A generalization of inverse probability weighting
Section 1. Introduction

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The usual inverse probability estimator of the total for a population of $N$  units is

$${\widehat{\theta }}_{\text{IP}}={\displaystyle \sum _{i=1}^N\frac{{\delta }_i{y}_i}{{\pi }_i}},(1.1)$$

where $y_i$ is the variable of interest for unit $i,$ ${\delta }_i$ is 1 or 0 depending on whether $i$ is in the sample $s$ or not, and ${\pi }_i>0$ is the probability that $i$ is in $s.$ Note that the expectation of ${\delta }_i$ is ${\pi }_i,$ this makes ${\widehat{\theta }}_{\text{IP}}$ unbiased for $\theta ={\displaystyle {\sum }_{i=1}^N{y}_i}.$ It is also known as the Horvitz-Thompson estimator, presented in Horvitz and Thompson (1952). In this paper, estimators that can draw some strength from units not in $s$ will be presented.

Here is an example of such an estimator for a population of $N$ units that is partitioned into $N_p=N/2$ pairs $\left\{2i-1,2i\right\}\left(i=1,2,\dots ,N_p\right),$

$${\widehat{\theta }}_{\text{LIM}}={\displaystyle \sum _{i=1}^{N_p}\frac{2y_{2i-1}{\delta }_{2i-1}+2y_{2i}{\delta }_{2i}-\left(y_{2i-1}+y_{2i}\right){\delta }_{2i-1}{\delta }_{2i}+\left(y_{2i-1}-y_{2i}\right){\delta }_{2i-1}{\delta }_{2i}{\pi }_{\text{diff}i}}{{\pi }_{2i-1}+{\pi }_{2i}-{\pi }_{2i-12i}}},(1.2)$$

where ${\pi }_{2i-12i}=E\left({\delta }_{2i-1}{\delta }_{2i}\right)$ is the probability that both units $2i-1$ and $2i$ are in $s,$ and ${\pi }_{\text{diff}i}=\left({\pi }_{2i}-{\pi }_{2i-1}\right)/{\pi }_{2i-12i}.$ It can be verified that ${\widehat{\theta }}_{\text{LIM}}$ is also unbiased.

It should be noted that the denominators in (1.2) correspond to the probability that at least one unit of the pair is in the sample. Thus, this estimator is reminiscent of inverse probability weighting, except it is based on pairs, instead of individual units. The numerators in (1.2) correspond to a value assigned to each pair with at least one sampled unit, and each observed pair is given a weight equal to the inverse of the probability of being observed. From the observation of only one unit of a pair, the estimator (1.2) assigns a value to the pair, and if the units of a pair are strongly correlated, this may be an efficient way to utilize this correlation. The estimator is a special case of a more general one that applies to more general populations, not only those with units grouped in pairs. Because it yields examples that give some insight into the general estimator, and because those examples can be given an explicit form that is simple to interpret and understand, Section 6 and Section 7 will also be about the case where the population, or a domain, is partitioned into pairs. The generalized inverse probability estimator is presented in Section 2; it depends on a parameter $\Sigma ,$ a positive definite $N\times N$ matrix. In Section 3, the new estimator is applied to the problem of calibration. The choice of the parameter $\Sigma$ is discussed in Section 4. In Section 5, it is seen that, with the right choice for $\Sigma ,$ the generalized calibration estimator is optimal, in the sense that it asymptotically attains a generalization of the Godambe-Joshi lower bound. Simple examples are given in Section 6, and the results of a simulation are presented in Section 7. Section 8 summarizes the paper.

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