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Multiscale testing of qualitative hypotheses. (English) Zbl 1029.62070

Summary: Suppose that one observes a process Y on the unit interval, where \[ dY(t) =n^{1/2} f(t)dt +dW (t) \] with an unknown function parameter \(f\), given scale parameter \(n \leq 1\) (“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.

MSC:

62M02 Markov processes: hypothesis testing
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference

Software:

SiZer
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References:

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