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Distributions generated by perturbation of symmetry with emphasis on a multivariate skew $$t$$-distribution. (English) Zbl 1065.62094
Summary: A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry and to generate a whole set of non-symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew $$t$$-density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H10 Multivariate distribution of statistics
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##### References:
 [1] Arnold B. C., Sankhya 62 pp 22– (2000) [2] DOI: 10.1016/S0167-7152(00)00059-6 · Zbl 0969.62037 [3] Arnold B. C., Test 11 pp 7– (2002) [4] Azzalini A., Scand. J. Statist. 12 pp 171– (1985) [5] Azzalini A., J. R. Statist. Soc. 61 pp 579– (1999) [6] Azzalini A., Biometrika 83 pp 715– (1996) [7] DOI: 10.1006/jmva.2000.1960 · Zbl 0992.62047 [8] Butler R. L., Rev. Econ. Statist. 72 pp 321– (1990) [9] Capitanio A., Scand. J. Statist. 30 (2003) [10] Cook R. D., An Introduction to Regression Graphics (1994) · Zbl 0925.62287 [11] David H. A., Order Statistics (1981) · Zbl 0553.62046 [12] Fang K.-T., Symmetric Multivariate and Related Distributions (1990) [13] Fernandez C., J. Am. Statist. Ass. 93 pp 359– (1998) [14] Fernandez C., Biometrika 86 pp 153– (1999) [15] Genton M. G., Statist. Probab. Lett. 51 pp 319– (2001) [16] M. G. Genton, and N. Loperfido (2002 ) Generalized skew-elliptical distributions and their quadratic forms.Mimeo 2539. Institute of Statistics, North Carolina State University, Raleigh. (Available fromhttp://www.stat.ncsu.edu/library/mimeo.html.) [17] Genz A., J. Statist. Computn Simuln 63 pp 361– (1999) · Zbl 0934.62020 [18] A. K. Gupta, G. Gonzales-Farias, and J. A. Dominguez-Molina (2001 ) A multivariate skew normal distribution .Report I-01-19. (Available fromhttp://www.cimat.mx/reportes.) [19] Healy M. J. R., Appl. Statist. 17 pp 157– (1968) [20] Jones M. C., Probability and Statistical Models with Applications: a Volume in Honor of Theophilos Cacoullos pp 269– (2001) [21] Jones M. C., Metrika 54 pp 215– (2002) [22] Jones M. C., J. R. Statist. Soc. 65 pp 159– (2003) [23] Kano Y., J. Multiv. Anal. 51 pp 139– (1994) [24] Loperfido N., Statist. Probab. Lett. 54 pp 381– (2001) [25] Loperfido N., Statist. Probab. Lett. 56 pp 13– (2002) [26] Roberts C., J. Am. Statist. Ass. 61 pp 1184– (1966) [27] Sahu S. K., Technical Report (2001) [28] Zuo Y., J. Statist. Planng Inf. 84 pp 55– (2000)
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