Improved Estimation in a Contingency Table: Independence Structure

Abstract

Abstract Estimation of the cell probabilities in a two-way contingency table is considered when it is plausible that the table might have an independence structure relating to the two traits. In a classical nonparametric setup, the unrestricted maximum likelihood estimators of the cell probabilities are the corresponding sample proportions; under the assumption of independence, the restricted estimators are the product of the respective row and column sample proportions. The latter estimators behave better than the former when independence actually holds, but a different picture may emerge for possible departure from the assumed independence structure; the restricted estimators may be heavily biased, inefficient, and even inconsistent. For this reason, a preliminary test on independence based on the classical contingency chi-squared statistic may be conveniently incorporated in the formulation of a preliminary test estimator of the matrix of cell probabilities. Since, typically, we have a multiparameter estimation problem, a pretest or shrinkage estimator based on the classical Stein rule may also be formulated in the same vein. The primary objective of this article is to focus on the asymptotic distribution theory of all four estimators under the null hypothesis of independence as well as local Pitman-type alternatives. The bias and risk based on suitable quadratic loss functions of these estimators are considered in the same asymptotic setup, and these are then incorporated in the study of the asymptotic relative performance of these estimators. In the light of their asymptotic risks, neither the preliminary test nor the shrinkage estimators dominates the other, though each fares well relative to the unrestricted or the restricted maximum likelihood estimators.