Risk-Optimal Curvature Selection for Finite-Sample Cressie-Read Moment Estimation

We propose a finite-sample risk-optimal selection criterion for Cressie-Read power divergence (CRPD) estimation in overidentified moment-based models. The CRPD family, dual to generalized empirical likelihood, is indexed by the power parameter $\gamma$. Although $\gamma$ is conventionally fixed at a researcher-chosen value, we argue that it should be interpreted as a data-tunable curvature parameter governing the finite-sample behavior of the CRPD objective. Through implied probability weights and associated Lagrange multipliers, $\gamma$ affects how the empirical distribution is reweighted to enforce the moment restrictions, even when population identification is unchanged. The proposed criterion selects $\gamma$ by minimizing an estimation- and system-oriented risk measure. It combines a structural component, which measures finite-sample distortion in the estimate of the structural parameter relative to a first-order GMM benchmark, with a multiplier-stability component, which measures the cost of moment enforcement in the full estimator-multiplier system. A researcher-specified weight determines the relative importance of the two components, allowing the selection rule to prioritize structural accuracy, multiplier stability, or a balance between them. The resulting selector is designed to reduce second-order finite-sample distortion while discouraging unstable multipliers, concentrated implied weights, and proximity to the feasible-probability boundary. Simulations show that the selected CRPD estimator remains approximately centered around the structural parameter while improving finite-sample stability. An empirical illustration using Owen's dairy-cow data shows that similar point estimates can correspond to different implied weighting schemes, highlighting the practical role of $\gamma$ as a curvature parameter in moment-based estimation.