ON CHEBYSHEV’S POLYNOMIALS OF NON-SMOOTH FUNCTIONAL AND ITS NONPARAMETRIC ESTIMATION - MUKHWANA, MOSES KOLOLI; OTIENO, ROMANUS O.; GEORGE, ORWA O.; JOSEPH, MUNG’ATU K.; OKELO, N. B. (IJRDO -JOURNAL OF MATHEMATICS, 2015)
In statistical inference, one of the basic problems is that of estimating functionals. This problemis considered in the nonparametric set-up. The quality of estimation depends on smoothnessproperties of the functional F. However, non smooth functionals lack some degree of propertiestraditionally relied upon in estimation. This highlights the reason why standard techniques fail toyield sharp results. In estimating non smooth functionals, the lower and upper bounds areconstructed for the MiniMax Risk. When working in the context of MiniMax estimation, thelower bounds are important. A single- value MiniMax lower bound is established by applyingthe general lower bound technique based on testing two composite hypotheses. A vital step is theconstruction of two special priors and bounding the chi-square distance between two normalmixtures. An estimator is constructed using approximation theory and Hermite polynomials andis shown to be asymptotically sharp MiniMax when the means are bounded by a given value.