Comparison of variance estimators in two-phase sampling: An empirical investigation

Two-phase sampling is useful when the sampling frame contains little or no useful auxiliary information on the population elements. to estimate a finite population total under a two-phase design,we consider two point estimators: (i) the double expansion estimator and (ii) the hajèk estimator. for each estimator,we consider two competing unbiased variance estimators: (i) the horvitz-thompson (ht) type variance estimator and (ii) the sen-yates-grundy (syg) type variance estimator. in this paper,we make an empirical study of the properties of these variance estimators in terms of stability as well as the tendency for the ht-type variance estimator to take on negative values. we provide simulation results under a two-phase sampling design,based on simple random sampling in the first phase to observe a size variable and probability proportional-to- size sampling without replacement in the second phase. © 2011 pakistan journal of statistics.