Optimal cluster selection probabilities to estimate the finite population distribution function under PPS cluster sampling

Abstract

We study the estimation of the finite population distribution function under several sampling strategies based on a PPS cluster sampling, that is to say, with cluster selection probabilities proportional to size. For the estimation of population means and totals, is well-known that this type of strategies give good results if the cluster selection probabilities are proportional to the total of the study variable or to a related auxiliary variable, over the cluster. We prove that, for the estimation of the distribution function using cluster sampling, this solution in general is not good and, under an appropriate criteria, we obtain the optimal cluster selection probabilities in order to minimize the variance of the estimation. We apply our methodology for two classical PPS sampling strategies: the sampling with replacement with the Hansen-Hurwitz estimator, and the random groups sampling procedure with the RaoHartley-Cochran estimator. We will present a small simulation to compare the efficiency of this approach with other methods.