The Rao-Hartley-Cochran (1962) method for selecting psus with unequal probabilities. To take a sample of size n, divide the population into n random groups of psus, U1, U2, . . . ,Un. Then select one psu from each group with probability proportional to size. Let Nk be the number of psus in group k. If psu i is in group k, it is selected with probability xki = Mi / Ʃj∈Uk Mj ; the estimator is

Show that ṫRHC is unbiased for t, and find its variance. Use two sets of indicator variables. Let Iki = 1 if psu i is in group k and 0 otherwise, and let Zi = 1 if psu i is selected to be in the sample.