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In practice, it is desired to have estimates that are invariant underreparameterization. The invariance property of the estimators helps toformulate a unified solution to the underlying estimation problem. In robustBayesian analysis, a frequent criticism is that the optimal estimators are notinvariant under smooth reparameterizations. This paper considers the problem ofposterior regret gamma-minimax (PRGM) estimation of the natural parameter ofthe exponential family of distributions under intrinsic loss functions. We showthat under the class of Jeffrey's Conjugate Prior (JCP) distributions, PRGMestimators are invariant to smooth one-to-one reparameterizations. We apply ourresults to several distributions and different classes of JCP, as well as theusual conjugate prior distributions. We observe that, in many cases, invariantPRGM estimators in the class of JCP distributions can be obtained by somemodifications of PRGM estimators in the usual class of conjugate priors.  Moreover, when the class of priors are convex or dependant on ahyper-parameter belonging to a connected set, we show that the PRGM estimatorunder the intrinsic loss function could be Bayes with respect to a priordistribution in the original prior class. Theoretical results are supplementedwith several examples and illustrations.

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