# Two-phase rejective sampling and its asymptotic properties

## TL;DR

This paper introduces two-phase rejective sampling, a method that improves sampling efficiency by using auxiliary data in two stages.

## Contribution

The paper proposes two-phase rejective sampling and establishes its asymptotic properties for various estimators.

## Key findings

- TPRS improves the efficiency of the double-expansion estimator to match that of a regression estimator.
- The method accommodates varying covariate importance and extends to multi-phase sampling.
- Asymptotic results for TPRS also apply to single-phase rejective sampling.

## Abstract

Rejective sampling improves design and estimation efficiency of single-phase sampling when auxiliary information in a finite population is available. When such auxiliary information is unavailable, we propose to use two-phase rejective sampling (TPRS), which involves measuring auxiliary variables for the sample of units in the first phase, followed by the implementation of rejective sampling for the outcome in the second phase. We explore the asymptotic design properties of double expansion and regression estimators under TPRS. We show that TPRS enhances the efficiency of the double-expansion estimator, rendering it comparable to a regression estimator. We further refine the design to accommodate varying importance of covariates and extend it to multi-phase sampling. We start with the theory for the population mean and then extend the theory to parameters defined by general estimating equations. Our asymptotic results for TPRS immediately cover the existing single-phase rejective sampling, under which the asymptotic theory has not been fully established.