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 eHT and the Murthy's (1957) estimator (eM) based on probability proportional to size without replacement (PPSWOR) sampling scheme under M and prove that the model expected variance is smaller for eM if g ≤ 1 and for e HT if g≥ 2. In particular, it follows that for g = 1, eM 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.