Improved minimax estimation of a multivariate normal mean under heteroscedasticity

TL;DR

This paper introduces a new minimax estimator for multivariate normal means with heteroscedastic variances, combining features of existing estimators to improve risk reduction and adaptivity.

Contribution

It proposes a novel minimax estimator that blends Berger's and empirical Bayes approaches, achieving scale adaptivity and better risk performance.

Findings

The estimator outperforms existing minimax estimators in numerical simulations.

It is scale adaptive, performing well across various prior scales.

The estimator effectively combines shrinkage directions for improved risk reduction.

Abstract

Consider the problem of estimating a multivariate normal mean with a known variance matrix, which is not necessarily proportional to the identity matrix. The coordinates are shrunk directly in proportion to their variances in Efron and Morris' (J. Amer. Statist. Assoc. 68 (1973) 117-130) empirical Bayes approach, whereas inversely in proportion to their variances in Berger's (Ann. Statist. 4 (1976) 223-226) minimax estimators. We propose a new minimax estimator, by approximately minimizing the Bayes risk with a normal prior among a class of minimax estimators where the shrinkage direction is open to specification and the shrinkage magnitude is determined to achieve minimaxity. The proposed estimator has an interesting simple form such that one group of coordinates are shrunk in the direction of Berger's estimator and the remaining coordinates are shrunk in the direction of the Bayes rule. Moreover, the proposed estimator is scale adaptive: it can achieve close to the minimum Bayes risk simultaneously over a scale class of normal priors (including the specified prior) and achieve close to the minimax linear risk over a corresponding scale class of hyper-rectangles. For various scenarios in our numerical study, the proposed estimators with extreme priors yield more substantial risk reduction than existing minimax estimators.