On admissible estimation of a mean vector when the scale is unknown

TL;DR

This paper extends the class of admissible generalized Bayes estimators for the mean of a multivariate normal distribution with unknown scale, identifying the boundary for admissibility and providing minimax estimators in all dimensions greater than 2.

Contribution

It enlarges the admissible class of priors by considering improper conditional priors with a hyperparameter in [-2, -1], completing the characterization of admissibility for this class.

Findings

Admissibility holds for hyperparameter a ≥ -2.

Inadmissibility for a < -2.

Provides minimax estimators in all dimensions > 2.

Abstract

We consider admissibility of generalized Bayes estimators of the mean of a multivariate normal distribution when the scale is unknown under quadratic loss. The priors considered put the improper invariant prior on the scale while the prior on the mean has a hierarchical normal structure conditional on the scale. This conditional hierarchical prior is essentially that of Maruyama and Strawderman (2021, Biometrika) (MS21) which is indexed by a hyperparameter $a$. In that paper $a$ is chosen so this conditional prior is proper which corresponds to $a>-1$. This paper extends MS21 by considering improper conditional priors with $a$ in the closed interval $[-2, -1]$, and establishing admissibility for such $a$. The authors, in Maruyama and Strawderman (2017, JMVA), have earlier shown that such conditional priors with $a < -2$ lead to inadmissible estimators. This paper therefore completes the determination of admissibility/inadmissibility for this class of priors. It establishes the the boundary as $a = -2$, with admissibility holding for $a\geq -2$ and inadmissibility for $a < -2$. This boundary corresponds exactly to that in the known scale case for these conditional priors, and which follows from Brown (1971, AOMS). As a notable benefit of this enlargement of the class of admissible generalized Bayes estimators, we give admissible and minimax estimators in all dimensions greater than $2$ as opposed to MS21 which required the dimension to be greater than $4$. In one particularly interesting special case, we establish that the joint Stein prior for the unknown scale case leads to a minimax admissible estimator in all dimensions greater than $2$.