Statistica Sinica 20 (2010), 263-280

EMPIRICAL LIKELIHOOD ESTIMATION FOR SAMPLES

WITH NONIGNORABLE NONRESPONSE

Fang Fang$^1$, Quan Hong$^2$ and Jun Shao$^{3,4}$

$^1$GE Consumer Finance, $^2$Eli Lilly and Company,

$^3$University of Wisconsin-Madison and $^4$East China Normal University

Abstract: Nonresponse is very common in survey sampling. Nonignorable nonresponse, a response mechanism in which the response probability of a survey variable $Y$ depends directly on the value of $Y$ regardless of whether $Y$ is observed or not, is the most difficult type of nonresponse to handle. The population mean estimators ignoring the nonrespondents typically have heavy biases. This paper studies an empirical likelihood-based estimation method, with samples under nonignorable nonresponse, when an observed auxiliary categorical variable $Z$ is available. The likelihood is semiparametric: we assume a parametric model on the response mechanism and the conditional probability of $Z$ given $Y$, and a nonparametric model on the distribution of $Y$. When the number of $Z$ categories is not small, a pseudo empirical likelihood method is applied to reduce the computational intensity. Asymptotic distributions of the proposed population mean estimators are derived. For variance estimation, we consider a bootstrap procedure and its consistency is established. Some simulation results are provided to assess the finite sample performance of the proposed estimators.

Key words and phrases: Empirical likelihood, nonignorable nonresponse, pseudo likelihood, sample survey, semiparametric likelihood, stratified samples.