A note on multiply robust predictive mean matching imputation with complex survey data
Section 2. Basic setup

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Consider a finite population $F_N\mathrm{<mo>=</mo>}\left\{\left(x_i\mathrm{,}{y}_i\right)\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N\right\},$ assumed to have been generated from the following superpopulation model:

$$y_i\mathrm{<mo>=</mo>}m\left(x_i\right)+{\epsilon }_i\mathrm{,}(2.1)$$

where $m\left(\text{}\cdot \text{}\right)$ is an unknown functional, $x_i$ is a vector of fully observed variables attached to unit $i,$ and the ${\epsilon }_i’s$ are mutually independent random variables such that $E\left({\epsilon }_i|x_i\right)\mathrm{<mo>=</mo>}0$ and $V\left({\epsilon }_i|x_i\right)\mathrm{<mo>=</mo>}{\sigma }^2.$ For simplicity, we assume that the variance structure is homoscedastic but our method can be easily extended to the case of unequal variances.

The interest lies in estimating the population mean, $\theta \mathrm{<mo>=</mo>}E\left(y\right).$ Given the finite population, a probability sample $S,$ of size $n,$ is selected according to a sampling design with first-order inclusion probabilities ${\pi }_i$ and second-order inclusion probabilities ${\pi }_{ij}.$ The sampling weight attached to unit $i$ is denoted by $w_i\mathrm{<mo>=</mo>}{\pi }_i^{-1}.$

Let $r_i$ be response indicator attached to unit $i$ such that $r_i\mathrm{<mo>=</mo>}1$ if $y_i$ is observed, and $r_i\mathrm{<mo>=</mo>}0$ if $y_i$ is missing. Let $S_r\mathrm{<mo>=</mo>}\left\{i\in S\mathrm{<mo>:</mo>}{r}_i\mathrm{<mo>=</mo>}1\right\}$ denote the set of respondents to the survey variable $y.$ We assume that the data are Missing At Random (MAR):

$$\mathrm{Pr}\left(r_i\mathrm{<mo>=</mo>}1|x_i\mathrm{,}{y}_i\right)=\mathrm{Pr}\left(r_i\mathrm{<mo>=</mo>}1|x_i\right)\mathrm{.}(2.2)$$

The customary PMM procedure can be described as follows. We first postulate a parametric outcome regression model $M\mathrm{<mo>=</mo>}\left\{m\left(x_i\mathrm{<mo>;</mo>}\beta \right)\right\}\mathrm{,}$ where $\beta$ is a vector of unknown parameters (Yang and Kim, 2020). For $i\in S,$ we compute the score ${\widehat{m}}_i\mathrm{<mo>=</mo>}m\left(x_i\mathrm{<mo>;</mo>}\widehat{\beta }\right),$ where $\widehat{\beta }$ is a suitable estimator of $\beta$ based on the responding units. Then, the imputed value for the missing $y_i$ is $y_i^{*}\mathrm{<mo>=</mo>}{y}_j,$ where $j$ is the index of the nearest-neighbour of unit $i,$ which satisfies $D\left({\widehat{m}}_j,{\widehat{m}}_i\right)\le D\left({\widehat{m}}_{j^{\prime }},{\widehat{m}}_i\right)$ for any $j^{\prime }\in S_r,$ where $D\left(\cdot \mathrm{,}\cdot \right)$ denotes a distance function; e.g., the Euclidean distance. In order for PMM to be robust against misspecification, the specified parametric model must satisfy the Lipschitz continuity condition (Yang and Kim, 2020). This condition may not be satisfied for some commonly used models and functional forms, including quadratic models; see Yang and Kim (2020) for a discussion.

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