Optimal confidence bands for shape-restricted regression in multidimensions

In this paper, we propose and study construction of confidence bands for shape-constrained regression functions when the predictor is multivariate. In particular, we consider the continuous multidimensional white noise model given by dY(t)=n1∕2f(t)dt+dW(t), where Y is the observed stochastic process on [0,1]d (d≥1), W is the standard Brownian sheet on [0,1]d, and f is the unknown function of interest assumed to belong to a (shape-constrained) function class, e.g., coordinate-wise monotone functions or convex functions. The constructed confidence bands are based on local kernel averaging with bandwidth chosen automatically via a multivariate multiscale statistic. The confidence bands have guaranteed coverage for every n and for every member of the underlying function class. Under monotonicity/convexity constraints on f, the proposed confidence bands automatically adapt (in terms of width) to the global and local (Hölder) smoothness and intrinsic dimensionality of the unknown f; the bands are also shown to be optimal in a certain sense. These bands have (almost) parametric (n−1∕2) widths when the underlying function has “low-complexity” (e.g., piecewise constant/affine). We also develop an optimal testing methodology for checking if the underlying regression function is actually coordinate-wise monotone.