A study of the properties of computationally simple rules in estimation problems
Branislav D Vidakovic · Purdue e-Pubs (Purdue University System) · 1992
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.