)], and derives formulae for it’s characteristics; mean and covariance. The extended GSE family is particularly relevant whenever the need arises to model data by skewed distributions, as is the case in actuarial science, risk management and other branches of science. The paper also generalizes the skew-normal distribution in the sense of A. Azzalini and A. Dalla Valle [Biometrika 83, No. 4, 715–726 (1996; Zbl 0885.62062)]. Furthermore, for estimation purposes, the maximum likelihood equations are derived. Finally, a numerical examples is provided which demonstrates that the extended GSE distribution offers a better fit in comparison to the GSE distribution.

MSC:

60E05 Probability distributions: general theory
62E15 Exact distribution theory in statistics
62F10 Point estimation

Software:

sn; AnDarl
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References:

[1] Arellano-Valle, R. B., Branco, M. D., and Genton, M. G. (2006). A unified view on skewed distributions arising from selections. Canadian Journal of Statistics, 34, 581-601. · Zbl 1121.60009
[2] Arellano-Valle, R. B., and Genton, M. G. (2010a). Multivariate unified skew-elliptical distributions. Chilean Journal of Statistics, 1, 17-33. · Zbl 1213.62087
[3] Arellano-Valle, R. B., and Genton, M. G. (2010b). Multivariate extended skew-t distributions and related families. Metron, 68, 201-234. · Zbl 1301.62016
[4] Arnold, B. C., Beaver, R. J., (2002). Skewed multivariate models related to hidden truncation and/or selective reporting. Test, 11, 7-54. · Zbl 1033.62013
[5] Azzalini A, Dalla Valle A. (1996). The multivariate skew-normal distribution. Biometrika, 83, 715-726. · Zbl 0885.62062
[6] Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew-normal distribution. Journal of the Royal Statistical Society B, 61, 579-602. · Zbl 0924.62050
[7] Azzalini, A. and Capitanio, A. (2003). “Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution”. J. Roy. Statist. Soc. Ser. B, 65, 367-389. · Zbl 1065.62094
[8] Azzalini, A. (2005). The Skew-normal Distribution and Related Multivariate Families. Scandinavian Journal of Statistics, 32, 159-188. · Zbl 1091.62046
[9] Azzalini, A., and Capitanio, A. (2014). The Skew-Normal and Related Families. Institute of Mathematical Statistics Monographs. · Zbl 0924.62050
[10] Branco, Marcia D., and Dipak K. Dey. (2001). “A general class of multivariate skew-elliptical distributions.” Journal of Multivariate Analysis, 79, 99-113. · Zbl 0992.62047
[11] Canale, A. (2011). Statistical aspects of the scalar extended skew-normal distribution. Metron, 69, 279-295. · Zbl 1260.62009
[12] Exton, H. (1976). Multiple hypergeometric functions and applications. Chichester: Ellis Horwood. · Zbl 0337.33001
[13] Fang, K. T., Kotz, S. and Ng, K. W. (1987). “Symmetric multivariate and related distributions”. London: Chapman and Hall. · Zbl 0699.62048
[14] Genton, M. G. (Ed.). (2004). Skew-elliptical distributions and their applications: a journey beyond normality. CRC Press. · Zbl 1069.62045
[15] Genton, M. G., and Loperfido, N. M. (2005). Generalized skew-elliptical distributions and their quadratic forms. Annals of the Institute of Statistical Mathematics, 57, 389-401. · Zbl 1083.62043
[16] Kelker, D. (1970). Distribution theory of spherical distributions and location-scale parameter generalization. Sankhya, 32, 831-869. · Zbl 0223.60008
[17] Landsman, Z. (2006). On the generalization of Stein’s Lemma for elliptical class of distributions. Statistics & Probability Letters, 76, 1012-1016. · Zbl 1089.62055
[18] Landsman, Z., and Neslehova, J. (2008). Stein’s Lemma for elliptical random vectors. Journal of Multivariate Analysis, 99, 912-927. · Zbl 1286.62018
[19] Landsman, Z. M., and Valdez E. A. (2003). Tail Conditional Expectations for Elliptical Distributions. North American Actuarial Journal, 7, 55-71. · Zbl 1084.62512
[20] Marsaglia, G., and Marsaglia, J. (2004). Evaluating the anderson-darling distribution. Journal of Statistical Software, 9, 1-5.
[21] Nadarajah, S., and Kotz S. (2003). Skewed distributions generated by the normal kernel. Statistics & Probability Letters, 65, 269-277. · Zbl 1048.62014
[22] Nadarajah, S. (2005). Linear combination, product and ratio of normal and logistic random variables. Kybernetika, 41, 787-798. · Zbl 1243.62020
[23] Vernic, R. (2005). On the multivariate Skew-Normal distribution and its scale mixtures. An. St. Univ. Ovidius Constanta, 13, 83-96. · Zbl 1108.62052
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