A generalization of inverse probability weighting
Section 8. Summary

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The concept of inverse probability estimation can be generalized with a positive definite matrix $\Sigma .$  There is then a whole family of unbiased estimators parameterized by $\Sigma$  where one member, with $\Sigma =I_{N\times N}$  is the usual inverse probability estimator. The concept of calibration can also be generalized so that weights close to those of the generalized inverse probability estimator are sought. The Godambe and Joshi lower bound of $E_{\xi }{V}_p\left(\widehat{\theta }\right)$  can also be generalized to a model $\xi$  where the variance matrix $V_{\xi }\left(y\right)$  is not necessarily diagonal. The calibrated generalized inverse probability estimator, with $\Sigma =V_{\xi }\left(y\right),$  asymptotically attains the generalized lower bound for any linear unbiased estimator $\widehat{\theta }.$  The new estimators are model assisted, not model based. They remain unbiased, or at least asymptotically unbiased, even if $\Sigma \ne V_{\xi }\left(y\right).$

Examples where the new estimators can be given an explicit form have been presented. Simulations comparing those new estimators with the usual ones have been done. Those simulations show that, while remaining asymptotically unbiased, significant improvements in variance can be obtained in situations where there is significant correlation between some units of the population, as for example there would be, between persons of a same household with regards to vaccination status. Improvements in variance can still be made, even with $\Sigma \ne V_{\xi }\left(y\right).$

Acknowledgements

I would like to thank the Associate Editor and the referees for their constructive comments and suggestions to improve the paper.

Appendix

Proof that with $\widehat{y}=X\widehat{\beta },$ $\widehat{\beta }=T^{1/2}{\left(T^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}X{T}^{1/2}\right)}^{†}{T}^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}y,$ where $T$ and $U$ are positive definite, then for any $\alpha \in$ ${ℝ}^N$ , if ${\Delta }_s\alpha$ is in the range of ${\Delta }_{s}X,$ then the weighted sum of residuals, ${\left(y-\widehat{y}\right)}^{\prime }{\left({\Delta }_{s}U{\Delta }_s\right)}^{†}\alpha ,$ is zero.

First,

$$\begin{array}{ll}{\left(y-\widehat{y}\right)}^{\prime }{\left({\Delta }_{s}U{\Delta }_s\right)}^{†}\alpha \hfill & =y\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}\left[I-X{T}^{1/2}{\left(T^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}X{T}^{1/2}\right)}^{†}{T}^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}\right]\alpha \hfill \\ \hfill & =y\prime M\alpha .(\text{A}.1)\hfill \end{array}$$

With ${\Delta }_s$ being an orthogonal projection, note that by Lemma 2 of Théberge (2017), $M=M{\Delta }_s,$ and that by the properties of the Moore-Penrose inverse, $M{\Delta }_{s}X{T}^{1/2}{\left(T^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}X{T}^{1/2}\right)}^{†}=0.$ For $T$ and $U$ of full rank, one has that the rank of ${\Delta }_{s}X{T}^{1/2}{\left(T^{1/2}X\prime {\left({\text{Δ}}_{s}U{\Delta }_s\right)}^{†}X{T}^{1/2}\right)}^{†}$ equals the rank of ${\Delta }_{s}X{T}^{1/2}.$ It then follows that the range of ${\Delta }_{s}X,$ which equals the range of ${\Delta }_{s}X{T}^{1/2},$ equals the range of ${\Delta }_{s}X{T}^{1/2}{\left(T^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}X{T}^{1/2}\right)}^{†}$ by exercise 1.10 of Ben-Israel and Greville (2002). Therefore, if ${\Delta }_s\alpha$ is in the range of ${\Delta }_{s}X$ which equals the range of ${\Delta }_{s}X{T}^{1/2}{\left(T^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}X{T}^{1/2}\right)}^{†},$ then we will have $M{\Delta }_s\alpha =M\alpha =0.$

References

Ben-Israel, A., and Greville, T.N.E. (2002). Generalized Inverses: Theory and Applications (Second Ed.). New York: Springer-Verlag.

Cassel, C.-M., Särndal, C.-E. and Wretman, J.H. (1977). Foundations of Inference in Survey Sampling. New York: John Wiley & Sons, Inc.

Deville, J.-C., and Särndal, C.-E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association, 87, 376-382.

Godambe, V.P., and Joshi, V.M. (1965). Admissibility and Bayes estimation in sampling finite populations, 1. Annals of Mathematical Statistics, 36, 1707-1722.

Horvitz, D.G., and Thompson, D.J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663-685.

Lunn, A.D., and Davies, S.J. (1998). A note on generating correlated binary variables. Biometrika, 85, 487-490.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. New York: Springer-Verlag.

Théberge, A. (1999). Extensions of calibration estimators in survey sampling. Journal of the American Statistical Association, 94, 635-644.

Théberge, A. (2017). Estimation when the covariance structure of the variable of interest is positive definite. Journal of Official Statistics, 33, 275-299.

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