Universally Optimal Robustness-Efficiency Tradeoffs for a General Class of Minimum Divergence Estimators
Balancing the efficiency of an estimator under ideal conditions against its robustness under contamination remains a central challenge in robust statistics. While minimum divergence methods offer a flexible alternative to traditional M-estimation, choosing the appropriate discrepancy measure has historically relied on heuristic or empirical justifications. This manuscript introduces a rigorous optimality criterion for this selection process. By investigating the comprehensive Generalized Alpha-Beta Divergence (GABD) family, we explicitly characterize the Pareto frontier dictating the lowest possible asymptotic variance for any strictly enforced asymptotic breakdown point. Our main theoretical results establish that the estimator achieving this mathematical optimum invariably falls within the extended $(\phi, \gamma)$-divergence class. Crucially, the derived optimal tuning parameter, $\phi^*$, given other parameters, depends solely on the desired breakdown threshold and is entirely invariant to both the assumed parametric model and the exact nature of the data contamination. Supported by comprehensive derivations of asymptotic normality, influence functions, and breakdown thresholds for both continuous and discrete settings, this work offers a unified, theoretical resolution to the long-standing problem of optimal divergence selection in robust inference.