Empirical likelihood inference for missing survey data under unequal probability sampling
Section 1. Introduction

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Item nonresponse is commonly seen in sample surveys. A popular method of handling item nonresponse is hot-deck imputation because (i) it preserves the distribution of item values as opposed to mean imputation which leads to a “spike” at the mean of respondent values, (ii) it provides a complete data file and allows the same survey weight to be used for all items, and (iii) results from different analyses based on imputed data are consistent with each other (Rao and Shao, 1992).

Our focus is on fractional hot-deck imputation, where a few values are drawn randomly from the set of respondent values (donors) and the average or weighted average of the drawn values is used to fill in a missing value. For validity and accuracy of inference based on imputation, the observed sample is usually grouped into homogeneous classes, called imputation classes, according to auxiliary variables that are observed for all the sample units (Brick and Kalton, 1996). Haziza and Beaumont (2007) gave a comprehensive review on different methods of constructing imputation classes. Missing values are imputed using the donors within classes and independently across classes.

We consider the case where imputation classes are formed according to a categorical variable $z$ with finite support $\left\{\mathrm{1,}\dots \mathrm{,}K\right\}$ and whose value is observed on all the units of a probability sample, denoted $s,$ of fixed size $n,$ selected from a finite population $U$ of size $N.$ We shall focus on the probability proportional to size (PPS) sampling with replacement or without replacement with negligible sampling fractions in the paper, but our theory applies to any fixed-size unequal probability sampling design with replacement. Let $y$ be a variable (or item) of interest with value $y_i$ observed only for some but not all $i\in s$ due to nonresponse. Let ${\delta }_i$ be the response indicator for $y_i$ taking the value 1 when the associated $y_i$ is observed and 0 otherwise. The ${\delta }_i$ are assumed to be independent random variables across $i\in s.$ We assume that $y_i$ is missing at random (MAR) with uniform response rate within each imputation class, that is, given the value of $z_i,$ the probability of response of unit $i$ does not depend on the value of $y_i.$ More precisely, this means that, for $k\mathrm{<mo>=</mo>\; 1,}\dots ,K,$

$$P\left({\delta }_i\mathrm{<mo>=</mo>\; 1}|y_i\mathrm{,}{z}_i\mathrm{<mo>=</mo>}k\right)\mathrm{<mo>=</mo>}P\left({\delta }_i\mathrm{<mo>=</mo>\; 1}|z_i\mathrm{<mo>=</mo>}k\right)\mathrm{<mo>=</mo>}{P}_k$$

for all $i\in s$ such that $z_i\mathrm{<mo>=</mo>}k,$ where $P_k\in \left(\mathrm{0,}1\right)$ is a constant.

We aim to construct reliable confidence intervals (CIs) for the population mean $\mu$ of a given function $h$ of $y,$ i.e.,

$$\mu \mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in U}h\left(y_i\right)}\mathrm{.}$$

For example, taking $h\left(y\right)\mathrm{<mo>=</mo>}y$ yields the population mean of $y,$ and taking $h\left(y\right)\mathrm{<mo>=</mo>}1\left(y\le t\right),$ where $1(\cdot )$ is the indicator function, gives the finite-population distribution function of $y$ at a given value $t.$ Note that $\mu$ can be alternatively defined as the solution to the population-level estimating equation ${\displaystyle {\sum }_{i\in U}}\left(h\left(y_i\right)-\mu \right)\mathrm{<mo>=</mo>\; 0}$ in $\mu ,$ and similarly, our theory can be readily extended to constructing CIs for a population parameter $\theta$ defined by the solution to the equation ${\displaystyle {\sum }_{i\in U}g\left(y_i\mathrm{,}\theta \right)\mathrm{<mo>=</mo>\; 0}}$ for a general smooth estimating function $g\left(y\mathrm{,}\theta \right)$ in  $\theta .$

The empirical likelihood (EL) method, proposed by Owen (1988) for independent and identically distributed (IID) complete data, has received much attention since it provides a non-parametric approach to constructing likelihood-ratio-type confidence intervals. The EL ratio intervals have several desirable properties: shape and orientation of these intervals are determined entirely by the data and the intervals are range preserving and transformation invariant (Owen, 2001). Qin and Lawless (1994) studied EL inference for parameters defined by smooth estimating equations. Wang and Chen (2009) used EL and imputation to handle IID data that are subject to a MAR assumption. Tang and Qin (2012) proposed an efficient EL estimator based on an inverse-probability-weighted imputation for IID MAR data. In the sample survey context, several variants of EL have been proposed. Chen and Sitter (1999) developed a pseudo EL for complex surveys with auxiliary information. Chen and Kim (2014) proposed a population EL. However, neither papers handles the case of missing data. Cai, Qin, Rao and Winiszewska (2019) proposed an EL method based on imputation for missing survey data under stratified random sampling. Our paper adapts the method of Cai et al. (2019) to accommodate unequal probability sampling designs.

We define a fractionally imputed estimating function of the mean parameter $\mu$ and propose an EL method based on this imputed estimating function for PPS sampling with negligible sampling fractions or with replacement. We derive the asymptotic distributions of the associated maximum EL estimator (MELE) and EL ratio. Based on these limiting distributions, we propose two asymptotically correct bootstrap methods for constructing CIs on $\mu .$ Additionally, we show that the usual bootstrap procedures lead to asymptotically incorrect coverage probabilities when the number of random draws in the fractional imputation is fixed. Simulation studies show that the proposed EL-ratio-based bootstrap interval clearly outperforms the proposed MELE-based bootstrap interval especially when the inclusion probabilities vary considerably or when the sample size is not too large.

Section 2 introduces the proposed fractional imputation on a function of the population mean. Section 3 presents an EL under imputation and its asymptotic properties. Section 4 gives the proposed bootstrap-EL procedures for constructing CIs. Section 5 presents results from simulation studies. Lengthy technical details and proofs are delegated to the Appendices.

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