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Adaptive and spatially adaptive testing of a nonparametric hypothesis. (English) Zbl 1103.62345

Summary: This paper continues to study the problem of adaptive nonparametric hypothesis testing started by O. V. Lepski and V. G. Spokoiny [Bernoulli 5, No. 2, 333–358 (1999; Zbl 0946.62050)] and V. G. Spokoiny [Ann. Stat. 24, 2477–2498 (1996; Zbl 0898.62056)]. Let a function \(f\) be observed with noise. A simple null hypothesis \(f\equiv f_0\) is tested against a composite alternative of the form \(\|f-f_0\|_r\geq \rho\). In addition, it is assumed that the underlying function \(f\) possesses some smoothness properties, namely, that \(f\) belongs to some Besov (or Sobolev) ball \(B_{s,p,q}(M)= \{f:\|f\|_{B_{s,p,q}}\leq M\}\). The aim is to evaluate the fastest rate of decay of the radius \(\rho\) to zero as the noise level tends to zero (or equivalently, as the number of observations tends to infinity) for which testing with prescribed error probabilities is still possible. The previous results show that the answer depends heavily on the smoothness parameters \(s,p,q,M\). We consider the problem of adaptive (assumption free) testing when these parameters are unknown. A test \(\varphi^*\) is proposed which is nearly minimax and adaptive at the same time. Compared with the optimal (minimax) rate, this test has a performance which is worse within a log log-factor which is inessential but an unavoidable price for adaptation. The proposed test procedure makes use of the wavelet transform.

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
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