An empirical likelihood approach under cluster sampling with missing observations

The parameter of interest considered is the unique solution to a set of estimating equations, such as regression parameters of generalised linear models. We consider a design-based approach; that is, the sampling distribution is specified by stratification, cluster (multi-stage) sampling, unequal selection probabilities, side information and a response mechanism. The proposed empirical likelihood approach takes into account of these features. Empirical likelihood has been mostly developed under more restrictive settings, such as independent and identically distributed assumption, which is violated under a design-based framework. A proper empirical likelihood approach which deals with cluster sampling, missing data and multidimensional parameters is absent in the literature. This paper shows that a cluster-level empirical log-likelihood ratio statistic is pivotal. The main contribution of the paper is to provide the rigorous asymptotic theory and underlining regularity conditions which imply root n-consistency and the Wilks's theorem or self-normalisation property. Negligible and large sampling fractions are considered.