zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nonparametric hierarchical Bayes via sequential imputations. (English) Zbl 0880.62038

Summary: We consider the empirical Bayes estimation of a distribution using binary data via the Dirichlet process. Let 𝒟(α) denote a Dirichlet process with α being a finite measure on [0,1]. Instead of having direct samples from an unknown random distribution F from 𝒟(α), we assume that only indirect binomial data are observable.

This paper presents a new interpretation of Lo’s formula [A. Y. Lo, ibid. 12, 351-357 (1984; Zbl 0557.62036)] and thereby relates the predictive density of the observations based on a Dirichlet process model to likelihoods of much simpler models. As a consequence, the log-likelihood surface, as well as the maximum likelihood estimate of c=α([0,1]), is found when the shape of α is assumed known, together with a formula for the Fisher information evaluated at the estimate.

The sequential imputation method of A. Kong, J. S. Liu and W. H. Wong [J. Am. Stat. Assoc. 89, No. 425, 278-288 (1994)] is recommended for overcoming computational difficulties commonly encountered in this area. The related approximation formulas are provided. An analysis of the tack data of L. Beckett and P. Diaconis [Adv. Math. 103, No. 1, 107-128 (1994; Zbl 0805.62085)] which motivated this study, is supplemented to illustrate our methods.

MSC:
62G05Nonparametric estimation
62C12Empirical decision procedures; empirical Bayes procedures
65C05Monte Carlo methods
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)