| In non-parametric function estimation, providing a confidence band with the right coverage is a challenging problem. This is especially the case when the underlying function has a wide range of unknown degrees of smoothness. Here we propose two methods of constructing an average coverage confidence band built from block shrinkage estimation methods. One is based on the James-Stein shrinkage estimator; the other begins with a Bayesian perspective and is based on a modification of the harmonic estimator. Simulation shows that these confidence bands have its coverage close to or above the nominal coverage even when the underlying function is rough and/or the signal to noise ratio is small. Both of the confidence bands perform consistently well across all test functions even though they have very different shapes and smoothness.;Furthermore, a minimax theory is developed for nonparametric average coverage confidence bands. We show that for the average coverage confidence band, the minimax rate of the expected average length in Sobolev space is of the order n-b1+2b , where beta is the smoothness order of the Sobolev space. It is known that for many minimax optimality criteria, adaptivity is impossible. We also show that even for the average coverage confidence band, adaptivity is still impossible. |
