Estimating Non-Smooth Functional Using Non-Parametric Procedure in the Hilbert Sample Space

Abstract

One of the problems in statistical inference is that of estimating functional. A functional is a mathematical relation that maps two or more functions in one number. They are either smooth or non-smooth. The smoothness properties of functional determine the quality of estimation. However, non-smooth functional lack some degree of properties traditionally relied upon in estimation. Lack of these properties highlights the reason why standard techniques fail to give sharp results. In this paper, an estimator for an arbitrary non-smooth functional is proposed in the nonparametric set-up using robust stochastic Hilbert sample spaces. The estimator is based on the Mini Max criterion where lower and upper bounds are constructed. However, while working in the context of Mini Max estimation, the lower bounds are most important. The approximation theory is used to construct an estimator that is asymptotically sharp Mini Max when the means are bounded. The procedure used allows analysis and presentation data at hand without making any assumption about the underlying distribution. Therefore, the predictions do not depend on whether or not the underlying distribution is normal