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2013 Intrinsic posterior regret gamma-minimax estimation for the exponential family of distributions
Mohammad Jafari Jozani, Nahid Jafari Tabrizi
Electron. J. Statist. 7: 1856-1874 (2013). DOI: 10.1214/13-EJS828

Abstract

In practice, it is desired to have estimates that are invariant under reparameterization. The invariance property of the estimators helps to formulate a unified solution to the underlying estimation problem. In robust Bayesian analysis, a frequent criticism is that the optimal estimators are not invariant under smooth reparameterizations. This paper considers the problem of posterior regret gamma-minimax (PRGM) estimation of the natural parameter of the exponential family of distributions under intrinsic loss functions with Kullback-Leibler distance. We show that under the class of Jeffrey’s Conjugate Prior (JCP) distributions, PRGM estimators are invariant to smooth one-to-one reparameterizations. We apply our results to several distributions and different classes of JCP, as well as the usual conjugate prior distributions. We observe that, in many cases, invariant PRGM estimators in the class of JCP distributions can be obtained by some modifications of PRGM estimators in the usual class of conjugate priors. Moreover, when the class of priors are convex or dependant on a hyper-parameter belonging to a connected set, we show that the PRGM estimator under the intrinsic loss function could be Bayes with respect to a prior distribution in the original prior class. Theoretical results are supplemented with several examples and illustrations.

Citation

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Mohammad Jafari Jozani. Nahid Jafari Tabrizi. "Intrinsic posterior regret gamma-minimax estimation for the exponential family of distributions." Electron. J. Statist. 7 1856 - 1874, 2013. https://doi.org/10.1214/13-EJS828

Information

Published: 2013
First available in Project Euclid: 10 July 2013

zbMATH: 1293.62048
MathSciNet: MR3084674
Digital Object Identifier: 10.1214/13-EJS828

Subjects:
Primary: 62F10, 62F15

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society

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