For a comparison between the
efficiencies of the ratio estimator and the randomized PPS
estimator, the reader is referred to Foreman and Brewer
(1971), Cochran (1977) and the references given therein. A
drawback of these comparisons is that finite populations
corrections are ignored. Hartley and Rao (1962) take the
finite population correction into account but without an
explicit formula for the efficiency. Elaborating on the results
of Gabler (1984), Qualité (2008) shows that the related HT
estimator from a rejective Poisson sample of size n is more
efficient than the Hansen-Hurwitz estimator for a sampling
scheme with replacement. No formula for the increased
efficiency is given, however