Conditional MSE-based discrimination of the sample mean and the post-stratification estimator in population sampling

Abstract

Whenever a random sample is drawn from a stratified population, the post-stratification estimator$\tilde X$ usually is preferred to the sample mean$\tilde X$, when the population mean is to be estimated. This is due to the fact that the variance of$\tilde X$ is asymptotically smaller than that of$\tilde X$, while both estimators are asymptotically unbiased. However, this only holds looking at post-stratification unconditionally, when strata sample sizes are random. Conditioned on the realized sample sizes, the MSE of$\tilde X$ can be higher than that of$\tilde X$ which means that$\tilde X$ should be preferred to$\tilde X$, even if it is biased. The conditional MSE difference of$\tilde X$ and$\tilde X$ is estimated, and using this estimation and its variance a heuristic test based on the Vysochanskiî-Petunin inequality is derived.