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

摘要

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.