Reduced Mean Square Error Estimation in Contingency Tables

A two-stage estimator is proposed for the cell probabilities in a two-dimensional contingency table. Using a multivariate mean square error criterion, we show that the decision to use either the observed cell probabilities or the estimators assuming row-column independence is approximately dependent upon the noncentrality parameter of the Pearson chi-square statistic x 2. We then derive a decision rule that says to use the independence-assumption estimators when X 2 is less than twice the number of degrees of freedom for testing independence.