Summary
For an inclusion probability proportional to size (IPPS) sampling scheme recently proposed by Saxena, Singh and Srivastava (1986), it is shown that under certain simple verifiable conditions (1) the Horvitz-Thompson (1952) estimator based on it has a smaller variance than the variance of the Hansen-Hurwitz (1943) estimator based on probability proportional to size (PPS) sampling with replacement (WR) both involving the same size-measures and the expected sample size in the former being equal to the number of draws in the latter and (2) the Yates-Grundy (1953) estimator for the variance of the Horvitz-Thompson estimator based on this IPPS scheme is uniformly non-negative.
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References
Hansen MH, Hurwitz WN (1943) On the theory of sampling from finite populations. Ann Math Statist 14:333–362
Horvitz DG, Thompson DJ (1952) A generalization of sampling without replacement from a finite universe. J Amer Statist Assoc 47:663–685
Saxena RR, Singh P, Srivastava AK (1986) An unequal probability sampling scheme. Biometrika 73:761–763
Yates F, Grundy PM (1953) Selection without replacement from within strata with probability proportional to size. J Roy Statist Soc B 15:253–261
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Adhikary, A.K., Chaudhuri, A. On two properties of an unequal probability sampling scheme. Metrika 36, 161–166 (1989). https://doi.org/10.1007/BF02614089
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DOI: https://doi.org/10.1007/BF02614089