Survey Methodology
A generalization of inverse probability weighting
- Release date: June 21, 2022
Abstract
In finite population estimation, the inverse probability or Horvitz-Thompson estimator is a basic tool. Even when auxiliary information is available to model the variable of interest, it is still used to estimate the model error. Here, the inverse probability estimator is generalized by introducing a positive definite matrix. The usual inverse probability estimator is a special case of the generalized estimator, where the positive definite matrix is the identity matrix. Since calibration estimation seeks weights that are close to the inverse probability weights, it too can be generalized by seeking weights that are close to those of the generalized inverse probability estimator. Calibration is known to be optimal, in the sense that it asymptotically attains the Godambe-Joshi lower bound. That lower bound has been derived under a model where no correlation is present. This too, can be generalized to allow for correlation. With the correct choice of the positive definite matrix that generalizes the calibration estimators, this generalized lower bound can be asymptotically attained. There is often no closed-form formula for the generalized estimators. However, simple explicit examples are given here to illustrate how the generalized estimators take advantage of the correlation. This simplicity is achieved here, by assuming a correlation of one between some population units. Those simple estimators can still be useful, even if the correlation is smaller than one. Simulation results are used to compare the generalized estimators to the ordinary estimators.
Key Words: Calibration estimator; Godambe-Joshi lower bound; Horvitz-Thompson estimator; Moore-Penrose inverse; Vaccination rate.
Table of contents
- Section 1. Introduction
- Section 2. The generalized inverse probability estimator
- Section 3. The generalized calibration estimator
- Section 4. The choice of the positive definite matrix
- Section 5. The generalized Godambe-Joshi lower bound
- Section 6. Examples
- Section 7. Simulation results
- Section 8. Summary
- Acknowledgements
- Appendix
- References
How to cite
Théberge, A. (2022). A generalization of inverse probability weighting. Survey Methodology, Statistics Canada, Catalogue No. 12-001-X, Vol. 48, No. 1. Paper available at http://www.statcan.gc.ca/pub/12-001-x/2022001/article/00009-eng.htm.
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