Limits of experiments associated with sampling plans

Starting from Milbrodt (1985), the asymptotic behaviour of experiments associated with Poisson sampling, Rejective sampling and its Sampford-Durbin modification is investigated. As superpopulation models so-called L^r-generated regression parameter families (1⩽r⩽2) are considered, allowing also the presence of nuisance parameters. Under some assumptions on the first order probabilities of inclusion it can be shown that the sampling experiments converge weakly if the underlying shift parameter families do so. In case of convergence the limit of the sampling experiments is characterized in terms of its Hellinger transforms and its Lévy-Khintchine representation, leading to criteria for the limit to be a pure Gaussian or a pure Poisson experiment respectively. These results are then applied to the situation of sampling in the presence of random non-response, and to establish local asymptotic normality (LAN) under more restrictive conditions. Applications also include asymptotic optimality properties of tests based on Horvitz-Thompson-type statistics, and LAM bounds and criteria for adaptivity, when testing or estimating a continuous linear functional in LAN situations. They especially cover the case of sampling from an unknown symmetric distribution, which has been subject to detailed investigations in the i.i.d. case.