Discussion Papers, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes 1999,99
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.
adaptivity concavity Lévy's modulus of continuity monotonicity multiple test nonparametric positivity