Shrinkage and modification techniques in estimation of variance and the related problems: A review

Abstract

One of the surprising decision-theoretic results Charles Stein discovered is the inadmissibility of the uniformly minimum variance unbiased estirnator(UMVUE) of the variance of a normal distribution with an unknown mean. Some methods for deriving estimators better than the UMVUE were given by Stein. Brown, Brewster and Zidek. Recently Kubokawa established a novel approach, called the IERD method, by use of which one gets a unified class of improved estimators including their previous procedures. This paper gives a review for a series of these decision-theoretical developments as well as surveys the study of the variance-estimation problem from various aspects. Related to this issue, the paper enumerates several topics with the situations where the usual plain estimators are required to be shrunken or modified, and gives reasonable procedures improving the usual ones through the IERD method.

Keywords:

• shrinkage procedure
• statistical decision theory
• empirical and generation Bayes rules
• estimation of covariance