## The Annals of Statistics

### Adaptive hypothesis testing using wavelets

V. G. Spokoiny

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 24, Number 6 (1996), 2477-2498.

Dates
First available in Project Euclid: 16 September 2002

https://projecteuclid.org/euclid.aos/1032181163

Digital Object Identifier
doi:10.1214/aos/1032181163

Mathematical Reviews number (MathSciNet)
MR1425962

Zentralblatt MATH identifier
0898.62056

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

#### Citation

Spokoiny, V. G. Adaptive hypothesis testing using wavelets. Ann. Statist. 24 (1996), no. 6, 2477--2498. doi:10.1214/aos/1032181163. https://projecteuclid.org/euclid.aos/1032181163

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