A General Approach for Calibration Weighting under Missing at Random

We propose a unified class of calibration weighting methods based on weighted generalized entropy to handle missing at random (MAR) data with improved stability and efficiency. The proposed generalized entropy calibration (GEC) formulates weight construction as a convex optimization program that unifies entropy-based approaches and generalized regression weighting. Double robustness is achieved by augmenting standard covariate balancing with a debiasing constraint tied to the propensity score model and a Neyman-orthogonal constraint that removes first-order sensitivity to nuisance estimation. Selection of the weights on the entropy function can lead to the optimal calibration estimator under a correctly specified outcome regression model. The proposed GEC weighting ha a nice geometric characterization: the GEC solution is the Bregman projection of the initial weights onto a constraint set, which yields a generalized Pythagorean identity and a nested decomposition that quantifies the incremental distance paid for additional constraints. We also develop a high-dimensional extension with soft calibration and a projection calibration constraint that preserves doubly robust inference. Two simulation studies are presented to compare the performance of the proposed method with the existing methods.