Adaptive hypothesis testing using wavelets. (English) Zbl 0898.62056

Summary: 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 Yu. I. Ingster [Math. Methods Stat. 2, No. 2, 85-114 (1993; Zbl 0798.62057); ibid. No. 3, 171-189 (1993; Zbl 0798.62058); ibid., No. 4, 249-268 (1993; Zbl 0798.62059)] and by O. V. Lepski and the author in an unpublished manuscript [see also the later paper Ann. Stat. 25, No. 6, 2512-2546 (1997; Zbl 0894.62041)] 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.


62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62G35 Nonparametric robustness
62M99 Inference from stochastic processes
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