Calibrated estimators in two-stage sampling

We consider the problem of the estimation of the population mean of a study variable by assuming that the population means of an auxiliary variable are known at both stages of sample selection. The design weights at the first and second stages of sample selection are calibrated by optimizing the chi-squared type distance between the design weights and the new weights at both the first and second stages of sample selection. The regression type estimator in two-stage sampling is shown to be a special case. An application of the proposed estimators using a real data set is discussed.