Multiscale testing of qualitative hypotheses - Lutz Dümbgen; Vladimir G. Spokoiny (Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes, 1999)
Suppose that one observes a process Y on the unit interval, where dY(t) = n 1/ 2f(t)dt + dW(t) with an unknown function parameter f, given scale parameter n 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.
urn:nbn:de:kobv:11-10046865
Keywords: 330; adaptivity; concavity; Lévy's modulus of continuity; monotonicity; multiple test; nonparametric; positivity