On inclusion probabilities for order πps sampling

Abstract

Abstract The order πps schemes introduced in Rosen (1997b. J. Statist. Plann. Inference 62, 159-191) are based on limit considerations. Their (asymptotic) πps property has so far been established only on a “macro” level, via consistency of a quasiHT-estimator. Here the property is studied on a “micro” level, notably for three particular schemes, called uniform, exponential and Pareto order πps. With λ i ( n ) and π i ( n ) for target, respectively, factual inclusion probabilities for population unit i when the sample size is n , the following is shown to hold under general conditions; π i ( n )/ λ i ( n )→1, as the sample size n tends to infinity. The basic proof tools are error bounds for the approximation π i ( n )≈ λ i ( n ). Exact formulae for π i ( n ) are also considered, the main message being that they mostly are numerically unmanageable. However, formulae for a special case provide basis for a small numerical study of the approximation accuracy, with the following tentative conclusions. (i) The approximation accuracy is good for all the three order πps schemes, and best for Pareto πps. (ii) For Pareto πps the approximation error can be regarded as negligible for quite small sample sizes.