Multiscale testing of qualitative hypotheses

Lutz Dümbgen, Vladimir G. Spokoiny

Research output: Contribution to journalArticlepeer-review

116 Citations (Scopus)

Abstract

Suppose that one observes a process Y on the unit interval, where dY(t) = n1/2f(t)dt+dW(t) with an unknown function parameter f, given scale parameter n ≥ 1 ("sample size") and standard Brownian motion W. We propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity. These tests are asymptotically optimal and adaptive in a certain sense. They are constructed via a new class of multiscale statistics and an extension of Lévy's modulus of continuity of Brownian motion.

Original languageEnglish
Pages (from-to)124-152
Number of pages29
JournalAnnals of Statistics
Volume29
Issue number1
DOIs
Publication statusPublished - Feb 2001
Externally publishedYes

Keywords

  • Adaptivity
  • Concavity
  • Lévy's modulus of continuity
  • Monotonicity
  • Multiple test
  • Nonparametric
  • Positivity

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