Abstract
This paper studies the case where the observations come from a unimodal and skew density function with an unknown mode. The skew-symmetric representation of such a density has a symmetric component which can be written as a scale mixture of uniform densities. A Dirichlet process (DP) prior is assigned to mixing distribution. We also assume prior distributions for the mode and the skewed component. A computational approach is used to obtain the Bayes estimate of the components. An example is given to illustrate the approach.
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References
Arellano-Valle RB, Bolfarine H, Lachos VH (2007) Bayesian inference for skew-normal linear mixed models. J Applied Stat 34(6): 663–682
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 2: 171–178
Azzalini A (1986) Further results on a class of distributions which includes the normal ones. Statistica 2: 199–208
Azzalini A (2005) The skew-normal distribution and related multivariate families (with discussion). Scand J Stat 32: 159–188 (C/R 189–200)
Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew normal distributions. J R Stat Soc B 61: 579–602
Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83: 715–726
Brunner LJ (1995) Bayesian linear regression with error terms that have symmetric unimodal densities. J Nonparametr Stat 4: 335–348
Chiogna M (1998) Some results on the scalar skew-normal distribution. J Ital Stat Soc 7: 1–13
Feller W (1971) An introduction to probability theory and its applications. Wiley, New York
Ferguson TS (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1: 209–230
Fruhwirth-Schnatter S, Pyne S (2010) Bayesian inference for finite mixtures of univariate and multivariate skew normal and skew-t distributions. Biostatistics 11: 317–336
Gelfand AE, Mukhopadhyay S (1995) On nonparametric Bayesian inference for the distribution of a random sample. Can J Stat 23: 411–420
Gelfand AE, Smith AFM (1990) Sampling-based approaches to calculating marginal densities. J Am Stat Assoc 85: 398–409
Genton MG, Loperfido N (2005) Generalized skew-elliptical distributions and their quadratic forms. Ann Inst Stat Math 57: 389–401
Gupta RC, Gupta RD (2004) Generalized skew normal model. Test 13: 501–524
Gupta AK, Nguyen TT, Sanqui JAT (2004) Characterization of the skew-normal distribution. Ann Inst Stat Math 56(2): 351–360
Henze N (1986) A probabilistic representation of the ‘skew-normal’ distribution. Scand J Stat 13: 271–275
Liseo B, Loperfido N (2003) A Bayesian interpretation of the multivariate skew-normal distribution. Stat Probab Lett 61: 395–401
Nadarajah S, Kotz S (2003) Skewed distributions generated by the normal kernel. Stat Probab Lett 65: 269–277
Neal RM (2000) Markow chain sampling methods for Dirichlet process mixture models. J Comput Graph Stat 9: 249–265
Pewsey A (2000) Problems of inference for Azzalini’s skew-normal distribution. J Appl Stat 27: 659–770
Roberts HV (1988) Data analysis for managers with Minitab. Scientific Press, Redwood City, CA
Sharafi M, Behboodian J (2008) The Balakrishnan skew-normal density. Stat Pap 49: 769–778
Walker SG, Damien P, Laud PW, Smith AFM (1999) Bayesian nonparametric inference for random distributions and related functions. J R Stat Soc B 61: 485–527
Wang T, Boyer J, Genton MG (2004) A skew-symmetric representation of multivariate distributions. Stat Sin 14: 1259–1270
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Ghalamfarsa Mostofi, A., Kharrati-Kopaei, M. Bayesian nonparametric inference for unimodal skew-symmetric distributions. Stat Papers 53, 821–832 (2012). https://doi.org/10.1007/s00362-011-0385-2
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DOI: https://doi.org/10.1007/s00362-011-0385-2