Rate of Convergence in the Sequence-Compound Squared-Distance Loss Estimation Problem for a Family of $m$-Variate Normal Distributions

Abstract

This paper is concerned with rates of convergence in the sequence-compound decision problem when the component problem is the squared-distance loss estimation of the mean of $m$-variate normal distribution with covariance matrix I. Section 1 introduces some notation and discusses the earlier work related to this problem. In Section 2, we prove two lemmas which are required in later sections. Sections 3, 4 and 5 exhibit sequence-compound decision procedures whose modified regrets are $O(n^{-1/m+4})$, near $O(n^{-\frac{1}{4}})$ and near $O(n^{-\frac{1}{2}})$ respectively, all the orders being uniform in parameter sequences concerned. In Section 6, comparisons have been made between the procedures given in Sections 3, 4 and 5. We conclude the paper with a few remarks.