Adaptive hypothesis testing using wavelets

V. G. Spokoiny

Research output: Contribution to journalArticlepeer-review

122 Citations (Scopus)


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., L2) norm and also possess some smoothness properties. The minimax rate of testing for this problem was evaluated in earlier papers by Ingster and by Lepski and Spokoiny 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.

Original languageEnglish
Pages (from-to)2477-2498
Number of pages22
JournalAnnals of Statistics
Issue number6
Publication statusPublished - Dec 1996
Externally publishedYes


  • Adaptive testing
  • Minimax hypothesis testing
  • Nonparametric alternative
  • Signal detection
  • Thresholding
  • Wavelet decomposition


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