Equal probability selection is a special case of the general theory of probability sampling in which population units may be selected with unequal probabilities. Unequal selection probabilities are often based on auxiliary variable values which are measures of the sizes of population units, thus leading to the acronym (PPS)—“Probability Proportional to Size”. The Horvitz–Thompson (1953) theorem provides a unifying framework for design-based sampling theory. A sampling design specifies the sample space (set of all possible samples) and associated first and second order inclusion probabilities (probabilities that unit i, or units i and j, respectively, are included in a sample of size n selected from N according to some selection method). A valid probability sampling scheme must have all first order inclusion probabilities > 00 (i.e., every population unit must have a chance of being in the sample). Unbiased variance estimation is possible only for those schemes that guarantee that all second order inclusion probabilities exceed zero, thus providing theoretical justification for the absence of unbiased estimators of sampling variance in systematic sampling and other schemes for which some second order inclusion probabilities are zero. Numerous generalized Horvitz–Thompson (HT) estimators can be formed and all are consistent estimators because they are functions of consistent HT estimators. Unequal probability systematic sampling and Poisson sampling (the unequal probability counterpart to Bernoulli sampling for which sample size is a random variable) are also considered. Several R programs for selecting unequal probability samples and for calculating first and second order inclusion probabilities are posted at http://global.oup.com/uk/companion/hankin.
Variance Estimation Using Median of the Auxiliary Variable
