Bootstrapping probability-proportional-to-size samples via calibrated empirical population

A collection of six novel bootstrap algorithms, applied to probability-proportional-to-size samples, is explored for variance estimation, confidence interval and p-value production. Developed according to bootstrap fundamentals such as the mimicking principle and the plug-in rule, these algorithms make use of an empirical bootstrap population informed by sampled units each with assigned weight. Starting from the natural choice of Horvitz–Thompson (HT)-type weights, improvements based on calibration to known population features are fostered. Focusing on the population total as the parameter to be estimated and on the distribution of the HT estimator as the target of bootstrap estimation, simulation results are presented with the twofold objective of checking practical implementation and of investigating the statistical properties of the bootstrap estimates supplied by the algorithms explored.