×

On posterior consistency of tail index for Bayesian kernel mixture models. (English) Zbl 1466.62268

Summary: Asymptotic theory of tail index estimation has been studied extensively in the frequentist literature on extreme values, but rarely in the Bayesian context. We investigate whether popular Bayesian kernel mixture models are able to support heavy tailed distributions and consistently estimate the tail index. We show that posterior inconsistency in tail index is surprisingly common for both parametric and nonparametric mixture models. We then present a set of sufficient conditions under which posterior consistency in tail index can be achieved, and verify these conditions for Pareto mixture models under general mixing priors.

MSC:

62F15 Bayesian inference
62G32 Statistics of extreme values; tail inference
62H30 Classification and discrimination; cluster analysis (statistical aspects)

Software:

BNPdensity; plfit
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Barrios, E., Lijoi, A., Nieto-Barajas, L.E. and Prünster, I. (2013). Modeling with normalized random measure mixture models. Statist. Sci.28 313–334. · Zbl 1331.62120
[2] Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. Chichester: Wiley. With contributions from Daniel De Waal and Chris Ferro. · Zbl 1070.62036
[3] Bottolo, L., Consonni, G., Dellaportas, P. and Lijoi, A. (2003). Bayesian analysis of extreme values by mixture modeling. Extremes6 25–47. · Zbl 1053.62061
[4] Boucheron, S. and Thomas, M. (2015). Tail index estimation, concentration and adaptivity. Electron. J. Stat.9 2751–2792. · Zbl 1352.60025
[5] Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Teor. Veroyatn. Primen.10 351–360. · Zbl 0147.37004
[6] Carpentier, A. and Kim, A.K.H. (2015). Adaptive and minimax optimal estimation of the tail coefficient. Statist. Sinica25 1133–1144. · Zbl 1415.62029
[7] Clauset, A., Shalizi, C.R. and Newman, M.E.J. (2009). Power-law distributions in empirical data. SIAM Rev.51 661–703. · Zbl 1176.62001
[8] Cormann, U. and Reiss, R.-D. (2009). Generalizing the Pareto to the log-Pareto model and statistical inference. Extremes12 93–105. · Zbl 1221.62039
[9] de Haan, L. and Resnick, S.I. (1980). A simple asymptotic estimate for the index of a stable distribution. J. Roy. Statist. Soc. Ser. B42 83–87. · Zbl 0422.62017
[10] Diebolt, J., El-Aroui, M.-A., Garrido, M. and Girard, S. (2005). Quasi-conjugate Bayes estimates for GPD parameters and application to heavy tails modelling. Extremes8 57–78. · Zbl 1091.62009
[11] do Nascimento, F.F., Gamerman, D. and Lopes, H.F. (2012). A semiparametric Bayesian approach to extreme value estimation. Stat. Comput.22 661–675. · Zbl 1322.62049
[12] Doss, H. and Sellke, T. (1982). The tails of probabilities chosen from a Dirichlet prior. Ann. Statist.10 1302–1305. · Zbl 0515.62008
[13] Drees, H. (1998). Optimal rates of convergence for estimates of the extreme value index. Ann. Statist.26 434–448. · Zbl 0934.62047
[14] Drees, H. (2001). Minimax risk bounds in extreme value theory. Ann. Statist.29 266–294. · Zbl 1029.62046
[15] Escobar, M.D. and West, M. (1995). Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc.90 577–588. · Zbl 0826.62021
[16] Favaro, S. and Teh, Y.W. (2013). MCMC for normalized random measure mixture models. Statist. Sci.28 335–359. · Zbl 1331.62138
[17] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist.1 209–230. · Zbl 0255.62037
[18] Frigessi, A., Haug, O. and Rue, H. (2002). A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes5 219–235. · Zbl 1039.62042
[19] Fristedt, B.E. (1967). Sample function behavior of increasing processes with stationary, independent increments. Pacific J. Math.21 21–33. · Zbl 0189.50802
[20] Fristedt, B.E. and Pruitt, W.E. (1971). Lower functions for increasing random walks and subordinators. Z. Wahrsch. Verw. Gebiete18 167–182. · Zbl 0197.44204
[21] Fúquene Patiño, J.A. (2015). A semi-parametric Bayesian extreme value model using a Dirichlet process mixture of gamma densities. J. Appl. Stat.42 267–280.
[22] Ghosal, S., Ghosh, J.K. and Ramamoorthi, R.V. (1999). Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist.27 143–158. · Zbl 0932.62043
[23] Ghosal, S., Ghosh, J.K. and van der Vaart, A.W. (2000). Convergence rates of posterior distributions. Ann. Statist.28 500–531. · Zbl 1105.62315
[24] Ghosal, S. and van der Vaart, A. (2007). Posterior convergence rates of Dirichlet mixtures at smooth densities. Ann. Statist.35 697–723. · Zbl 1117.62046
[25] Ghosh, J.K. and Ramamoorthi, R.V. (2003). Bayesian Nonparametrics. Springer Series in Statistics. New York: Springer.
[26] Green, P.J. and Richardson, S. (2001). Modelling heterogeneity with and without the Dirichlet process. Scand. J. Stat.28 355–375. · Zbl 0973.62031
[27] Haeusler, E. and Teugels, J.L. (1985). On asymptotic normality of Hill’s estimator for the exponent of regular variation. Ann. Statist.13 743–756. · Zbl 0606.62019
[28] Hall, P. (1982). On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B44 37–42. · Zbl 0521.62024
[29] Hall, P. and Welsh, A.H. (1984). Best attainable rates of convergence for estimates of parameters of regular variation. Ann. Statist.12 1079–1084. · Zbl 0539.62048
[30] Hall, P. and Welsh, A.H. (1985). Adaptive estimates of parameters of regular variation. Ann. Statist.13 331–341. · Zbl 0605.62033
[31] Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist.3 1163–1174. · Zbl 0323.62033
[32] James, L.F., Lijoi, A. and Prünster, I. (2009). Posterior analysis for normalized random measures with independent increments. Scand. J. Stat.36 76–97. · Zbl 1190.62052
[33] Kruijer, W., Rousseau, J. and van der Vaart, A. (2010). Adaptive Bayesian density estimation with location-scale mixtures. Electron. J. Stat.4 1225–1257. · Zbl 1329.62188
[34] Lange, K. (1973). Borel sets of probability measures. Pacific J. Math.48 141–161. · Zbl 0236.28002
[35] Li, C., Lin, L. and Dunson, D.B. (2019). Supplement to “On posterior consistency of tail index for Bayesian kernel mixture models.” DOI:10.3150/18-BEJ1043SUPP.
[36] Lijoi, A., Mena, R.H. and Prünster, I. (2007). Controlling the reinforcement in Bayesian non-parametric mixture models. J. R. Stat. Soc. Ser. B. Stat. Methodol.69 715–740.
[37] Lijoi, A. and Prünster, I. (2010). Models beyond the Dirichlet process. In Bayesian Nonparametrics. Camb. Ser. Stat. Probab. Math.28 80–136. Cambridge: Cambridge Univ. Press.
[38] Lo, A.Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist.12 351–357. · Zbl 0557.62036
[39] MacEachern, S.N. (1994). Estimating normal means with a conjugate style Dirichlet process prior. Comm. Statist. Simulation Comput.23 727–741. · Zbl 0825.62053
[40] Mason, D.M. (1982). Laws of large numbers for sums of extreme values. Ann. Probab.10 754–764. · Zbl 0493.60039
[41] Neal, R.M. (2000). Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Statist.9 249–265.
[42] Novak, S.Y. (2014). Lower bounds to the accuracy of inference on heavy tails. Bernoulli20 979–989. · Zbl 1400.62103
[43] Pickands, J. III (1975). Statistical inference using extreme order statistics. Ann. Statist.3 119–131. · Zbl 0312.62038
[44] Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist.31 560–585. Dedicated to the memory of Herbert E. Robbins. · Zbl 1068.62034
[45] Richardson, S. and Green, P.J. (1997). On Bayesian analysis of mixtures with an unknown number of components. J. Roy. Statist. Soc. Ser. B59 731–792. · Zbl 0891.62020
[46] Schwartz, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete4 10–26. · Zbl 0158.17606
[47] Shen, W., Tokdar, S.T. and Ghosal, S. (2013). Adaptive Bayesian multivariate density estimation with Dirichlet mixtures. Biometrika100 623–640. · Zbl 1284.62183
[48] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley. · Zbl 1170.62365
[49] Stephenson, A. and Tawn, J. (2004). Bayesian inference for extremes: Accounting for the three extremal types. Extremes7 291–307. · Zbl 1090.62025
[50] Tokdar, S.T. (2006). Posterior consistency of Dirichlet location-scale mixture of normals in density estimation and regression. Sankhyā68 90–110. · Zbl 1193.62056
[51] Tressou, J. (2008). Bayesian nonparametrics for heavy tailed distribution. Application to food risk assessment. Bayesian Anal.3 367–391. · Zbl 1330.62183
[52] Wang, Z., Rodriguez, A. and Kottas, A. (2012). A nonparametric mixture modeling framework for extreme value analysis. Technical report. https://www.soe.ucsc.edu/research/technical-reports/UCSC-SOE-11-26/download.
[53] Watanabe, T. (1960). A probabilistic method in Hausdorff moment problem and Laplace–Stieltjes transform. J. Math. Soc. Japan12 192–206. · Zbl 0214.44602
[54] Wu, Y. and Ghosal, S. (2008). Kullback Leibler property of kernel mixture priors in Bayesian density estimation. Electron. J. Stat.2 298–331. · Zbl 1135.62022
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.