Evaluation of some approximate variance estimators under the Rao-Sampford unequal probability sampling design

Inclusion probability proportional to size (IPPS) sampling designs are often used in surveys, especially for the selection of primary sampling units in the context of multi-stage sampling. Under IIPS sampling, the exact second-order inclusion probabilities are often difficult to obtain, in which case the usual design-unbiased Horvitz-Thompson variance estimator and Sen-Yates-Grundy variance estimator cannot be computed. This led researchers to develop alternative variance estimators based on approximations of the second-order inclusion probabilities in terms of the first-order inclusion probabilities. However, the resulting variance estimators are generally design-biased. In this paper, we first show that all the approximate variance estimtors can be written using one common form. Then, using the Rao-Sampford IPPS sampling design, we conduct an extensive simulation study to investigate the performance of the alternative variance estimators in terms of relative bias and relative stability.