On Asymptotically Exact Testing of Nonparametric Hypotheses

Oleg V. Lepskii
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Oleg V. Lepskii: CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium and Institute for System Analysis, Russian Academy of Sciences, Moscow, Russian

No 1993029, LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

Abstract: 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.

Date: 1993-07-01
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