An adaptive algorithm for estimating inclusion probabilities and performing the Horvitz-Thompson criterion in complex designs

An algorithm for estimating inclusion probabilities and applying the Horvitz–Thompson criterion is considered for complex sampling designs, when the computation of the actual inclusion probabilities is prohibitive. The inclusion probabilities are estimated by means of independent replications of the sampling scheme. In turn, the number of replications is determined on the basis of the stability of the resulting estimates as well as on the basis of their precision, checked by means of the Bennet inequality. The number of replications is adaptively increased until a suitable level of precision is reached in a sustainable computational time. Details on the FORTRAN routines adopted for implementing the algorithm are given. The procedure is checked using three artificial examples