A study of the properties of computationally simple rules in estimation problems

Branislav D Vidakovic, Purdue University

Abstract

A difficulty in the implementation of Bayes type procedures is that they are frequently not computationally simple and a study of their theoretical properties is therefore notoriously difficult. In such a case, it is very natural to ask if we can limit ourselves to consideration of computationally simple rules. We consider three problems in which we address this issue. The first problem is the $\Gamma$-minimax estimation of a multivariate normal mean. An underlying family of priors $\Gamma$ is the family of mixtures of zero mean normals with a covariance matrix $\tau I,$ where the mixing distribution for $\tau$ belongs to a family ${\cal G}$. It is shown that the optimal linear rule is "good" whenever sup$\sb\tau$ ${E\tau\over1+E\tau}$ is "close" to sup$\sb\tau$ $E{\tau\over1+\tau},$ irrespective of the dimensions of the model. The second problem is the estimation of a normal variance. Vast literature exists on this important problem, but the concerns raised here have not been addressed. We take i.i.d. observations $X\sb1, X\sb2,\...,X\sb{n}$ from a normal distribution with mean 0 and unknown variance $\sigma\sp2,$ and consider estimators of the form $aT+b,$ where $T=\sum X\sbsp{i}{2}.$ For prior distributions on $\sigma\sp2$ which are appropriate mixtures of inverse gamma distributions, we derive analogous upper bounds on the loss of efficiency due to the use of rules linear in T. A lower bound on Bayes risks of independent interest is also obtained. Finally, the third problem is the estimation of a bounded normal mean. Building on a recent work of Donoho, Liu and MacGibbon (1990) on this practically important problem, we prove the surprising result that if the mean is known to lie in a bounded interval (a, b), then uniformly over all such bounded intervals, the optimal linear rule is at most 7.3% worse than the unconstrained optimal rule if the performance of any rule is measured by its maximum Bayes risk over the class of priors which are symmetric and unimodal about ${a+b\over2}.$ The fact that the loss of efficiency is at most 7.3% irrespective of exactly which bounded interval contains the mean lends further strength to linear rules in this case. Extensions to higher dimensions are given.

Degree

Ph.D.

Advisors

DasGupta, Purdue University.

Subject Area

Statistics

COinS