×

On the small sample behavior of Dirichlet process mixture models for data supported on compact intervals. (English) Zbl 1489.62119

Summary: Bayesian nonparametric models provide a general framework for flexible statistical modeling of modern complex data sets. We compare a rate-optimal and rate-suboptimal Bayesian nonparametric model for density estimation for data supported on a compact interval, by means of the analyses of simulated and real data. The results show that rate-optimal models are not uniformly better, across sample sizes, with respect to the way in which the posterior mass concentrates around a true model and that suboptimal models can outperform the optimal ones, even for relatively large sample sizes.

MSC:

62G07 Density estimation
62F15 Bayesian inference
62G20 Asymptotic properties of nonparametric inference

Software:

DPpackage; boa
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bernstein, S. N., Démonstration du théoreme de weierstrass fondée sur le calcul des probabilités, Communications of the Kharkov Mathematical Society, 13, 1-2 (1912) · JFM 43.0301.03
[2] Chen, M. H.; Shao, Q. M., Monte Carlo estimation of Bayesian credible and HPD intervals, Journal of Computational and Graphical Statistics, 8, 69-92 (1999) · doi:10.2307/1390921
[3] Escobar, M. D.; West, M., Bayesian density estimation and inference using mixtures, Journal of the American Statistical Association, 90, 430, 577-88 (1995) · Zbl 0826.62021 · doi:10.2307/2291069
[4] Ferguson, T. S., A bayesian analysis of some nonparametric problems, The Annals of Statistics, 1, 2, 209-30 (1973) · Zbl 0255.62037 · doi:10.1214/aos/1176342360
[5] Ferguson, T. S., Prior distribution on the spaces of probability measures, The Annals of Statistics, 2, 4, 615-29 (1974) · Zbl 0286.62008 · doi:10.1214/aos/1176342752
[6] Geisser, S.; Eddy, W., A predictive approach to model selection, Journal of the American Statistical Association, 74, 365, 153-60 (1979) · Zbl 0401.62036 · doi:10.2307/2286745
[7] Gelfand, A. E.; Dey, D., Bayesian model choice: Asymptotics and exact calculations, Journal of the Royal Statistical Society, Series B, 56, 501-14 (1994) · Zbl 0800.62170 · doi:10.1111/j.2517-6161.1994.tb01996.x
[8] Ghosal, S., Convergence rates for density estimation with Bernstein polynomials, The Annals of Statistics, 29, 5, 1264-80 (2001) · Zbl 1043.62024 · doi:10.1214/aos/1013203453
[9] Ghosal, S.; Ghosh, J. K.; Ramamoorthi, R. V., Posterior consistency of Dirichlet mixtures in density estimation, The Annals of Statistics, 27, 143-58 (1999) · Zbl 0932.62043 · doi:10.1214/aos/1018031105
[10] Ghosal, S.; Ghosh, J. K.; Van der Vaart, A. W., Convergence rates of posterior distributions, The Annals of Statistics, 28, 2, 500-31 (2000) · Zbl 1105.62315 · doi:10.1214/aos/1016218228
[11] Ghosal, S.; Van der Vaart, A. W., Posterior convergence rates of Dirichlet mixtures at smooth densities, The Annals of Statistics, 35, 2, 697-723 (2007) · Zbl 1117.62046 · doi:10.1214/009053606000001271
[12] Jara, A.; Hanson, T.; Quintana, F.; Müller, P.; Rosner, G. L., DPpackage: Bayesian semi- and nonparametric modeling in R, Journal of Statistical Software, 40, 1-30 (2011)
[13] Klinger, R.; Olaya, J.; Marmolejo, L.; Madera, C., A sampling plan for residentially generated solid waste quantification at urban zones of Middle sized cities, Revista Facultad de Ingeniería Universidad de Antioquia, 48, 76-86 (2009)
[14] Kottas, A.2006a. Dirichlet process mixtures of beta distributions, with applications to density and intensity estimation. Tech. rep., department of applied mathematics and statistics. University of California, Santa Cruz.
[15] Kottas, A., Nonparametric Bayesian survival analysis using mixtures of Weibull distributions, Journal of Statistical Planning and Inference, 136, 3, 578-96 (2006) · Zbl 1079.62095 · doi:10.1016/j.jspi.2004.08.009
[16] Kruijer, W.; Van der Vaart, A., Posterior convergence rates for Dirichlet mixtures of beta densities, Journal of Statistical Planning and Inference, 138, 7, 1981-92 (2008) · Zbl 1134.62023 · doi:10.1016/j.jspi.2007.07.012
[17] Lijoi, A.; Prünster, I.; Walker, S., On consistency of non-parametric normal mixtures for Bayesian density estimation, Journal of the American Statistical Association, 100, 472, 1292-6 (2005) · Zbl 1117.62387 · doi:10.1198/016214505000000358
[18] Lo, A. Y., On a class of Bayesian nonparametric estimates I: Density estimates, The Annals of Statistics, 12, 1, 351-7 (1984) · Zbl 0557.62036 · doi:10.1214/aos/1176346412
[19] Müller, P.; Quintana, F.; Jara, A.; Hanson, T., Bayesian nonparametric data analysis (2015), New York: Springer, New York · Zbl 1333.62003
[20] Petrone, S., Bayesian density estimation using Bernstein polynomials, Canadian Journal of Statistics, 27, 1, 105-26 (1999) · Zbl 0929.62044 · doi:10.2307/3315494
[21] Petrone, S., Random Bernstein polynomials, Scandinavian Journal of Statistics, 26, 3, 373-93 (1999) · Zbl 0939.62046 · doi:10.1111/1467-9469.00155
[22] Petrone, S.; Wasserman, L., Consistency of Bernstein polynomial posterior, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64, 1, 79-100 (2002) · Zbl 1015.62033 · doi:10.1111/1467-9868.00326
[23] Rivoirard, V.; Rousseau, J., Posterior concentration rates for infinite dimensional exponential families, Bayesian Analysis, 7, 2, 311-34 (2012) · Zbl 1330.62179 · doi:10.1214/12-BA710
[24] Rousseau, J., Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density, The Annals of Statistics, 38, 1, 146-80 (2010) · Zbl 1181.62047 · doi:10.1214/09-AOS703
[25] Sethuraman, J., A constructive definition of dirichlet priors, Statistica Sinica, 2, 639-50 (1994) · Zbl 0823.62007
[26] Shen, X.; Wasserman, L., Rates of convergence of posterior distributions, The Annals of Statistics, 29, 687-714 (2001) · Zbl 1041.62022 · doi:10.1214/aos/1009210686
[27] Smith, B. J., Boa: An r package for MCMC output convergence assessment and posterior inference, Journal of Statistical Software, 21, :1-37 (2007)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.