On minimax risk of non-smooth functional, its asymptotic properties and polynomial estimation

Abstract

Nonparametric estimation of non-smooth functionals deals with highly structured problems which arise, modeled or cast differently from the ones for which mainline numerical methods have been designed. Non-smooth functional estimation problems show some features that are different from those of estimating smooth functionals. This is in terms of the optimal rates of convergence as well as the technical tools needed for the analysis of the MiniMax lower bounds and the construction of the optimal estimators. The main difficulty of estimating the non-smooth functionals is traced back to the non differentiability of the absolute value function at the origin. This is reflected both in the derivation of the lower bounds and the construction of optimal estimators. The construction of the optimal estimators of the non-smooth functionals is more complicated than those for linear and quadratic functionals. In this study we consider asymptotic properties, polynomial estimation and MiniMax risk involving non-smooth functionals.