On the relative efficiency of Murthy's sampling strategy under a super population model

Abstract

Rao (1966), Hanurav (1967), Rao (1967) and Chaudhuri and Arnab (1979) had compared the model expected variances of the Horvitz - Thompson (1952) estimator (e _HT ) based on inclusion probability proportional to size(IPPS) sampling design, the Rao - Hartley - Cochran (1962) estimator(e _RHC ) and the ratio estimator (e _MS ) based on Midzuno - Sen (1952, 1953) sampling scheme for estimating a finite population total under a super-population model M involving a parameter g and had shown that the model expected variance is least for e _HT if g > 1 and for e _MS if g < 1 while for g = 1, the relative efficiencies are equal for all the three estimators. In this note we compare the relative efficiencies of e_HT and the Murthy's (1957) estimator (e_M) based on probability proportional to size without replacement (PPSWOR) sampling scheme under M and prove that the model expected variance is smaller for e_M if g ≤ 1 and for e _HT if g≥ 2. In particular, it follows that for g = 1, e_M is more efficient than each of e _HT , e _RHC and e _MS which had been proved in Rao(1966) only for samples of size two.