Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative. (English) Zbl 0946.62050

Lepski, Oleg V.; Spokoiny, Vladimir G.

Bernoulli 5, No. 2, 333-358 (1999).

The aim of the paper is to study the problem of testing a simple hypothesis for a non-parametric “signal + white-noise” model. Under the null hypothesis it is assumed that the “signal” is completely specified (e.g., that no signal is present). This hypothesis is tested against a composite alternative where the underlying function (the signal) is separated from the null in the \(L_2\)-norm and, furthermore, it possesses some smoothness properties. The case of an inhomogeneous alternative is investigated, when the smoothness properties of the signal are measured in a \(L_p\)-norm, with \(p< 2\). Tests whose errors have probabilities not exceeding prescribed values are considered, and the quality of testing by the minimal distance between the null and the alternative set is measured (when possible). The optimal rate of decay of this distance to zero is also evaluated, as the noise level tends to zero. Finally, a rate-optimal test is proposed, which essentially uses a pointwise-adaptive estimation procedure.

Reviewer: Neculai Curteanu (Iaşi)

MSC:

62G10	Nonparametric hypothesis testing
62C20	Minimax procedures in statistical decision theory
62M99	Inference from stochastic processes

Keywords:

bandwidth selection; error probabilities; minimax hypothesis testing; nonparametric alternative; pointwise daptive estimation; signal detection; white-noise