Bernoulli

  • Bernoulli
  • Volume 5, Number 2 (1999), 333-358.

Minimax nonparametric hypothesis testing: the case of an inhomogeneous alternative

Oleg V. Lepski and Vladimir G. Spokoiny

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Abstract

We study the problem of testing a simple hypothesis for a nonparametric ''signal + white-noise'' model. It is assumed under the null hypothesis that the ''signal'' is completely specified, e.g., that no signal is present. This hypothesis is tested against a composite alternative of the following form: the underlying function (the signal) is separated away from the null in the L2 norm and, in addition, it possesses some smoothness properties. We focus on the case of an inhomogeneous alternative when the smoothness properties of the signal are measured in an Lp norm with p<2. We consider tests whose errors have probabilities which do not exceed prescribed values and we measure the quality of testing by the minimal distance between the null and the alternative set for which such testing is still possible. We evaluate the optimal rate of decay for this distance to zero as the noise level tends to zero. Then a rate-optimal test is proposed which essentially uses a pointwise-adaptive estimation procedure.

Article information

Source
Bernoulli, Volume 5, Number 2 (1999), 333-358.

Dates
First available in Project Euclid: 5 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1173147910

Mathematical Reviews number (MathSciNet)
MR1681702

Zentralblatt MATH identifier
0946.62050

Keywords
bandwidth selection error probabilities minimax hypothesis testing nonparametric alternative pointwise adaptive estimation signal detection

Citation

Lepski, Oleg V.; Spokoiny, Vladimir G. Minimax nonparametric hypothesis testing: the case of an inhomogeneous alternative. Bernoulli 5 (1999), no. 2, 333--358. https://projecteuclid.org/euclid.bj/1173147910


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