New goodness-of-fit tests and their application to nonparametric confidence sets

Suppose one observes a process V on the unit interval, where dV = fₒ + dW with an unknown parameter fₒ ε L₁[0, 1] and standard Brownian motion W. We propose a particular test of one-point hypotheses about fₒ which is based on suitably standardized increments of V. This test is shown to have desirable consistency properties if, for instance, fₒ is restricted to various Hölder classes of functions. The test is mimicked in the context of nonparametric density estimation, nonparametric regression and interval-censored data. Under shape restrictions on the parameter, such as monotonicity or convexity, we obtain confidence sets for fₒ adapting to its unknown smoothness.