During the last twenty years several results on the conditional properties of statistical procedures have appeared in the literature. However, apart from results linking conditional properties and Bayesianity, few general results have been found. Here a systematic set of definitions of conditional properties is given and a systematic investigation of their consequences is started. An analogy between interval and point estimation is used both to extend definitions of conditional properties to point estimation and to extend the usual definition of admissibility for point estimation to provide a new definition of admissibility for interval estimation.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176344725

Conditional Properties of Statistical Procedures

This article traces the history of the problem of estimating the variance, $\sigma^2$, based on a random sample from a normal distribution with mean $\mu$ unknown. Considered are both the point estimation and confidence interval cases. We see that improvement over both usual estimators follows a remarkably parallel development and stemmed from the innovative ideas presented in Stein (1964). We examine developments through the most recent dealing with improved confidence intervals and conditional evaluations of interval estimators.

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Developments in Decision-Theoretic Variance Estimation

In this review of estimation problems in restricted parameter spaces, we focus through a series of illustrations on a number of methods that have proven to be successful. These methods relate to the decision-theoretic aspects of admissibility and minimaxity, as well as to the determination of dominating estimators of inadmissible procedures obtained for instance from the criteria of unbiasedness, maximum likelihood, or minimum risk equivariance. Finally, we accompany the presentation of these methods with various related historical developments.

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Estimation in restricted parameter spaces: a review

Microarray technology allows a scientist to study genomewide patterns of gene expression. Thousands of individual genes are measured with a relatively small number of replications, which poses challenges to traditional statistical methods. In particular, the gene-specific estimators of variances are not reliable and gene-by-gene tests have low powers. In this article we propose a family of shrinkage estimators for variances raised to a fixed power. We derive optimal shrinkage parameters under both Stein and squared loss functions. Our results show that the standard sample variance is inadmissible under either loss function. We propose several estimators for the optimal shrinkage parameters and investigate their asymptotic properties under two scenarios: large number of replications and large number of genes. We conduct simulations to evaluate the finite sample performance of the data-driven optimal shrinkage estimators and compare them with some existing methods. We construct F-like statistics using these...

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Optimal Shrinkage Estimation of Variances With Applications to Microarray Data Analysis

Questions of admissibility of statistical estimators are reduced to considerations involving differential inequalities. The coefficients of these inequalities involve moments of the underlying distributions; and so are, in principle, not difficult to derive. The methods are "heuristic" because it is necessary to verify on an ad-hoc basis that error terms are small. Some conditions on the structure of the problem are given which we believe will guarantee that these error terms are small. Several different statistical estimation problems are discussed. Each problem is transformed (if necessary) so as to meet the above mentioned structure conditions. Then the heuristic method is applied in order to generate conjectures concerning the admissibility of certain generalized Bayes procedures in these problems.

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A Heuristic Method for Determining Admissibility of Estimators--With Applications

Techniques for improving on equivariant estimators are described. They may be applied, although without assurance of success, whatever be the family of underlying distributions. The loss function is required to satisfy an intuitively reasonable condition but is otherwise arbitrary. One of these techniques amounts to a sample space, orbit-by-orbit analysis of the conditional expected loss given the orbit. It yields, when successful, a "testimator". A second technique obtains the limit of a certain sequence of "testimator-like" estimators. The result is "smoother" than a testimator and often identical to a generalized Bayes estimator over much of its domain. Applications are presented. In the first we extend results of Stein (1964) and obtain a minimax estimator which is generalized Bayes, and in a univariate subcase, admissible within the class of scale-equivariant estimators. In the second, we extend a result of Srivastava and Bancroft (1967).

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J. F. Brewster

Papers

Improving on Equivariant Estimators