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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 \(\{1,\dots, n\}\).

MSC:

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

Citations:

Zbl 0697.62003
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References:

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