Empirical likelihood inference for missing survey data under unequal probability sampling
Section 2. Fractional imputation

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Following the notation in Section 1, for inference about the population mean $\mu$ of $h\left(y\right),$ we first impute the missing $y_i$ within imputation classes as follows. Let $p_i,$ $i\in U,$ be the probabilities induced by a positive size measure, according to which a PPS sample $s$ is selected. Define $d_i\mathrm{<mo>=</mo>}1/\left(n{p}_i\right)$ for all $i\in U.$ We use $d_i$ as design weights in the paper. Define the donor set ${\mathcal{R}}_k$ of class $k,$ $k\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}K,$ as the set of pairs $\left(y_i\mathrm{,}{d}_i\right)$ within class $k$ for which $y_i$ are observed, that is,

$${\mathcal{R}}_k\mathrm{<mo>=</mo>}\left\{\left(y_i\mathrm{,}{d}_i\right):i\in s\mathrm{,}{\delta }_i\mathrm{<mo>=</mo>\; 1,}{z}_i\mathrm{<mo>=</mo>}k\right\}\mathrm{.}$$

For an $i\in s$ with $z_i\mathrm{<mo>=</mo>}k,$ if ${\delta }_i\mathrm{<mo>=</mo>\; 0},$ we select $J\ge 1$ pairs of $\left(y_i\mathrm{,}{d}_i\right)$ at random with replacement from ${\mathcal{R}}_k$ and denote them as $\left(y_{ij}^{*}\mathrm{,}{d}_{ij}^{*}\right)$ for $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}J.$ We then define an imputed estimating function of $\mu$ for all $i\in s$ as

$${\tilde{h}}_i\left(\mu \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K}1\left(z_i\mathrm{<mo>=</mo>}k\right)}\left[{\delta }_i{d}_i\left\{h\left(y_i\right)-\mu \right\}+\left(1-{\delta }_i\right)J^{-1}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^J{d}_{ij}^{*}\left\{h\left(y_{ij}^{*}\right)-\mu \right\}}\right],(2.1)$$

where $1\left(z_i\mathrm{<mo>=</mo>}k\right)$ is an indicator function taking value 1 if $z_i\mathrm{<mo>=</mo>}k$ and 0 otherwise. This is essentially fractional imputation (Kalton and Kish, 1984) on the function $d_i\left\{h\left(y_i\right)-\mu \right\}.$ Note that the typical approach to imputing a missing value from an unequal probability sample, as proposed by Rao and Shao (1992), is to draw $y-$ values from the donor set within imputation class $k$ with probabilities $d_i/{\displaystyle {\sum }_{i\in {\mathcal{R}}_k}}{d}_i,$ then use the drawn values as the imputed values for the missing observation. As oppose to that approach, we draw $y$ values from the donor set with equal probabilities and attach the corresponding design weights as factors to obtain imputed values. Our approach agrees with that of Platek and Gray (1983).

The number of random draws $J$ in (2.1) is assumed to be a fixed integer that does not change with sample size $n.$ This is a typical setup that is used in most real-world applications. In fact, single random imputation with $J\mathrm{<mo>=</mo>\; 1}$ is often used in practice. This setting distinguishes our study from most studies in the literature (such as Wang and Chen (2009)), where $J$ is assumed to increase with $n$ to infinity in asymptotic studies. Note that, in the survey context, imputed values are reported along with the observed ones in data files. Having a small $J$ will keep the data file manageable, which is the primary reason that $J\mathrm{<mo>=</mo>\; 1}$ is preferred in practice. Having a fixed $J$ also frees users from choosing appropriate $J$ for asymptotic validity. In addition, using a small $J$ lessens computation time, although not substantially with modern computers unless both the sample size and the proportion of missing are very large.

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