Intrinsic posterior regret gamma-minimax estimation for the exponential family of distributions

TL;DR

This paper develops invariant posterior regret gamma-minimax estimators for the natural parameter in exponential family distributions, ensuring reparameterization invariance and linking to Bayesian estimators under certain prior classes.

Contribution

It introduces invariant PRGM estimators under Jeffrey's Conjugate Priors for exponential families, extending robustness and invariance in Bayesian estimation.

Findings

PRGM estimators are invariant under reparameterizations within JCP class

Invariant PRGM estimators can be derived by modifications of standard conjugate priors

PRGM estimators can be Bayes with respect to priors in the original class under certain conditions

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. 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.