On the convolution theorem for directionally pathwise differentiable functionals

Abstract

The classical semiparametric convolution theorem characterizes the limiting distributions of regular estimators of pathwise differentiable parameters. We extend this analysis to directionally pathwise differentiable functionals. Regularity along a subspace or convex cone forces the directional derivative to agree there with a bounded linear functional, and every such estimator has a Gaussian convolution factor determined by that functional. We characterize the relevant subspaces, compare their efficiency bounds, and identify the influence functions of efficient estimators. These comparisons show when a stronger regularity requirement increases the variance bound. Under an additivity condition, the Gaussian limit experiment separates into a linear estimation problem and a nonlinear one. Their minimax risks add under squared error; for more general losses, the Gaussian component enters through convolution of the loss. Applications to treatment values, instrumental-variable bounds, and calibration give explicit efficient influence functions, compare existing procedures, and establish feasible attainment in finite-stratum models.