Bayesian nonparametric methods for data from a unimodal density. (English) Zbl 0806.62038

Summary: A strongly unimodal density with mode \(\theta\) is one that is non- decreasing on \((-\infty, \theta)\) and non-increasing on \((\theta, \infty)\). L. J. Brunner and A. Y. Lo [Ann. Stat. 17, No. 4, 1550-1566 (1989; Zbl 0697.62003)] have described Bayesian procedures for sampling from a unimodal density, assuming only that it is symmetric about an unknown mode \(\theta\). Here, the case where the unimodal density need not be symmetric is considered. The unimodal density is first written as a mixture with mixing distribution \(G\). Placing a Dirichlet process prior on the unknown mixing distribution \(G\) and an arbitrary prior on the unknown model \(\theta\), the posterior distribution of the pair \((\theta, G)\) is obtained; the marginal posterior distribution of \(\theta\) and the posterior expectation of \(G\) are expressed in terms of sums over partitions of the set of integers \(\{1,\dots, n\}\).


62G99 Nonparametric inference
62C10 Bayesian problems; characterization of Bayes procedures


Zbl 0697.62003
Full Text: DOI


[1] Brunner, L. J.; Lo, A. Y., Bayes methods for a symmetric unimodal density and its mode, Ann. Statist., 17, 1550-1566 (1989) · Zbl 0697.62003
[2] Diaconis, P.; Freedman, D., On the consistency of Bayes estimates, Ann. Statist., 14, 1-26 (1986) · Zbl 0595.62022
[3] Doob, J., Application of the theory of martingales, Coll. Int. CNRS, Paris, 23-27 (1949)
[4] Dykstra, R. L.; Laud, P., A Bayesian nonparametric approach to reliability, Ann. Statist., 9, 356-367 (1981) · Zbl 0469.62077
[5] Feller, W., An Introduction to Probability Theory and its Applications, II (1971), Wiley: Wiley New York · Zbl 0219.60003
[6] Ferguson, T. S., A Bayesian analysis of some nonparametric problems, Ann. Statist., 1, 209-230 (1973) · Zbl 0255.62037
[7] Kuo, L., Computations of mixtures of Dirichlet process, SIAM J. Sci. Statist. Comput., 7, 60-71 (1986)
[8] Lawless, J. F., Statistical Models and Methods for Life-time Data (1982), Wiley: Wiley New York · Zbl 0541.62081
[9] Lindsey, B. G., The geometry of mixture likelihoods: A general theory, Ann. Statist., 11, 86-94 (1983) · Zbl 0512.62005
[10] Lo, A. Y., On a class of Bayesian nonparametric estimates: I. Density estimates, Ann. Statist., 12, 351-357 (1984) · Zbl 0557.62036
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