Optimal confidence bands for shape-restricted curves

Abstract

Let K be a stochastic process on [0, 1] satisfying dY(t) = n 1/2f(t)dt + dW(t), where in ≥ 1 is a given scale parameter ('sample size'), W is standard Brownian motion and f is an unknown function. Utilizing suitable multiscale tests, we construct confidence bands for f with guaranteed given coverage probability, assuming that f is isotonic or convex. These confidence bands are computationally feasible and shown to be asymptotically sharp optimal in an appropriate sense. © 2003 ISI/BS.