A note on multiply robust predictive mean matching imputation with complex survey data
Section 3. Proposed method

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The proposed method allows the user to specify multiple outcome regression models for the survey variable $y.$ This grants a greater probability of selecting a model that performs well at replicating the relationship between the response variable and the explanatory variables, making the approach multiply robust without requiring the Lipschitz continuity condition to hold. As long as one of the specified models is correctly specified, the resulting estimator will be consistent.

We consider a class of outcome regression models: $M\mathrm{<mo>=</mo>}\left\{m^{(k)}\left(x_i\mathrm{<mo>;</mo>}{\beta }^{(k)}\right)\mathrm{,}k\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}K\right\}\mathrm{.}$ To impute the missing values, we proceed as follows: 

$${\widehat{U}}_m^{\left(\text{}k\text{}\right)}\left({\widehat{\beta }}^{(k)}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in S}{w}_i{r}_i}\left\{y_i-m^{(k)}\left(x_i\mathrm{<mo>;</mo>}{\beta }^{(k)}\right)\right\}\frac{\partial m^{(k)}\left(x_i\mathrm{<mo>;</mo>}{\beta }^{(k)}\right)}{\partial {\beta }^{(k)}}\mathrm{<mo>=</mo>}0.(3.1)$$

$$V_i\mathrm{<mo>=</mo>}(m^{(1)}(x_i\mathrm{<mo>;</mo>}{\widehat{\beta }}^{(1)})\mathrm{,}{m}^{(2)}(x_i\mathrm{<mo>;</mo>}{\widehat{\beta }}^{(2)})\mathrm{,}\dots \mathrm{,}{m}^{(K)}(x_i\mathrm{<mo>;</mo>}{\widehat{\beta }}^{(K)}){)}^T\text{.}$$

$$\widehat{M}\left(x_i\mathrm{<mo>;</mo>}\widehat{\beta }\mathrm{,}\widehat{\eta }\right)\mathrm{<mo>=</mo>}{V}_i^T\widehat{\eta },$$

$$\widehat{\eta }\mathrm{<mo>=</mo>}{\left\{{\displaystyle \sum _{i\in S}{w}_i{r}_i{V}_i{V}_i^T}\right\}}^{-1}{\displaystyle \sum _{i\in S}{w}_i{r}_i{V}_i{y}_i}\mathrm{.}(3.2)$$

After applying (Step1)-(Step4), we construct the imputed estimator of $\theta \text{:}$

$${\widehat{\theta }}_{\text{MR}}\mathrm{<mo>=</mo>}\frac{1}{\widehat{N}}{\displaystyle \sum _{i\in S}{w}_i}\left\{r_i{y}_i+\left(1-r_i\right)y_i^{*}\right\}\mathrm{,}(3.3)$$

where $\widehat{N}={\displaystyle {\sum }_{i\in S}{w}_i}.$ Using an approach similar to the one used by Yang and Kim (2020), it can be shown that the estimator ${\widehat{\theta }}_{\text{MR}}$ is multiply robust in the sense that it is consistent if all but one model are misspecified.

Estimating the variance of ${\widehat{\theta }}_{\text{MR}}$ can be done through replication variance estimation procedures; see e.g., Rust and Rao (1996) and Wolter (2007). In the context of PMM for survey data, Yang and Kim (2020) also considered replication procedures. Let $L$ denote the number of replicates and $w_i^{(g)}$ be a replication weight attached to unit $i$ in the $g^{\text{th}}$ replicate. A replication variance estimator of ${\widehat{\theta }}_{\text{MR}}$ is given by

$${\widehat{V}}_{\text{rep}}\left({\widehat{\theta }}_{\text{MR}}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{g\mathrm{<mo>=</mo>\; 1}}^L{c}_g}{\left({\widehat{\theta }}_{\text{MR}}^{\left(\text{}g\text{}\right)}-{\widehat{\theta }}_{\text{MR}}\right)}^2\mathrm{,}(3.4)$$

where

$${\widehat{\theta }}_{\text{MR}}^{(g)}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in S}{w}_i^{(g)}}\left\{r_i{y}_i+(1-r_i)y_i^{*(g)}\right\}$$

denote the estimator ${\widehat{\theta }}_{\text{MR}}$ in the $g^{\text{th}}$ replicate with $y_i^{*(g)}$ denoting the imputed value attached to unit $i$ in the $g^{\text{th}}$ replicate, obtained from (Step1)-(Step4) above, based on the replication weight $w_i^{(g)}$ instead of the original weights $w_i.$ The factor $c_g$ in (3.4) is determined by the replication method. For instance, with the delete-one jackknife, we have $L\mathrm{<mo>=</mo>}n\mathrm{,}$ $c_g\mathrm{<mo>=</mo>}n/\left(n-1\right)$ and $w_i^{(g)}\mathrm{<mo>=</mo>}n/(n-1)w_i$ if $i\ne g$ and $w_i^{(g)}\mathrm{<mo>=</mo>}0$ if $i\mathrm{<mo>=</mo>}g.$

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