In this paper, the Horvitz-Thompson estimator is generalized for the presence of correlation. Since calibration estimation seeks weights that are close to the Horvitz-Thompson weights, it too can be generalized by seeking weights that are close to those of the generalized Horvitz-Thompson estimator. Calibration is known to be optimal, in the sense that it asymptotically attains the Godambe-Joshi lower bound. That lower bound can also be generalized to allow for correlation. Generalized calibration asymptotically attains the generalized lower bound.
Simple examples are given here to illustrate how the generalized estimators take advantage of the correlation. This simplicity is achieved by assuming a correlation of one between some population units. Those simple estimators can still be useful, even if the correlation is smaller than one. Simulation results are used to compare the generalized estimators to the ordinary Horvitz-Thompson estimator.