A generalization of inverse probability weighting
Section 6. Examples

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There are cases simple enough for ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right)$  to be given explicitly. Say $\Sigma \left(\rho \right)$  is a block-diagonal matrix where each of $N_p=N/2$  blocks equals ${\sigma }^2\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right),$  with $-1<\rho <1.$  Such a block-diagonal matrix corresponds to a model of a population which can be partitioned into pairs $\left\{2i-1,2i\right\}\left(i=1,2,\dots ,N_p\right)$  where, within a pair, the variable of interest is correlated. Then, (2.3) reduces to

$${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \left(\rho \right)\right)={\displaystyle \sum _{i=1}^{N_p}\frac{a_{2i-1}{y}_{2i-1}+a_{2i}{y}_{2i}}{\left({\pi }_{2i-1}{\pi }_{2i}\left(1-{\rho }^2\right)+\left({\pi }_{2i-1}+{\pi }_{2i}-{\pi }_{2i-12i}\right){\pi }_{2i-12i}{\rho }^2\right)}},(6.1)$$

where

$$\begin{array}{ll}{a}_{2i-1}\hfill & ={\delta }_{2i-1}\left[{\pi }_{2i}\left(1-{\rho }^2\right)+{\pi }_{2i-12i}\rho \left(1+\rho \right)\right]+{\delta }_{2i-1}{\delta }_{2i}\left[{\rho }^2{\pi }_{2i}-\rho {\pi }_{2i-1}-{\rho }^2{\pi }_{2i-12i}\right]\hfill \\ a_{2i}\hfill & ={\delta }_{2i}\left[{\pi }_{2i-1}\left(1-{\rho }^2\right)+{\pi }_{2i-12i}\rho \left(1+\rho \right)\right]+{\delta }_{2i-1}{\delta }_{2i}\left[{\rho }^2{\pi }_{2i-1}-\rho {\pi }_{2i}-{\rho }^2{\pi }_{2i-12i}\right].(6.2)\hfill \end{array}$$

Once again, this generalized inverse probability estimator is unbiased, for any value of $\rho ,$ “correct” or not. It is seen that as expected, when $\rho =0,$ the estimator reduces to the inverse probability estimator. The value of $\rho$ cannot simply be set to one in (6.1), because $\Sigma \left(\text{}1\text{}\right)$ is not positive definite. However, the limit of (6.1) as $\rho \to 1$ results in the estimator (1.2) given in the Introduction, ${\widehat{\theta }}_{\text{LIM}}.$ It can be calibrated so that the sum of the weights is equal to $N.$ If the probabilities of inclusion do not vary with $i=1,2,\dots ,N_p,$ the resulting estimator is

$${\widehat{\theta }}_{\text{LCAL}}=\frac{N_p}{{\nu }_p}{\displaystyle \sum _{i=1}^{N_p}2y_{2i-1}{\delta }_{2i-1}+2y_{2i}{\delta }_{2i}-\left(y_{2i-1}+y_{2i}\right){\delta }_{2i-1}{\delta }_{2i}+\left(y_{2i-1}-y_{2i}\right){\delta }_{2i-1}{\delta }_{2i}{\pi }_{\text{diff}i}},(6.3)$$

where ${\nu }_p={\displaystyle {\sum }_{i=1}^{N_p}\left({\delta }_{2i-1}+{\delta }_{2i}-{\delta }_{2i-1}{\delta }_{2i}\right)}$ is the number of pairs with at least one unit in the sample. It is easy to verify, by setting $y=1_{N\times 1}$ in (6.3), that the sum of the weights of ${\widehat{\theta }}_{\text{LCAL}}$ is equal to $2N_p=N.$ The generalized calibration estimator (6.3) is optimized for $\rho \to 1,$ but it can still have a lower variance than both, the inverse probability estimator and the ordinary calibration estimator, if the correlation between the units of a pair is strong (for example, race, religion or education level of a couple). Since a variable indicating which unit is paired with which, must be on the frame, a calibration at the pair level would be possible. The calibration would ensure that the sum of the weights of the sampled units of a pair would equal 2. However, the low number of observations per calibration group would not ensure the validity of asymptotic results and could result in significant biases.

There are modified versions of the generalized inverse probability estimator and of the generalized calibration estimator. The modified versions have the advantage of having a closed form; there is no need to compute the expectation of ${\left({\Delta }_s\Sigma {\Delta }_s\right)}^{†}.$ They also do not rely on the Moore-Penrose inverse. For a positive definite matrix $\Sigma ,$ they are defined as

$${\widehat{\theta }}_{\text{MGIP}}\left(\Sigma \right)=y^{\prime }{\Delta }_s{\Sigma }^{-1}{\Delta }_s{\left({\Sigma }^{-1}\circ \Pi \right)}^{-1}{1}_{N\times 1}(6.4)$$

and

$${\widehat{\theta }}_{\text{MGCAL}}\left(\Sigma \right)=\widehat{y}\prime 1_{N\times 1}+{\left(y-\widehat{y}\right)}^{\prime }{w}_{s\text{MGHT}}\left(\Sigma \right),(6.5)$$

where $\Pi =\left({\pi }_{kl}\right)=\left(E\left({\delta }_k{\delta }_l\right)\right)\in$ ${ℝ}^{N\times N}$ is the matrix of second order probabilities of inclusion, $w_{s\text{MGIP}}\left(\Sigma \right)$ is the vector of weights of ${\widehat{\theta }}_{\text{MGIP}}\left(\Sigma \right),$ and $\circ$ denotes the Hadamard product, i.e., element-wise multiplication. With ${\widehat{\theta }}_{\text{MGIP}}\left(\Sigma \right),$ the “probability” part of the phrase “inverse probability” is ${\Sigma }^{-1}\circ \Pi .$ The modified generalized estimators are also unbiased, or at least asymptotically unbiased in the case of ${\widehat{\theta }}_{\text{MGCAL}}\left(\Sigma \right).$ The usual estimators ${\widehat{\theta }}_{\text{IP}}$ and ${\widehat{\theta }}_{\text{CAL}}$ are obtained if $\Sigma =I.$

If $\rho \to 1,$ the modified generalized inverse probability estimator, ${\widehat{\theta }}_{\text{MGIP}}\left(\Sigma \left(\rho \right)\right),$ becomes:

$${\widehat{\theta }}_{\text{MLIM}}={\displaystyle \sum _{i=1}^{N_p}{w}_{2i-1}{y}_{2i-1}+w_{2i}{y}_{2i}},(6.6)$$

where

$$w_{2i-1}=\frac{{\delta }_{2i-1}\left({\pi }_{2i}+{\pi }_{2i-12i}\right)-{\delta }_{2i-1}{\delta }_{2i}\left({\pi }_{2i-1}+{\pi }_{2i-12i}\right)}{{\pi }_{2i-1}{\pi }_{2i}-{\pi }_{2i-12i}^2}(6.7)$$

and

$$w_{2i}=\frac{{\delta }_{2i}\left({\pi }_{2i-1}+{\pi }_{2i-12i}\right)-{\delta }_{2i-1}{\delta }_{2i}\left({\pi }_{2i}+{\pi }_{2i-12i}\right)}{{\pi }_{2i-1}{\pi }_{2i}-{\pi }_{2i-12i}^2}.(6.8)$$

If the sampling plan is such that ${\pi }_{2i}={\pi }_{2i-1}$ for any $i=1,2,\dots ,N_p,$ and if both units of that pair are sampled, then the weights of both units will be zero. That some sampled units may not contribute to the estimator, in some circumstances, is an undesirable property of ${\widehat{\theta }}_{\text{MLIM}}.$

One characteristic of the estimator ${\widehat{\theta }}_{\text{LIM}}$ is somewhat surprising. It is constructed in such a way that for each observed pair, that is each pair with at least one unit in the sample, the numerator in (1.2) corresponds to a value for the pair’s variable of interest total. The numerator of the $i^{\text{th}}$ term is 0 if neither unit $2i-1$ nor unit $2i$ are observed, it is $2y_{2i-1}$ if only unit $2i-1$ of the pair is sampled, it is $2y_{2i}$ if only unit $2i$ of the pair is sampled, and it is $\left(y_{2i-1}+y_{2i}\right)+\left(y_{2i-1}-y_{2i}\right){\pi }_{\text{diff}i}$ if both units of the pair are sampled. The unexpected characteristic is that when both units of a pair $i$ $\left(i=1,2,\dots ,N_p\right)$ are observed, the estimated value for the pair’s total is not the known total $y_{2i-1}+y_{2i}.$ This is the motivation for yet another estimator and its calibrated version, where the estimate for a pair, while still being unbiased, will agree with the known total when both units of the pair are sampled. The alternative estimators are

$${\widehat{\theta }}_{\text{ALIM}}={\displaystyle \sum _{i=1}^{N_p}\frac{\left(a_i{\delta }_{2i-1}+b_i{\delta }_{2i-1}{\delta }_{2i}\right)y_{2i-1}+\left(c_i{\delta }_{2i}+d_i{\delta }_{2i-1}{\delta }_{2i}\right)y_{2i}}{{\pi }_{2i-1}+{\pi }_{2i}-{\pi }_{2i-12i}}}(6.9)$$

and

$${\widehat{\theta }}_{\text{ALCAL}}=\widehat{y}\prime 1_{N\times 1}+{\left(y-\widehat{y}\right)}^{\prime }{w}_{s\text{ALIM}}\left(\Sigma \right),(6.10)$$

where $w_{s\text{ALIM}}$ is the vector of weights of ${\widehat{\theta }}_{\text{ALIM}},$ $a_i+b_i=c_i+d_i=1,$ motivated by what is wanted when both units of the pair are sampled, and in order to have ${\widehat{\theta }}_{\text{ALIM}}$ unbiased, one should have $a_i{\pi }_{2i-1}+b_i{\pi }_{2i-12i}=c_i{\pi }_{2i}+d_i{\pi }_{2i-12i}={\pi }_{2i-1}+{\pi }_{2i}-{\pi }_{2i-12i}.$ Therefore, for $i=1,2,\dots ,N_p,$

$$\begin{array}{}{\displaystyle a_i\hfill & =\frac{{\pi }_{2i-1}+{\pi }_{2i}-2{\pi }_{2i-12i}}{{\pi }_{2i-1}-{\pi }_{2i-12i}}\hfill }{\displaystyle b_i\hfill & =\frac{{\pi }_{2i-12i}-{\pi }_{2i}}{{\pi }_{2i-1}-{\pi }_{2i-12i}}\hfill }{\displaystyle c_i\hfill & =\frac{{\pi }_{2i-1}+{\pi }_{2i}-2{\pi }_{2i-12i}}{{\pi }_{2i}-{\pi }_{2i-12i}}\hfill }{\displaystyle d_i\hfill & =\frac{{\pi }_{2i-12i}-{\pi }_{2i-1}}{{\pi }_{2i}-{\pi }_{2i-12i}}.(6.11)\hfill }\end{array}$$

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