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A unified view on skewed distributions arising from selections. (English) Zbl 1121.60009
Summary: Parametric families of multivariate nonnormal distributions have received considerable attention in the past few decades. The authors propose a new definition of a selection distribution that encompasses many existing families of multivariate skewed distributions. Their work is motivated by examples that involve various forms of selection mechanisms and lead to skewed distributions. They give the main properties of selection distributions and show how various families of multivariate skewed distributions, such as the skew-normal and skew-elliptical distributions, arise as special cases. The authors further introduce several methods of constructing selection distributions based on linear and nonlinear selection mechanisms.

MSC:
60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
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[1] Arellano-Valle, On the unification of families of skew-normal distributions, Scandinavian Journal of Statistics 33 pp 561– (2006) · Zbl 1117.62051
[2] Arellano-Valle, Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality pp 113– (2004)
[3] Arellano-Valle, Definition and probabilistic properties of skew-distributions, Statistics & Probability Letters 58 pp 111– (2002) · Zbl 1045.62046
[4] Arellano-Valle, On fundamental skew distributions, Journal of Multivariate Analysis 96 pp 93– (2005) · Zbl 1073.62049
[5] R. B. Arellano-Valle & M. G. Genton (2006a). On the exact distribution of the maximum of dependent random variables. Statistics & Probability Letters, revised version submitted for publication. · Zbl 1134.60033
[6] R. B. Arellano-Valle & M. G. Genton (2006b). On the exact distribution of linear combinations of order statistics from dependent random variables. Journal of Multivariate Analysis, revised version submitted for publication. · Zbl 1139.62028
[7] R. B. Arellano-Valle, M. G. Genton, H. W. Gómez & P. Iglesias (2006). Shape mixtures of skewed distributions, with application in regression analysis. Submitted for publication.
[8] Arellano-Valle, A new class of skew-normal distributions, Communications in Statistics: Theory and Methods 33 pp 1465– (2004) · Zbl 1134.60304
[9] Arellano-Valle, Statistical inference for a general class of asymmetric distributions, Journal of Statistical Planning and Inference 128 pp 427– (2005) · Zbl 1095.62015
[10] Arnold, Hidden truncation models, Sankyã Series A 62 pp 22– (2000)
[11] Arnold, The skew-Cauchy distribution, Statistics & Probability Letters 49 pp 285– (2000) · Zbl 0969.62037
[12] Arnold, Skewed multivariate models related to hidden truncation and/or selective reporting, Test 11 pp 7– (2002) · Zbl 1033.62013
[13] Arnold, Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality pp 101– (2004)
[14] Arnold, The nontruncated marginal of a truncated bivariate normal distribution, Psychometrika 58 pp 471– (1993) · Zbl 0794.62075
[15] Arnold, Conditional Specification of Statistical Models (1999) · Zbl 0932.62001
[16] Arnold, Conditionally specified distributions: an introduction (with discussion), Statistical Science 16 pp 249– (2001)
[17] Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics 12 pp 171– (1985) · Zbl 0581.62014
[18] Azzalini, Further results on a class of distributions which includes the normal ones, Statistica 46 pp 199– (1986) · Zbl 0606.62013
[19] Azzalini, The skew-normal distribution and related multivariate families, Scandinavian Journal of Statistics 32 pp 159– (2005) · Zbl 1091.62046
[20] Azzalini, Statistical applications of the multivariate skew-normal distribution, Journal of the Royal Statistical Society Series B 61 pp 579– (1999) · Zbl 0924.62050
[21] Azzalini, Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution, Journal of the Royal Statistical Society Series B 65 pp 367– (2003) · Zbl 1065.62094
[22] Azzalini, The multivariate skew-normal distribution, Biometrika 83 pp 715– (1996) · Zbl 0885.62062
[23] Bayarri pp 17– (1992)
[24] J. L. Bázan, M. D. Branco & H. Bolfarine (2006). A skew item response model. Bayesian Analysis, in press. · Zbl 1331.62448
[25] Birnbaum, Effect of linear truncation on a multinormal population, Annals of Mathematical Statistics 21 pp 272– (1950) · Zbl 0038.09201
[26] Branco, A general class of multivariate skew-elliptical distributions, Journal of Multivariate Analysis 79 pp 99– (2001) · Zbl 0992.62047
[27] Branco, Regression model under skew elliptical error distribution, The Journal of Mathematical Sciences 1 pp 151– (2002) · Zbl 1076.62056
[28] Capitanio, Graphical models for skew-normal variates, Scandinavian Journal of Statistics 30 pp 129– (2003)
[29] Chen, Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality pp 131– (2004)
[30] Copas, Inference for non-random samples (with discussion), Journal of the Royal Statistical Society Series B 59 pp 55– (1997) · Zbl 0896.62003
[31] Crocetta, The exact sampling distribution of L-statistics, Metron 63 pp 1– (2005)
[32] DiCiccio, Inferential aspects of the skew exponential power distribution, Journal of the American Statistical Association 99 pp 439– (2004) · Zbl 1117.62318
[33] Fernández, On Bayesian modelling of fat tails and skewness, Journal of the American Statistical Association 93 pp 359– (1998) · Zbl 0910.62024
[34] Ferreira, Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality pp 175– (2004)
[35] M. G.Genton, ed. (2004a). Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality. Chapman & Hall/CRC, Boca Raton, Florida. · Zbl 1069.62045
[36] Genton, Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality pp 81– (2004) · Zbl 1069.62045 · doi:10.1201/9780203492000
[37] Genton, Discussion of: The skew-normal distribution and related multivariate families, by A. Azzalini, Scandinavian Journal of Statistics 32 pp 189– (2005)
[38] Genton, Generalized skew-elliptical distributions and their quadratic forms, Annals of the Institute of Statistical Mathematics 57 pp 389– (2005) · Zbl 1083.62043
[39] González-Farías, Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality pp 25– (2004)
[40] González-Farías, Additive properties of skew normal random vectors, Journal of Statistical Planning and Inference 126 pp 521– (2004)
[41] Gupta, A class of multivariate skew-normal models, Annals of the Institute Statistical Mathematics 56 pp 305– (2004) · Zbl 1056.62064
[42] Heekman, Sample selection bias as a specification error, Econometrica 47 pp 153– (1979)
[43] Johnson, Continuous Univariate Distributions 1 (1994)
[44] Liseo, A note on reference priors for the scalar skew-normal distribution, Journal of Statistical Planning and Inference 136 pp 373– (2006) · Zbl 1077.62017
[45] Loperfido, Statistical implications of selectively reported inferential results, Statistics & Probability Letters 56 pp 13– (2002) · Zbl 0994.62012
[46] Ma, A flexible class of skew-symmetric distributions, Scandinavian Journal of Statistics 31 pp 459– (2004) · Zbl 1063.62079
[47] Ma, Locally efficient semiparametric estimators for generalized skew-elliptical distributions, Journal of the American Statistical Association 100 pp 980– (2005) · Zbl 1117.62394
[48] Y. Ma & J. Hart (2006). Constrained local likelihood estimators for semiparametric skew-normal distributions. Biometrika, in press. · Zbl 1143.62019
[49] Malmquist, A study of the stars of spectral type A, Meddelande fràn Lunds Astronomiska Observatorium, Serie II 22 pp 1– (1920)
[50] Mudholkar, The epsilon-skew-normal distribution for analyzing near-normal data, Journal of Statistical Planning and Inference 83 pp 291– (2000) · Zbl 0943.62012
[51] Nelson, The sum of values from a normal and a truncated normal distribution, Technometrics 6 pp 469– (1964)
[52] O’Hagan, Bayes estimation subject to uncertainty about parameter constraints, Biometrika 63 pp 201– (1976)
[53] Pewsey, Problems of inference for Azzalini’s skew-normal distribution, Journal of Applied Statistics 27 pp 859– (2000) · Zbl 1076.62514
[54] Pewsey, Advances in Distribution Theory, Order Statistics and Inference pp 75– (2006)
[55] Rao, A Celebration of Statistics: The ISI Centenary Volume pp 543– (1985) · doi:10.1007/978-1-4613-8560-8_24
[56] Roberts, A correlation model useful in the study of twins, Journal of the American Statistical Association 61 pp 1184– (1966) · Zbl 0147.38001
[57] Sahu, A new class of multivariate skew distributions with application to Bayesian regression models, The Canadian Journal of Statistics 31 pp 129– (2003) · Zbl 1039.62047
[58] Sartori, Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions, Journal of Statistical Planning and Inference 136 pp 4259– (2006) · Zbl 1098.62023
[59] Wahed, The skew-logistic distribution, Journal of Statistical Research 35 pp 71– (2001)
[60] Wang, A skew-symmetric representation of multivariate distributions, Statistica Sinica 14 pp 1259– (2004) · Zbl 1060.62059
[61] Wang, A note on an equivalence between chi-square and generalized skew-normal distributions, Statistics & Probability Letters 66 pp 395– (2004) · Zbl 1075.62010
[62] Wang, The multivariate skew-slash distribution, Journal of Statistical Planning and Inference 136 pp 209– (2006) · Zbl 1081.60013
[63] Weinstein, The sum of values from a normal and a truncated normal distribution, Technometries 6 pp 104– (1964)
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