On the generalization of Rao, Hartley and Cochran's scheme (Q578781)

This brief paper contains a slight modification of the usual Rao, Hartley and Cochran's scheme [\textit{J. N. K. Rao}, \textit{H. O. Hartley} and \textit{W. G. Cochran}, J. R. Stat. Soc., Ser. B 24, 482-491 (1962; Zbl 0112.109)]. The proposed sampling scheme proceeds in three steps: 1. The population of N primary units is randomly divided into \(n+k\) groups of sizes \(N_ 1,N_ 2,...,N_{n+k}\), n being intended sample size and k a positive integer. First, \(N_ 1\) units are selected using SRSWOR, then \(N_ 2\) from the \((N-N_ 1)\) remaining ones etc. 2. n groups out of the \(n+k\) are selected using SRSWOR. 3. In each of these groups one unit is selected with probability proportional to the original probabilities \(p_ s.\) For this scheme an unbiased estimator of the population total is given together with its variance and an unbiased estimator of this variance. The main result of the paper is the consideration of the relative efficiency of the proposed estimator compared to that of Rao, Hartley and Cochran's scheme. The ``new'' estimator is more efficient whenever \(N\sigma^ 2<\sigma^ 2_ z\), \[ \sigma^ 2_ z=\sum^{N}_{s=1}y^ 2_ s/p_ s-Y^ 2,\quad \sigma^ 2=N^{- 1}(\sum^{N}_{s=1}Y^ 2_ s-Y^ 2/N) \] and the groups have equal size.