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.

翻译：我们考虑了在多变量非参数回归设置下某个点的回归函数的适应性推断问题。回归函数属于H\'older类,在部分或全部参数方面被认为是单一的。我们得出适应基本顺畅度的信任区间最小趋同率,并提供一个获得这一微缩速率的适应性推断程序。该程序不同于Cai和Low(2004年)的程序,后者旨在根据实际相关的规格产生较短的CI。拟议方法适用于回归函数的一般线性功能,并显示与现有的推断程序相比,其性能优于现有推论程序。