Estimation of Population Mean Under Unequal Probability Sampling with Unknown Selection Probabilities

Abstract

In many practical problems of biology, medicine, and social sciences, inference about population mean has to be made froma sample generated with unknown selection probabilities that may vary between the units in the population. The most common approach to analysis of such samples is to use the ordinary sample mean as an estimator of population mean, under the assumption that the sample is “representative” of the population, although the concept of representative sampling is not given a formal mathematical definition. The main objective of this paper is to identify a necessary and sufficient condition for unbiasedness of the ordinary sample mean under unequal probability sampling. It is shown that in the process of sampling with unequal selection probabilities with a fixed sample size, expectation of the ordinary sample mean is exactly equal to population mean plus the covariance of the unit-level outcome with the unit-level relative selection probability, where the relative selection probability is defined as the ratio of the unit’s selection probability to the mean selection probability in the population. Hence, ordinary sample mean is unbiased if and only if the unit-level outcome is uncorrelated with the unit-level relative selection probability. Samples generated under this condition may be considered “representative” of the population for the purpose of inference about population mean. It is also shown that under representative sampling, asymptotically conservative confidence intervals for population mean can be constructed based on the ordinary sample variance estimator as long as there are no positive correlations among the sample observations (negative correlations are allowed) and conditions for asymptotic normality of the sample mean are satisfied.