The Annals of Statistics

Multiscale Testing of Qualitative Hypotheses

Lutz Dümbgen and Vladimir G. Spokoiny

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Suppose that one observes a process Y on the unit interval, where dY(t) =n1/2 f(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.

Article information

Ann. Statist., Volume 29, Number 1 (2001), 124-152.

First available in Project Euclid: 5 August 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

adaptivity concavity Lévy's modulus of continuity monotonicity multiple test nonparametric positivity


Dümbgen, Lutz; Spokoiny, Vladimir G. Multiscale Testing of Qualitative Hypotheses. Ann. Statist. 29 (2001), no. 1, 124--152. doi:10.1214/aos/996986504.

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