Comparisons of sampling strategies for estimating finite population proportions in direct and randomized response surveys under a super population model

We consider the problem of unbiased estimation of a finite population proportion and compare the unequal probability sampling strategies due to Hansen-Hurwitz [9], Horvitz-Thompson [12], Rao-Hartley-Cochran [18] and Midzuno-Sen [16,20] under a super-population model. It is shown that the model expected variance is least for the Midzuno-Sen [16,20] strategy both when these sampling strategies are based on data obtained from (i) a direct survey and (ii) a randomized response (RR) survey employing some RR technique following a general RR model.