Abstract
Discrete mixture models are one of the most successful approaches for density estimation. Under a Bayesian nonparametric framework, Dirichlet process location–scale mixture of Gaussian kernels is the golden standard, both having nice theoretical properties and computational tractability. In this paper we explore the use of the skew-normal kernel, which can naturally accommodate several degrees of skewness by the use of a third parameter. The choice of this kernel function allows us to formulate nonparametric location–scale–shape mixture prior with desirable theoretical properties and good performance in different applications. Efficient Gibbs sampling algorithms are also discussed and the performance of the methods are tested through simulations and applications to galaxy velocity and fertility data. Extensions to accommodate discrete data are also discussed.
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References
Arellano-Valle RB, Azzalini A (2006) On the unification of families of skew-normal distributions. Scand J Stat 33(3):561–574
Arellano-Valle RB, Genton MG, Loschi RH (2009) Shape mixture of multivariate skew-normal distributions. J Multivar Anal 100:91–101
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178
Cabras S, Racugno W, Castellanos M, Ventura L (2012) A matching prior for the shape parameter of the skew-normal distribution. Scand J Stat 39:236–247
Canale A, Dunson DB (2011) Bayesian kernel mixtures for counts. J Am Stat Assoc 106(496):1528–1539
Canale A, Scarpa B (2013) Informative bayesian inference for the skew-normal distribution. arXiv:1305.3080
Cavatti Vieira C, Loschi RH, Duarte D (2013) Nonparametric mixtures based on skew-normal distributions: an application to density estimation. Commun Stat A Theor. doi:10.1080/03610926.2013.771745 (in press)
Chandola T, Coleman D, Hiorns RW (1999) Recent European fertility patterns: fitting curves to ‘distorted’ distributions. Pop Stud 53:317–329
Dunson DB, Pillai N, Park J-H (2007) Bayesian density regression. J R Stat Soc B Methodol 69:163–183
Efron B, Morris C (1972) Limiting the risk of Bayes and empirical Bayes estimators—part II: the empirical Bayes case. J Am Stat Assoc 67:130–139
Escobar MD, West M (1995) Bayesian density estimation and inference using mixtures. J Am Stat Assoc 90:577–588
Ferguson TS (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1:209–230
Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97:611–631
Freedman D, Diaconis P (1981) On the histogram as a density estimator: \(L_2\) theory. Probab Theory Relat 57:453–476
Frühwirth-Shnatter S, Pyne S (2010) Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-\(t\) distributions. Biostatistics 11:317–336
Ghosal S, Ghosh JK, Ramamoorthi RV (1999) Posterior consistency of Dirichlet mixtures in density estimation. Ann Stat 27(1):143–158
Gnedin A, Pitman J (2005) Exchangeable gibbs partitions and stirling triangles. Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 325:83–102
Kalli M, Griffin J, Walker S (2011) Slice sampling mixture models. Stat Comput 21(1):93–105
Lijoi A, Mena RH, Prünster I (2005) Hierarchical mixture modelling with normalized inverse Gaussian priors. J Am Stat Assoc 100:1278–1291
Lin TI, Lee JC, Yen SY (2007) Finite mixture modelling using the skew normal distribution. Stat Sin 17:909–927
Liseo B (1990) La classe delle densità normali sghembe: aspetti inferenziali da un punto di vista bayesiano. Statistica L:71–82
Liseo B, Loperfido N (2006) A note on reference priors for the scalar skew-normal distribution. J Stat Plan Infer 136:373–389
Lo AY (1984) On a class of Bayesian nonparametric estimates: I. Density estimates. Ann Stat 12:351–357
McCulloch RE (1989) Local model influence. J Am Stat Assoc 84:473–478
Pati D, Dunson DB, Tokdar ST (2013) Posterior consistency in conditional distribution estimation. J Multivar Anal 116:456–472
Peristera P, Kostaki A (2007) Modelling fertility in modern populations. Demogr Res 16:141–194
Perman M, Pitman J, Yor M (1992) Size-biased sampling of poisson point processes and excursions. Probab Theory Relat 92:21–39
Petralia F, Rao VA, Dunson DB (2012) Repulsive mixture. In: Bartlett P, Pereira F, Burges C, Bottou L, Weinberger K (eds) NIPS, vol 25, pp 1898–1906
Rodríguez CE, Walker SG (2014) Univariate Bayesian nonparametric mixture modeling with unimodal kernels. Stat Comput 24(1):35–49
Roeder K (1990) Density estimation with confidence sets emplified by superclusters and voids in galaxies. J Am Stat Assoc 85:617–624
Schwartz L (1965) On Bayes procedures. Z Wahrscheinlichkeit 4:10–26
Sethuraman J (1994) A constructive definition of Dirichlet priors. Stat Sin 4:639–650
Wu Y, Ghosal S (2008) Kullback Leibler property of kernel mixture priors in Bayesian density estimation. Electron J Stat 2:298–331
Acknowledgments
The authors thank Adelchi Azzalini for his shepherding into the skew world and Pierpaolo De Blasi, Stefano Mazzucco, and Igor Prünster for comments on early versions of the paper. The comments of two anonymous referees and of the Associate Editor are gratefully acknowledged.
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Canale, A., Scarpa, B. Bayesian nonparametric location–scale–shape mixtures. TEST 25, 113–130 (2016). https://doi.org/10.1007/s11749-015-0446-2
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DOI: https://doi.org/10.1007/s11749-015-0446-2
Keywords
- Discrete random probability measures
- Model-based clustering
- Skew-normal distribution
- Rounded mixture priors