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 $d Y(\mathbf{t}) = n^{1/2} f(\mathbf{t}) \,d\mathbf{t} + d W(\mathbf{t})$, where $Y$ is the observed stochastic process on $[0,1]^d$ ($d\ge 1$), $W$ 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 (H\"{o}lder) 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).

翻译：本文研究并提出了预测变量为多元时，形状约束回归函数的置信带构建方法。具体而言，我们考虑连续多维白噪声模型：$d Y(\mathbf{t}) = n^{1/2} f(\mathbf{t}) \,d\mathbf{t} + d W(\mathbf{t})$，其中 $Y$ 是定义在 $[0,1]^d$（$d\ge 1$）上的观测随机过程，$W$ 是 $[0,1]^d$ 上的标准布朗片，$f$ 属于特定（形状约束）函数类（如坐标单调函数或凸函数）的未知目标函数。所构建的置信带基于局部核平均，并通过多元多尺度统计量自动选择带宽。该置信带对任意样本量 $n$ 及函数类中的每个成员均保证覆盖概率。在 $f$ 满足单调/凸性约束时，所提置信带（在宽度上）能自动适应未知函数 $f$ 的全局与局部（Hölder）光滑性及内在维度；在特定意义上，该置信带也被证明是最优的。当底层函数具有“低复杂度”（如分段常数/仿射函数）时，这些置信带具有（近似）参数化 $n^{-1/2}$ 的宽度。