The variance of the estimate of the population total depends on the sampling method having different probabilities of sampling. In this paper, we provide some proof(s) or alternative proof(s) of statements about the Hansen-Hurwitz estimator (Hansen and Hurwitz [1]) and the Horvitz-Thompson estimator (Horvitz and Thompson [2]) in the book Sampling (Thompson [4]). We also analyze the asymptotic behaviors of these estimators.

probability sampling, sampling with replacement, sampling without replacement, unbiased estimates, variance of the estimators.