  We consider the problem of adaptive inference on a regression function at a
point under a multivariate nonparametric regression setting. The regression
function belongs to a H\"older class and is assumed to be monotone with respect
to some or all of the arguments. We derive the minimax rate of convergence for
confidence intervals (CIs) that adapt to the underlying smoothness, and provide
an adaptive inference procedure that obtains this minimax rate. The procedure
differs from that of Cai and Low (2004), intended to yield shorter CIs under
practically relevant specifications. The proposed method applies to general
linear functionals of the regression function, and is shown to have favorable
performance compared to existing inference procedures.
