Horvitz-Thompson estimator and generalized πPS designs

Generalized πPS designs were defined by T.J. Rao (1972). Working with a general super-population model θ(g), the strategy consisting of GπPS design together with the associated Horvitz-Thompson estimator of the population total was shown to be better than two other well known strategies in T.J. Rao (1971,1972). In this note we prove the θ(g)-optimality of the strategy consisting of GπPS design together with the associated Horvitz-Thompson estimator in the entire class of p-unbiased strategies of the population total with expected sample size fixed. In view of our theorem the results of T.J. Rao follow as special cases.