On Averaging Over Distinct Units in Sampling with Replacement

Abstract

Raj and Khamis [2], and Raj [3], have shown that in sampling with replacement the average over distinct units possesses lower variance than the average over the entire sample including repetitions. But no usable results are available regarding the amount of reduction in variance. The relevant expression given by Raj and Khamis is prohibitively cumbersome even for numerical computations. The present note provides the simplest possible exact expression as well as a convenient approximation. In particular, it follows readily that the relative benefit due to averaging over distinct units only is basically an increasing function of the sampling fraction.