2. Basic setup

Jae Kwang Kim and Shu Yang

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Consider a finite population of $N$ elements identified by a set of indices $U=\left\{\mathrm{1,2,}\dots \mathrm{,}N\right\}$ with $N$ known. Associated with each unit $i$ in the population are study variables, $x_i$ and $y_i$ , with $x_i$ always observed and $y_i$ subject to non-response. Let $A$ denote the set of indices for the elements in a sample selected by a probability sampling mechanism. We are interested in estimating $\eta ,$ defined as a (unique) solution to the population estimating equation ${\displaystyle {\sum }_{i=1}^N}U\left(\eta ;x_i\mathrm{,}{y}_i\right)=0.$ For example, a population mean can be obtained by letting $U\left(\eta ;x_i\mathrm{,}{y}_i\right)=\eta -y_i.$ Under complete response, a consistent estimator of $\eta$ is obtained by solving

$${\displaystyle \sum _{i\in A}}{w}_{i}U\left(\eta ;x_i\mathrm{,}{y}_i\right)=\mathrm{0,} (2.1)$$

where $w_i={\left\{Pr\left(i\in A\right)\right\}}^{-1}$ is the inverse of the first-order inclusion probability of unit $i$ . Binder and Patak (1994) and Rao, Yung and Hidiroglou (2002) considered the asymptotic properties of the estimator obtained from (2.1). Under the existence of missing data, we define

$${\delta }_i=\{\begin{array}{ll}1\hfill & \text{if }{y}_i\text{ is observed}\hfill \\ 0\hfill & \text{otherwise}\text{.}\hfill \end{array}$$

A consistent estimator of $\eta$ is then obtained by taking the conditional expectation and solving

$${\displaystyle \sum _{i\in A}}{w}_i\left[{\delta }_{i}U\left(\eta ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right)E\left\{U\left(\eta ;x_i\mathrm{,}Y\right)|x_i\mathrm{,}{\delta }_i=0\right\}\right]=0 (2.2)$$

for $\eta .$ Estimating equation (2.2) is sometimes referred to as expected estimating equation (Wang and Pepe 2000).

To compute the conditional expectation in (2.2), we assume that the finite population at hand is a realization from an infinite population, called superpopulation. In the superpopulation model, we often postulate a parametric conditional distribution of $y$ given $x,f\left(y|x;\theta \right),$ which is known up to the parameter $\theta$ with parameter space $\Omega .$ Under the specified model, we can compute a consistent estimator $\widehat{\theta }$ of $\theta$ and then use a Monte Carlo method to evaluate the conditional expectation in (2.2) given the estimate $\widehat{\theta }.$ If the response mechanism is missing at random (MAR) or ignorable in the sense of Rubin (1976), we can approximate the expected estimating equation in (2.2) by

$${\displaystyle \sum _{i\in A}}{w}_i\left\{{\delta }_{i}U\left(\eta ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right)\frac{1}{m}{\displaystyle \sum _{j=1}^m}U\left(\eta ;x_i\mathrm{,}{y}_i^{*(j)}\right)\right\}=\mathrm{0,} (2.3)$$

where

$$y_i^{*(1)}\mathrm{,}\dots \mathrm{,}{y}_i^{*(m)}\stackrel{i\mathrm{.}i\mathrm{.}d\mathrm{.}}{\sim }f\left(y_i|x_i;\widehat{\theta }\right)\mathrm{.}$$

Often, we use the maximum likelihood estimator $\widehat{\theta },$ which solves

$$S\left(\theta \right)={\displaystyle \sum _{i\in A}}{w}_i{\delta }_{i}S\left(\theta ;x_i\mathrm{,}{y}_i\right)=\mathrm{0,} (2.4)$$

where $S\left(\theta ;x\mathrm{,}y\right)=\partial \log f\left(y|x;\theta \right)/\partial \theta .$ Note that we use the sampling weights $w_i$ in the score equation (2.4). Thus, we are implicitly assuming that the imputation model, the model for generating the imputed values, is the model about the finite population values $f\left(y_i|x_i\right),$ not the model about the sample values. Thus, we allow that the sampling mechanism can be informative in the sense of Pfeffermann (2011). Multiple imputation, on the other hand, uses the sample model, $f_s\left(y_i|x_i\right)\equiv f\left(y_i|x_i\mathrm{,}i\in A\right)$ , to generate the imputed values and often assumes that the sampling mechanism is non-informative. Thus, in multiple imputation, MAR is assumed for the sample at hand, while, in fractional imputation, MAR is assumed for the population. Under informative sampling design, generating imputed values from the sample model $f_s\left(y_i|x\right)$ does not necessarily lead to valid inference even when sample MAR condition holds. See Section 8.4 of Kim and Shao (2013) for further discussion of MAR under informative sampling.

To compute the conditional expectation in (2.2) efficiently, the parametric fractional imputation (PFI) of Kim (2011) can be used. In PFI, the imputed values are generated from a suitable proposal distribution $h\left(y|x_i\right)$ and then the imputed estimating equation (2.3) is changed to

$${\displaystyle \sum _{i\in A}}{w}_i\left\{{\delta }_{i}U\left(\eta ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j=1}^m}{w}_{ij}^{*}U\left(\eta ;x_i\mathrm{,}{y}_i^{*(j)}\right)\right\}=\mathrm{0,} (2.5)$$

where

$$w_{ij}^{*}=\frac{f\left(y_i^{*(j)}|x_i;\widehat{\theta }\right)/h\left(y_i^{*(j)}|x_i\right)}{{\displaystyle {\sum }_{k=1}^m\left\{f\left(y_i^{*(k)}|x_i;\widehat{\theta }\right)/h\left(y_i^{*(k)}|x_i\right)\right\}}}\mathrm{.} (2.6)$$

The choice of the proposal distribution $h(\cdot )$ is somewhat arbitrary. We will discuss a particular choice that may lead to a robust estimation.

The consistency of the resulting estimator $\widehat{\eta }$ from (2.3) or (2.5) can be established under the assumption that the conditional distribution $f\left(y|x;\theta \right)$ is correctly specified (by similar argument in the proof of Corollary II.2 of Andersen and Gill (1982) and its proof is skipped here). In this paper, we consider an alternative approach of fractional imputation that is more robust against the failure of the assumption on the imputation model.

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