On Inadmissibility of the Yates-Grundy Variance Estimator in Unequal Probability Sampling

Abstract

Abstract An example is given to show that there exist fixed sample size designs for any sample size greater than two, for which the Yates-Grundy estimator is inadmissible in the class of nonnegative unbiased quadratic estimators of the variance of the Horvitz-Thompson estimator. A necessary condition for the Horvitz-Thompson variance estimator to be nonnegative definite is pointed out. It is also pointed out that a posterior lower bound can be obtained for any nonnegative definite quadratic function of a finite population. An example is given to show that the Yates-Grundy estimator, even when it is nonnegative, can take values smaller than this bound.