Conditional MSE-based discrimination of the sample mean and the post-stratification estimator in population sampling
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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.
KeywordsPost-stratification conditional inference Vysochanskiî-Petunin inequality superpopulation model
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