# Bootstrap inference for the finite population total under complex sampling designs

## TL;DR

This paper introduces a unified bootstrap method for complex survey designs, improving finite population total inference accuracy over traditional methods, especially with small samples.

## Contribution

It develops a bootstrap approach applicable to complex sampling schemes like Poisson and probability-proportional-to-size sampling, using studentization and multinomial bootstrapping.

## Key findings

- The proposed bootstrap method is second-order accurate.
- It outperforms Wald-type methods in coverage rate with limited samples.
- Simulation results confirm improved inference accuracy.

## Abstract

Bootstrap is a useful tool for making statistical inference, but it may provide erroneous results under complex survey sampling. Most studies about bootstrap-based inference are developed under simple random sampling and stratified random sampling. In this paper, we propose a unified bootstrap method applicable to some complex sampling designs, including Poisson sampling and probability-proportional-to-size sampling. Two main features of the proposed bootstrap method are that studentization is used to make inference, and the finite population is bootstrapped based on a multinomial distribution by incorporating the sampling information. We show that the proposed bootstrap method is second-order accurate using the Edgeworth expansion. Two simulation studies are conducted to compare the proposed bootstrap method with the Wald-type method, which is widely used in survey sampling. Results show that the proposed bootstrap method is better in terms of coverage rate especially when sample size is limited.