The paper studies the problem of constructing nonparametric simultaneous confidence bands with nonasymptotic and distribition-free guarantees. The target function is assumed to be band-limited and the approach is based on the theory of Paley-Wiener reproducing kernel Hilbert spaces. The starting point of the paper is a recently developed algorithm to which we propose three types of improvements. First, we relax the assumptions on the noises by replacing the symmetricity assumption with a weaker distributional invariance principle. Then, we propose a more efficient way to estimate the norm of the target function, and finally we enhance the construction of the confidence bands by tightening the constraints of the underlying convex optimization problems. The refinements are also illustrated through numerical experiments.

翻译：本文研究具有非渐近和无分布保证的非参数同步置信带的构建问题。目标函数假定为带限函数，该方法基于Paley-Wiener再生核希尔伯特空间理论。本文的出发点是近期开发的一种算法，我们针对该算法提出了三类改进。首先，通过将对称性假设替换为更弱的分布不变性原理，放宽了对噪声的假设。其次，提出了一种更高效的目标函数范数估计方法。最后，通过收紧底层凸优化问题的约束条件，增强了置信带的构建。数值实验也验证了这些改进措施的有效性。