A generalization of inverse probability weighting
Section 5. The generalized Godambe-Joshi lower bound

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For any unbiased estimator $\widehat{\theta }$  of the population total $\theta ,$  if $V_p\left(\widehat{\theta }\right)$  is the variance of $\widehat{\theta }$  under the sampling plan, Godambe and Joshi (1965) have given a lower bound for the value of $E_{\xi }{V}_p\left(\widehat{\theta }\right)$  under the assumption that the variance matrix $V_{\xi }\left(y\right)$  was diagonal. That lower bound is the sum of the elements of the diagonal matrix $\left({\left(E\left({\Delta }_s\right)\right)}^{-1}-I\right)V_{\xi }\left(y\right).$  That result is generalized in the following paragraph.

For any linear unbiased total estimator, $\widehat{\theta },$ if $V_{\xi }\left(y\right)$ is positive definite, then $E_{\xi }{V}_p\left(\widehat{\theta }\right)$ is not lower than the sum of the elements of the matrix ${\left(E{\left({\Delta }_s{V}_{\xi }\left(y\right){\Delta }_s\right)}^{†}\right)}^{-1}-V_{\xi }\left(y\right).$ It is easily verified that the usual Godambe-Joshi lower bound is obtained if $V_{\xi }\left(y\right)$ is diagonal.

Just as the calibration estimator asymptotically attains the Godambe-Joshi lower bound, the generalized calibration estimator with $\Sigma =V_{\xi }\left(y\right),$ asymptotically attains the generalized Godambe-Joshi lower bound, regardless of the value of the matrices $X,$ $T$ and $U.$ The link between the value of those three matrices and the value of $V_{\xi }\left(y\right)$ is not examined in this paper, but the calibration problem stated in Section 3 does clarify the role of each of those matrices. The derivation of the generalized lower bound and the proof of the optimality of the generalized calibration estimator are given in Théberge (2017).

The fact that ${\widehat{\theta }}_{\text{GCAL}}\left(V_{\xi }\left(y\right)\right)$ asymptotically attains the generalized Godambe-Joshi lower bound shows that the generalized inverse probability estimator performs well when applied to residuals, as it does in (3.1), even though it is not recommended in general. Similarly, the ordinary inverse probability estimator can be inefficient if the sample size is random, but will perform well if applied to residuals.

It should be noted that, contrary to the ordinary Godambe-Joshi lower bound, the generalized lower bound applies only to linear unbiased estimators. In fact, an example with $V_{\xi }\left(y\right)$ not diagonal, of a non-linear unbiased estimator which does better than the lower bound is given in Théberge (2017).

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