A Simple Variance Estimator for Unequal Probability Sampling without Replacement

Abstract

Survey sampling textbooks often refer to the Sen–Yates–Grundy variance estimator for use with without-replacement unequal probability designs. This estimator is rarely implemented because of the complexity of determining joint inclusion probabilities. In practice, the variance is usually estimated by simpler variance estimators such as the Hansen–Hurwitz with replacement variance estimator; which often leads to overestimation of the variance for large sampling fractions that are common in business surveys. We will consider an alternative estimator: the Hájek (1964) variance estimator that depends on the first-order inclusion probabilities only and is usually more accurate than the Hansen–Hurwitz estimator. We review this estimator and show its practical value. We propose a simple alternative expression; which is as simple as the Hansen–Hurwitz estimator. We also show how the Hájek estimator can be easily implemented with standard statistical packages.

Keywords:

• Design-based inference
• Hansen–Hurwitz variance estimator
• inclusion probabilities
• π-estimator
• Sen–Yates–Grundy variance estimator

Acknowledgments

The 1998–99 UK Family Expenditure Survey data were made available by the Office for National Statistics (UK). This work was supported by the European research project entitled ‘Data Quality in Complex Surveys within the New European Information Society’ (DACSEIS). The author is grateful to Chris Skinner (University of Southampton, UK), Dan Hedlin (Statistics, Sweden) and Pascal Rivière (INSEE, France) and to the referee for helpful comments.