Open Access
December 1996 Adaptive hypothesis testing using wavelets
V. G. Spokoiny
Ann. Statist. 24(6): 2477-2498 (December 1996). DOI: 10.1214/aos/1032181163


Let a function f be observed with a noise. We wish to test the null hypothesis that the function is identically zero, against a composite nonparametric alternative: functions from the alternative set are separated away from zero in an integral (e.g., $L_2$) norm and also possess some smoothness properties. The minimax rate of testing for this problem was evaluated in earlier papers by Ingster and by Lepski and Spokoiny under different kinds of smoothness assumptions. It was shown that both the optimal rate of testing and the structure of optimal (in rate) tests depend on smoothness parameters which are usually unknown in practical applications. In this paper the problem of adaptive (assumption free) testing is considered. It is shown that adaptive testing without loss of efficiency is impossible. An extra log log-factor is inessential but unavoidable payment for the adaptation. A simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of Besov classes.


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V. G. Spokoiny. "Adaptive hypothesis testing using wavelets." Ann. Statist. 24 (6) 2477 - 2498, December 1996.


Published: December 1996
First available in Project Euclid: 16 September 2002

zbMATH: 0898.62056
MathSciNet: MR1425962
Digital Object Identifier: 10.1214/aos/1032181163

Primary: 62G10
Secondary: 62G20

Keywords: Adaptive testing , minimax hypothesis testing , nonparametric alternative , signal detection , thresholding , wavelet decomposition

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 6 • December 1996
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