Asymptotic behavior of Bayes estimates under possibly incorrect models
Olaf Bunke and Xavier Milhaud
Source: Ann. Statist.
Volume 26, Number 2
(1998), 617-644.
Abstract
We prove that the posterior distribution in a possibly incorrect parametric model a.s. concentrates in a strong sense on the set of pseudotrue parameters determined by the true distribution. As a consequence, we obtain in the case of a unique pseudotrue parameter the strong consistency of pseudo-Bayes estimators w.r.t. general loss functions.
Further, we present a simple example based on normal distributions and having two different pseudotrue parameters, where pseudo-Bayes estimators have an essentially different asymptotic behavior than the pseudomaximum likelihood estimator. While the MLE is strongly consistent, the sequence of posterior means is strongly inconsistent and a.s. almost all its accumulation points are not pseudotrue. Finally, we give conditions under which a pseudo-Bayes estimator for a unique pseudotrue parameter has an asymptotic normal distribution.
Primary Subjects: 62F12, 62F15
Keywords: Consistency; asymptotic normality; incorrect parametric models; inconsistent Bayes estimates
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1028144851
Mathematical Reviews number (MathSciNet):
MR1626075
Digital Object Identifier: doi:10.1214/aos/1028144851
Zentralblatt MATH identifier:
0929.62022
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