Abstract
The relationships among normal hidden truncation models, closed skew normal families, fundamental skew normal families and extended skew normal families are explored. The models of Arnold and Beaver in (Test 11(1):7–54, 2002) include all of these absolutely continuous models. Slightly more general absolutely continuous models are available with the label of selection models in Arellano-Valle and Genton (Journal of Multivariate Analysis 96:93–116, 2005). The hidden truncation paradigm provides a convenient description of models that subsumes the full spectrum of these skewed models, including singular and absolutely continuous versions.
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References
R. B. Arellano-Valle and A. Azzalini, “On the unification of families of skew-normal distributions,” Scandinavian Journal of Statistics vol. 33 pp. 561–574, 2006.
R. B. Arellano-Valle and M. G. Genton, “On fundamental skew distributions,” Journal of Multivariate Analysis vol. 96 pp. 93–116, 2005.
R. B. Arellano-Valle, M. D. Branco and M. G. Genton, “A unified view on skewed distributions arising from selections,” Canadian Journal of Statistics vol. 31 pp. 129–150, 2006.
B. C. Arnold, R. J. Beaver, R. A. Groeneveld, and W. Q. Meeker, “The nontruncated marginal of a truncated bivariate normal distribution,” Psychometrika vol. 58 pp. 471–478, 1993.
B. C. Arnold and R. J. Beaver, “Hidden truncation models,” Sankhȳa vol. 62 pp. 22–35, 2000a.
B. C. Arnold and R. J. Beaver, “The skew-Cauchy distribution,” Statistics and Probability Letters vol. 49 pp. 285–290, 2000b.
B. C. Arnold and R. J. Beaver, “Some skewed multivariate distributions,” American Journal of Mathematical and Management Sciences vol. 20 pp. 27–38, 2000c.
B. C. Arnold and R. J. Beaver, “Skewed multivariate models related to hidden truncation and/or selective reporting,” Test vol. 11(1) pp. 7–54, 2002.
B. C. Arnold and R. J. Beaver, “Alternative constructions of skewed multivariate distributions,” Acta et Commentationes Universitatis Tartuensis de Mathematica vol. 8 pp. 1–9, 2004.
A. Azzalini, “A class of distributions that includes the normal ones,” Scandinavian Journal of Statistics vol. 12 pp. 171–178, 1985.
A. Azzalini and M. Chiogna, “Some results on the stress-strength model for skew-normal variates,” Metron vol. LXII pp. 315–326, 2004.
A. Azzalini and A. Dalla Valle, “The multivariate skew-normal distribution,” Biometrika vol. 83 pp. 715–726, 1996.
N. Balakrishnan and R. S. Ambagaspitiya, “On skew-Laplace distributions,” Technical Report, Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada, 1994.
C. Crocetta and N. Loperfido, “On the exact sampling distribution of L-statistics,” Quaderni DSEMS 06-2003, Dipartimento di Scienze Economiche, Mathematiche e Statistiche, Universita di Foggia, 2003a.
C. Crocetta and N. Loperfido, “Sampling distribution of the Gini index from a skew normal,” Quaderni, DSEMS 07-2003, Dipartimento di Scienze Economiche, Mathematiche e Statistiche, Universita di Foggia, 2003b.
G. Gonzales-Farias, J. A. Dominguez-Molina, and A. K. Gupta, “The closed skew-normal distribution.” In M. G. Genton (ed.), Skew-elliptical Distributions and Their Applications, chap. 2 pp. 25–42, Chapman and Hall: London, 2004.
C. G. Gupta and N. Brown, “Reliability studies of the skew-normal distribution and its application to a strength-stress model,” Communications in Statistics-Theory and Methods vol. 30(11) pp. 2427–2445, 2001.
N. Henze, “A probabilistic representation of the ’skew-normal’ distribution,” Scandinavian Journal of Statistics vol. 13 pp. 271–275, 1986.
N. Loperfido, “Statistical implications of selectively reported inferential results,” Statistics and Probability Letters vol. 56 pp. 13–22, 2002.
T. Miwa, A. J. Hayter, and S. Kiriki, “The evaluation of general non-centered orthant probabilites,” Journal of the Royal Statistical Society, B vol. 65 pp. 223–234, 2003.
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Arnold, B.C., Beaver, R.J. Skewing Around: Relationships Among Classes of Skewed Distributions. Methodol Comput Appl Probab 9, 153–162 (2007). https://doi.org/10.1007/s11009-007-9028-4
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DOI: https://doi.org/10.1007/s11009-007-9028-4