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 & Särndal, 1992) and includes the pseudo empirical likelihood estimator (Chen & Sitter, 1999; Wu & 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 chi-square distribution, which extends the main result in Wu & 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 chi-square 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.