Quantile processes and their applications in finite populations

TL;DR

This paper investigates the weak convergence of quantile processes in finite populations under various sampling designs, applying the results to derive asymptotic distributions and confidence intervals for key population parameters.

Contribution

It establishes weak convergence results for quantile processes based on different estimators under multiple sampling schemes, and explores their implications for statistical inference.

Findings

Auxiliary information can sometimes negatively impact estimator performance.

Certain estimators perform worse under complex sampling designs than simple random sampling.

Confidence intervals for population parameters are derived from asymptotic distributions.

Abstract

The weak convergence of the quantile processes, which are constructed based on different estimators of the finite population quantiles, is shown under various well-known sampling designs based on a superpopulation model. The results related to the weak convergence of these quantile processes are applied to find asymptotic distributions of the smooth $L$-estimators and the estimators of smooth functions of finite population quantiles. Based on these asymptotic distributions, confidence intervals are constructed for several finite population parameters like the median, the $\alpha$-trimmed means, the interquartile range and the quantile based measure of skewness. Comparisons of various estimators are carried out based on their asymptotic distributions. We show that the use of the auxiliary information in the construction of the estimators sometimes has an adverse effect on the performances of the smooth $L$-estimators and the estimators of smooth functions of finite population quantiles under several sampling designs. Further, the performance of each of the above-mentioned estimators sometimes becomes worse under sampling designs, which use the auxiliary information, than their performances under simple random sampling without replacement (SRSWOR).