Uniform confidence bands: Characterization and optimality

This paper studies optimal uniform confidence bands for functions g(x,β0), where β0 is an unknown parameter vector. We provide a simple characterization of a general class of taut 1−α uniform confidence bands, allowing for both nonlinear functions and nonparametrically estimated functions. Specifically, we show that all taut bands can be obtained from projections on confidence sets for β0 and we characterize the class of sets which yield taut bands. Using these results, we then present a computational method for selecting an approximately optimal confidence band for a given objective function. We illustrate the applicability of these results in numerical applications.