Empirical and Constrained Empirical Bayes Variance Estimation Under A One Unit Per Stratum Sample Design

A single primary sampling unit (PSU) per stratum design is a popular design for estimating the parameter of interest. Although, the point estimator of the design is unbiased and efficient, an unbiased variance estimator does not exist. A common practice to solve this is to collapse or combine the two adjacent strata, but the attained estimator of variance is not design-unbiased, and the bias increases as the population means of the collapsed strata become more variant. Therefore, the one PSU per stratum design with collapsed stratum variance estimator might not be a good choice, and some statisticians prefer a design in which two PSUs per stratum are selected. In this paper, we first compare a one PSU per stratum design to a two PSUs per stratum design. Then, we propose an empirical Bayes estimator for the variance of one PSU per stratum design, where it over-shrinks towards the prior mean. To protect against this, we investigate the potential of a constrained empirical Bayes estimator. Through a simulation study, we show that the empirical Bayes and constrained empirical Bayes estimators outperform the classical collapsed one in terms of empirical relative mean squared error.