Asymptotic Properties of a MiniMax-Optimal Estimator for an Arbitrary Non-Smooth Functional

Abstract

This paper revisits the estimation of the non-smooth functional from a Gaussian observation , and studies the finite-sample and asymptotic behaviour of a minimax-optimal estimator constructed by best polynomial approximation and Hermite-polynomial moment estimation. Building on the sharp minimax framework of Cai and Low, we (i) give a self-contained derivation of the general minimax lower bound obtained by testing a pair of composite hypotheses; (ii) construct the estimator and derive its bias and variance with the cutoff-dependence carried through explicitly, showing that the mean squared error attains the minimax rate over the bounded parameter space ; and (iii) establish asymptotic normality under an undersmoothed cutoff, clarifying the conditions under which interval estimation is valid and the bias correction it requires. We then extend the estimator to finite-population estimation in a model-based survey-sampling setting and compare it, by simulation, with the Nadaraya–Watson and modified Nadaraya– Watson estimators. An independent reproduction across a range of sample sizes exhibits the predicted behaviour and quantifies the severe variance growth of the estimator with the cutoff. The paper is an applied and expository consolidation rather than a new rate result.