Empirical likelihood estimation for samples with nonignorable nonresponse

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.