On the Feasibility of Basing Horvitz and Thompson's Estimator on a Sample by Rao,Hartley, and Cochran's Scheme

Formulae for the first and second order inclusion probabilities for the Rao et al. (1962 Rao , J. N. K. , Hartley , H. O. , Cochran , W. G. ( 1962 ). On a simple procedure of unequal probability sampling without replacement . J. Roy. Statist. Soc. B 24 : 482 – 491 . [CSA]  [Google Scholar]) (RHC) scheme of sampling are derived. They enable one to evaluate, for a sample drawn according to the RHC scheme, the Horvitz and Thompson's (1952 Horvitz , D. G. , Thompson , D. J. ( 1952 ). A generalization of sampling without replacement from a finite universe . J. Amer. Statist. Assoc. 47 : 663 – 685 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) estimator (HTE) along with its unbiased variance estimator given by Yates and Grundy (1953 Yates , F. , Grundy , P. M. ( 1953 ). Selection without replacement from within strata with probability proportional to size . J. Amer. Statist. Assoc. 75 : 206 – 211 . [CSA]  [Google Scholar]). So, for a sample at hand thus drawn one may choose between the RHCE and the HTE for use on finding which one has the smaller coefficient of variation.