Bootstrap of means under stratified sampling

TL;DR

This paper investigates the asymptotic behavior of bootstrap estimators of means in stratified sampling, showing how weighted resampling approximates the distribution effectively depending on sample size convergence.

Contribution

It introduces a bootstrap method for means under stratified sampling and analyzes its asymptotic properties, highlighting the role of population sample size convergence.

Findings

Bootstrap estimators are asymptotically Gaussian.

Weighted resampling provides accurate approximation.

Accuracy depends on the convergence rate of sample sizes.

Abstract

In a two-stage cluster sampling procedure, $n$ random populations are drawn independently from independent populations and a sub-sample of observations is taken in each of them. The estimator of the general mean of the observed variables is asymptotically Gaussian and the asymptotic distributions of several bootstrap versions of the normalized and studentized statistics are studied. A weighted population resampling provides a good approximation and its accuracy depends on the convergence rate of the sample size of the populations.