A Gaussian sequence approach for proving minimaxity: A Review

TL;DR

This paper reviews minimax best equivariant estimators for invariant problems like location, scale, and covariance matrix estimation, introducing a unified Gaussian sequence approach that simplifies existing proofs of minimaxity.

Contribution

It presents a novel, unified Gaussian sequence method for proving minimaxity of best equivariant estimators, simplifying traditional proofs.

Findings

Unified Gaussian approach simplifies minimaxity proofs

Revisits minimaxity of estimators for location, scale, covariance

Provides a smooth prior sequence for theoretical analysis

Abstract

This paper reviews minimax best equivariant estimation in these invariant estimation problems: a location parameter, a scale parameter and a (Wishart) covariance matrix. We briefly review development of the best equivariant estimator as a generalized Bayes estimator relative to right invariant Haar measure in each case. Then we prove minimaxity of the best equivariant procedure by giving a least favorable prior sequence based on non-truncated Gaussian distributions. The results in this paper are all known, but we bring a fresh and somewhat unified approach by using, in contrast to most proofs in the literature, a smooth sequence of non truncated priors. This approach leads to some simplifications in the minimaxity proofs.