Bregman projection for calibration estimation in Survey Sampling

Calibration weighting is a fundamental tool in survey sampling for incorporating auxiliary population information into design-based estimators. Classical formulations measure distance between calibrated and design weights on the multiplicative ratio scale. We develop a unified framework based on Bregman divergence defined directly on the weight vector. The framework reveals a primal--dual symmetry in which both the weight-space and multiplier-space optimization problems are themselves Bregman projections, and the calibrated weights satisfy a generalized Pythagorean decomposition with respect to the constraint manifold. The resulting estimator is asymptotically equivalent to a debiased prediction estimator whose regression coefficient depends explicitly on the Bregman generator, in contrast to the generalized regression estimator equivalent of classical calibration. Exploiting this dependence, we identify a contrast-entropy generator that achieves design-optimality under Poisson sampling. Two extensions are developed: cross-fitted estimation under non-probability sampling, yielding doubly robust inference under standard product-rate conditions; and a regularized extension whose Lagrangian dual produces a H\"older-conjugate penalty for soft balance under high-dimensional auxiliary variables. Simulations and an analysis of National Oceanic and Atmospheric Administration (NOAA)'s Large Pelagics Intercept Survey illustrate the framework.