Generalized pseudo empirical likelihood inferences for complex surveys

We consider generalized pseudo empirical likelihood inferences for complex surveys. The method is based on a weighted version of the Kullback-Leibler (KL) distance for calibration estimation (Deville and Särndal, 1992) and includes the pseudo empirical likelihood estimator (Chen and Sitter, 1999; Wu and Rao, 2006) and the calibrated likelihood estimator (Tan, 2013) as special cases. We show that a suitably formulated empirical likelihood ratio-type statistic follows asymptotically a scaled chisquare distribution, which extends the main result in Wu and Rao (2006) and makes the likelihood ratio-type confidence intervals available for calibration estimation using arbitrary choices of the weighting factor in the weighted KL distance. We further show that the scaling factor for the scaled chisquare distribution can be circumvented either through a particular choice of the weighting factor for the KL distance or using a bootstrap method. The proposed bootstrap procedure is justified for single-stage sampling designs with negligible sampling fractions. Finite sample performances of confidence intervals constructed using our proposed methods are investigated and compared with existing ones through two simulation studies.