Honest adaptive confidence bands and self-similar functions

Confidence bands are confidence sets for an unknown function f, containing all functions within some sup-norm distance of an estimator. In the density estimation, regression, and white noise models, we consider the problem of constructing adaptive confidence bands, whose width contracts at an optimal rate over a range of H\"older classes. While adaptive estimators exist, in general adaptive confidence bands do not, and to proceed we must place further conditions on f. We discuss previous approaches to this issue, and show it is necessary to restrict f to fundamentally smaller classes of functions. We then consider the self-similar functions, whose H\"older norm is similar at large and small scales. We show that such functions may be considered typical functions of a given H\"older class, and that the assumption of self-similarity is both necessary and sufficient for the construction of adaptive bands. Finally, we show that this assumption allows us to resolve the problem of undersmoothing, creating bands which are honest simultaneously for functions of any H\"older norm

Spatially-adaptive sensing in nonparametric regression

While adaptive sensing has provided improved rates of convergence in sparse regression and classification, results in nonparametric regression have so far been restricted to quite specific classes of functions. In this paper, we describe an adaptive-sensing algorithm which is applicable to general nonparametric-regression problems. The algorithm is spatially adaptive, and achieves improved rates of convergence over spatially inhomogeneous functions. Over standard function classes, it likewise retains the spatial adaptivity properties of a uniform design