WIAS Preprint No. 389, (1998)

Optimal Nonparametric Testing of Qualitative Hypotheses


  • Dümbgen, Lutz
  • Spokoiny, Vladimir
    ORCID: 0000-0002-2040-3427

2010 Mathematics Subject Classification

  • 62G10


  • adaptivity, convexity, Lévy´s modulus of continuity, monotonicity, positivity


Suppose one observes a process Y on the unit interval, where dY = ƒ + n-1/2dW with an unknown function parameter ƒ, given scale parameter n ≥ 1 ("sample size") and standard Brownian motion W. We propose two classes of tests of qualitative nonparametric hypotheses about ƒ such as monotonicity or convexity. These tests are asymptotically optimal and adaptive with respect to two different criteria. As a by-product we obtain an extension of Lévy´s modulus of continuity of Brownian motion. It is of independent interest because of its potential applications to simultaneous confidence intervals in nonparametric curve estimation.

Appeared in

  • Ann. Statist., 29, no. 1 (2001), pp. 124 - 152 under the title: Multiscale testing of qualitative hypotheses.

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