Optimal confidence bands for shaperestricted curves

Let Y be a stochastic process on [0, 1] satisfying dY(t) = n(1/2) f(t)dt + dW(t), where n greater than or equal to 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.