A generalization of inverse probability weighting
Section 3. The generalized calibration estimator

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The sum of the weights of an estimator is an estimate of the known population size, $N.$ When the sampling plan is such that the sample size is not fixed, the ordinary inverse probability estimator of the population size will have a variance greater than zero. The sum of the weights of ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right),$ noted $S\left(\Sigma \right),$ is often a worse estimator of the population size than the sum of the weights of ${\widehat{\theta }}_{\text{IP}};$ it will often vary, even when the sample size is fixed. An estimator whose estimates of the population size vary, cannot be seen as very reliable.

To fix the problem that the ordinary inverse probability estimator experiences when the sample size is variable, calibration can be used. The weights of ${\widehat{\theta }}_{\text{CAL}}$ are calibrated so that their sum equals the population size, $N$. A similar fix can be made to the generalized estimator: ${\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right)=\left(N/S\left(\Sigma \right)\right){\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right).$ The definition of ${\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right)$ will be expanded to include the possibility of more calibration equations involving more auxiliary variables. The use of calibration equations was presented in Deville and Särndal (1992).

With an auxiliary variable matrix $X\in$ ${ℝ}^{N\times q}$  assumed to be of full rank and noting ${\Vert v\Vert }_M={\left(v\prime Mv\right)}^{1/2}$ the weighted Euclidean norm of the vector $v,$ the following problem is addressed:

Calibration Problem: Among the weight vectors $w_s\in$ ${ℝ}^N$  in the range of ${\Delta }_s,$ i.e., non-sampled units should have a weight of 0, which minimize ${\Vert X\prime w_s-X\prime 1_{N\times 1}\Vert }_T,$ i.e., which “best” satisfy the $q$ calibration equations, seek one that minimizes ${\Vert w_s-w_{s\text{GIP}}\left(\Sigma \right)\Vert }_U,$ i.e., as close as possible to the weights of ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right),$ where $T\in$ ${ℝ}^{q\times q}$  and $U\in$${ℝ}^{N\times N}$  are positive definite matrices.

Weights, $w_s,$ that satisfy the calibration equations, $X\prime w_s=X\prime 1_{N\times 1},$ do not always exist, especially if the number of equations, $q,$ is high relative to the sample size. To prepare for this eventuality, the matrix $T$ is at the statistician’s disposal for specifying the relative importance of the $q$ calibration equations. The matrix $U$ specifies the relative importance given to each unit when measuring the distance from $w_{s\text{GIP}}\left(\Sigma \right).$ This formulation of the calibration problem generalizes that of Théberge (1999), where $T$ and $U$ were diagonal matrices, and the inverse probability, or Horvitz-Thompson, weights were used instead of the generalized inverse probability weights.

The solution to the calibration problem yields

$$\begin{array}{ll}{\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right)\hfill & =y\prime w_{s\text{GCAL}}\left(\Sigma \right)\hfill \\ \hfill & =\widehat{y}\prime 1_{N\times 1}+{\left(y-\widehat{y}\right)}^{\prime }{w}_{s\text{GIP}}\left(\Sigma \right),(3.1)\hfill \end{array}$$

where $\widehat{y}=X\widehat{\beta }$ with

$$\widehat{\beta }=T^{1/2}{\left(T^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}X{T}^{1/2}\right)}^{†}{T}^{1/2}X\prime {\left({\Delta }_{s}U{\Delta }_s\right)}^{†}y.(3.2)$$

The estimator ${\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right)$ is asymptotically unbiased. Also, if ${\widehat{\Sigma }}_s\to \Sigma$ in probability, then the bias and variance of ${\widehat{\theta }}_{\text{GCAL}}\left({\widehat{\Sigma }}_s\right)$ are asymptotically the same as those of ${\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right).$ The rate at which ${\widehat{\Sigma }}_s\to \Sigma$ will depend on the estimator ${\widehat{\Sigma }}_s$ and on the number of parameters in $\Sigma .$

The difference between ${\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right)$ and the ordinary calibration estimator, ${\widehat{\theta }}_{\text{GCAL}}\left(I_{N\times N}\right),$ is simply the use of generalized inverse probability weights to estimate the sum of the residues, rather than the usual inverse probability weights. This was to be expected given that in one case we are, in the calibration problem, seeking weights that minimize ${\Vert w_s-w_{s\text{GIP}}\left(\Sigma \right)\Vert }_U,$ instead of weights that minimize ${\Vert w_s-w_{s\text{IP}}\Vert }_U,$ where $w_{s\text{IP}}=w_{s\text{GIP}}\left(I_{N\times N}\right)$ are the usual inverse probability weights.

The following result is proven in the Appendix: 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. A vector $v$ is said to be in the range of a matrix $F$ is there exists a vector $\lambda$ such that $v=F\lambda .$ In particular, if the matrix $U$ is diagonal and written $U=A^{-1}D,$ where $A={\left(E\left({\Delta }_s\right)\right)}^{-1}$ is the diagonal matrix of the ordinary inverse probability weights and $D\in$ ${ℝ}^{N\times N}$  is an arbitrary positive diagonal matrix, then with $\alpha =Dc$ the result gives that ${\left(y-\widehat{y}\right)}^{\prime }{\left({\Delta }_s{A}^{-1}D{\Delta }_s\right)}^{†}Dc={\left(y-\widehat{y}\right)}^{\prime }A{\Delta }_{s}c$ is zero if ${\Delta }_{s}Dc$ is in the range of ${\Delta }_{s}X.$ This is similar to result 6.5.1 of Särndal, Swensson and Wretman (1992), for example, where $c$ is a vector of ones and the diagonal elements of $D$ are variances.

It can be seen from the form of (3.1), that ${\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right)$ is also a regression estimator that uses a model $\xi$ such that $E_{\xi }\left(y\right)=X\beta .$ Despite the notation used in (3.2), calibration estimators do not use models, instead there are calibration equations. When viewed as a regression estimator, it is important to realize that ${\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right)$ is asymptotically design unbiased, regardless of the choice of the model parameter $\beta ,$ and regardless of the choice of the positive definite matrix $\Sigma .$

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