Some design properties of a rejective sampling procedure

Occasionally, a selected probability sample may appear undesirable with respect to the available auxiliary information. In such a situation, the practitioner might consider rejecting the sample and selecting a new set of sample elements. We consider a procedure in which the probability sample is rejected unless the sample mean of an auxiliary vector is within a specified distance of the population mean. It is proven that the large sample mean and variance of the regression estimator for the rejective sample are the same as those of the regression estimator for the original selection procedure. Likewise, the usual estimator of variance for the regression estimator is appropriate for the rejective sample. In a Monte Carlo experiment, the large sample properties hold for relatively small samples and the Monte Carlo results are in agreement with the theoretical orders of approximation. The efficiency effect of the described rejective sampling is o(n(N)(-1), where n(N) is the expected sample size, but the effect can be important for particular samples. For example, rejective sampling can be used to eliminate those samples that give negative weights for the regression estimator.