zbMATH — the first resource for mathematics

Skewed multivariate models related to hidden truncation and/or selective reporting. With discussion and a rejoinder by the authors. (English) Zbl 1033.62013
Summary: The univariate skew-normal distribution was introduced by A. Azzalini [Scand. J. Stat., Theory Appl. 12, 171–178 (1985; Zbl 0581.62014)] as a natural extension of the classical normal density to accommodate asymmetry. He extensively studied the properties of this distribution and, in conjunction with coauthors, extended this class to include the multivariate analog of the skew-normal. B. C. Arnold et al. [Psychometrika 58, 471–488 (1993; Zbl 0794.62075)] introduced a more general skew-normal distribution as the marginal distribution of a truncated bivariate normal distribution in which \(X\) was retained only if \(Y\) satisfied certain constraints. Using this approach, more general univariate and multivariate skewed distributions have been developed. A survey of such models is provided together with discussion of related inference questions.

62E10 Characterization and structure theory of statistical distributions
62H05 Characterization and structure theory for multivariate probability distributions; copulas
Full Text: DOI
[1] Arnold, B. C. andBeaver, R. J. (2000a). Hidden truncation models.Sankhya Series A, 62:23–35. · Zbl 0973.62041
[2] Arnold, B. C. andBeaver, R. J. (2000b). The skew-Cauchy distribution.Statistics and Probability Letters, 49:285–290. · Zbl 0969.62037
[3] Arnold, B. C. andBeaver, R. J. (2000c). Some skewed multivariate distributions.American Journal of Mathematical and Management Sciences, 20:27–38. · Zbl 1189.62087
[4] Arnold, B. C. andBeaver, R. J. (2002a). An alternative construction of skewed multivariate distributions. Technical Report 270, Department of Statistics, University of California, Riverside.
[5] Arnold, B. C. andBeaver, R. J. (2002b). Multivariate survival models incorporating hidden truncation. Technical Report. 269, Department of Statistics, University of California, Riverside. · Zbl 1137.62382
[6] Arnold, B. C., Beaver, R. J., Groeneveld, R. A., andMeeker, W. Q. (1993). The nontruncated marginal of a truncated bivariate normal distribution.Psychometrika, 58:471–188. · Zbl 0794.62075
[7] Arnold, B. C., Castillo, E., andSarabia, J. M. (2002). Conditionally specified multivariate skewed distributions.Sankhya Series A (to appear). · Zbl 1192.60039
[8] Azzalini, A. (1985). A class of distributions which includes the normal ones.Scandinavian Journal of Statistics, 12:171–17. · Zbl 0581.62014
[9] Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones.Statistica, 46:199–208. · Zbl 0606.62013
[10] Azzalini, A. andCapitanio, A. (1999). Statistical applications of the multivariate skew normal distribution.Journal of the Royal Statistical Society B, 61:579–602. · Zbl 0924.62050
[11] Azzalini, A. andDalla Valle, A. (1996). The multivariate skew-normal distribution.Biometrika, 83:715–726. · Zbl 0885.62062
[12] Balakrishnan, N. andAmbagaspitiya, R. S. (1994). On skew-Laplace distributions. Technical report, Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada.
[13] Birnbaum, Z. W. (1950). Effect of linear truncation on a multinormal population.The Annals of Mathematical Statitics, 21:272–279. · Zbl 0038.09201
[14] Branco, C. D. andDey, D. K. (2001). A general class of multivariate skew-elliptical distributions.Journal of Multivariate Analysis, 79:99–113. · Zbl 0992.62047
[15] Cain, M. (1994). The moment-generating function of the minimum of bivariate normal random variables.The American Statistician 48:124–125.
[16] Capitanio, A., Azzalini, A., andStanghellini, E. (2002). Graphical models for skew normal variates.Scandinavian Journal of Statistics, 29 (to appear). · Zbl 0924.62050
[17] Chiogna, M. (1998). Some results on the scalar skew-normal distribution.Journal of the Italian Statistical Society, 7:1–13.
[18] Genton, M. G., Li, H., andLiu, X. (2001). Moments of skew-normal random vectors and their quadratic forms.Statistics and Probability Letters, 51:319–325. · Zbl 0972.62031
[19] Gupta, A. K. andChen, T. (2001). Goodness-of-fit tests for the skewnormal distribution.Communications in Statistics, Part C: Computing and Simulation, 30:907–930. · Zbl 1008.62590
[20] Gupta, R. C. andBrown, N. (2001). Reliability studies of the skewnormal distribution and its application to a strength-stress model.Communication in Statistics: Theory and Methods, 30:2427–2445. · Zbl 1009.62513
[21] Henze, N. (1986). A probabilistic representation of the ’skew-normal’ distribution.Scandinavian Journal of Statistics, 13:271–275. · Zbl 0648.62016
[22] Loperfido, N. (2001). Quadratic forms of skew-normal random vectors.Statistics and Probability Letters, 54:381–387. · Zbl 1002.62039
[23] Loperfido, N. (2002). Statistical implications of selectively reported inferential results.Statistics and Probability Letters, 56:13–22. · Zbl 0994.62012
[24] Mukhopadhyay, S. andVidakovic, B. (1995). Efficiency of linear Bayes rules for a normal mean: skewed prior class.Journal of the Royal Statistical Society D, 44:469–471.
[25] Nelson, L. S. (1964). The sum of values from a normal and a truncated normal distribution.Technometrics, 6:469–470.
[26] O’Hagan, A. andLeonard, T. (1976). Bayes estimation subject to uncertainty about parameter constraints.Biometrika, 63:201–203. · Zbl 0326.62025
[27] Pewsey, A. (2000). Problems of inference for Azzalini’s skew-normal distribution.Journal of Applied Statistics, 27:859–870. · Zbl 1076.62514
[28] Roberts, C. (1966). A correlation model useful in the study of twins.Journal of the American Statistical Association, 61:1184–1190. · Zbl 0147.38001
[29] Weinstein, M. A. (1964). The sum of values from a normal and a truncated normal distribution.Technometrics, 6:104–105.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.