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 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.