On Asymptotically Exact Testing of Nonparametric Hypotheses

This paper deals with testing of nonparametric hypotheses when the model of observation is unknown function [ sigma (.)] plus a Gaussian White Noise with a small diffusion [ epsilon > 0 ]. It is required to distinguish the simple hypothesis H_0 : [ sigma(.) ] = 0 against the composite alternative H_[ epsilon] : [ sigma(.) ][ is an element of ][ summation_epsilon], where [summation_epsilon] is a certain class of smooth functions, separated from zero by the value [ C psi (epsilon)], that is described by some functional ( the function [ psi (epsilon)] and the constant C depend on the smoothness parameter). We consider two kings of such fonctional namely functional that is a uniform norm on [0,1] and the functional that is the vaue of function at a given point, belonging to [0,1]. Using the minimax approach, we find the exact value of constant C for these two problems in situation of Hölder classes of functions.