ON MINIMAX RISK OF NON-SMOOTH FUNCTIONAL, ITS ASYMPTOTIC PROPERTIES AND POLYNOMIAL ESTIMATION - MUKHWANA, MOSES KOLOLI; OTIENO, ROMANUS O.; GEORGE, ORWA O.; JOSEPH, MUNG’ATU K.; OKELO, N. B. (IJRDO -JOURNAL OF MATHEMATICS, 2015)
Nonparametric estimation of non-smooth functionals deals with highly structuredproblems which arise, modeled or cast differently from the ones for whichmainline numerical methods have been designed. Non-smooth functionalestimation problems show some features that are different from those of estimatingsmooth functionals. This is in terms of the optimal rates of convergence as well asthe technical tools needed for the analysis of the MiniMax lower bounds and theconstruction of the optimal estimators. The main difficulty of estimating the nonsmooth functionals is traced back to the non differentiability of the absolute valuefunction at the origin. This is reflected both in the derivation of the lower boundsand the construction of optimal estimators. The construction of the optimalestimators of the non-smooth functionals is more complicated than those for linearand quadratic functionals. In this study we consider asymptotic properties,polynomial estimation and MiniMax risk involving non-smooth functionals.