On second-order inclusion probabilities and variances among large randomized PPS samples

This paper presents and discusses some new results on the second-order inclusion probabilities of a systematic probability proportional to size sample drawn from a randomly ordered list, also called randomized PPS sampling. It is shown that some standard approximations of these second-order inclusion probabilities meant for relatively small sample sizes, need not be valid when the sample size n is of the same order as the population size N. In addition, it is shown that under a number of assumptions the variance formulas for rejective Poisson sampling can be applied to randomized PPS sampling designs when both n and N-n are large.