Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative

Oleg V. Lepski, Vladimir G. Spokoiny

Research output: Contribution to journalArticlepeer-review

57 Citations (Scopus)


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 a inhomogeneous alternative when the smoothness properties of the signal arc measured in a 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 of 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.

Original languageEnglish
Pages (from-to)333-358
Number of pages26
Issue number2
Publication statusPublished - 1999
Externally publishedYes


  • Bandwidth selection
  • Error probabilities
  • Minimax hypothesis testing
  • Nonparametric alternative
  • Pointwise adaptive estimation
  • Signal detection


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