Multiscale Testing of Qualitative Hypotheses

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 su...

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Bibliographic Details
Published in:The Annals of statistics 2001-02, Vol.29 (1), p.124-152
Main Authors: Dumbgen, Lutz, Spokoiny, Vladimir G.
Format: Article
Language:English
Subjects:
62G10
62G20
adaptivity
concavity
Estimators
Exact sciences and technology
Gaussian distributions
Linear regression
Lévy's modulus of continuity
Mathematical independent variables
Mathematics
monotonicity
multiple test
Musical intervals
nonparametric
Nonparametric inference
Nonparametric Testing
Null hypothesis
positivity
Probability and statistics
Random variables
Sciences and techniques of general use
Signal bandwidth
Statistical theories
Statistics
White noise
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Description
Summary: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 Levy's modulus of continuity of Brownian motion.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/996986504