We propose a new framework for the simultaneous inference of monotone and smoothly time-varying functions under complex temporal dynamics utilizing the monotone rearrangement and the nonparametric estimation. We capitalize the Gaussian approximation for the nonparametric monotone estimator and construct the asymptotically correct simultaneous confidence bands (SCBs) by carefully designed bootstrap methods. We investigate two general and practical scenarios. The first is the simultaneous inference of monotone smooth trends from moderately high-dimensional time series, and the proposed algorithm has been employed for the joint inference of temperature curves from multiple areas. Specifically, most existing methods are designed for a single monotone smooth trend. In such cases, our proposed SCB empirically exhibits the narrowest width among existing approaches while maintaining confidence levels, and has been used for testing several hypotheses tailored to global warming. The second scenario involves simultaneous inference of monotone and smoothly time-varying regression coefficients in time-varying coefficient linear models. The proposed algorithm has been utilized for testing the impact of sunshine duration on temperature which is believed to be increasing by the increasingly severe greenhouse effect. The validity of the proposed methods has been justified in theory as well as by extensive simulations.

翻译：本文提出一种新框架，用于在复杂时间动态下利用单调重排和非参数估计对单调和光滑时变函数进行联立推断。我们利用非参数单调估计量的高斯逼近性质，通过精心设计的bootstrap方法构建渐近正确的联立置信带（SCBs）。研究考察了两种通用且实用的场景。第一个场景是针对中等高维时间序列中单调光滑趋势的联立推断，所提算法已用于多区域温度曲线的联合推断。具体而言，现有方法多针对单一单调光滑趋势设计。在此类情形下，我们提出的SCB在保持置信水平的同时实证展现出最窄宽度，并已用于检验针对全球变暖的若干假设。第二个场景涉及时变系数线性模型中单调和光滑时变回归系数的联立推断。所提算法已被用于检验日照时长对温度的影响——该影响被认为随日益严重的温室效应而增强。本文所提方法的有效性在理论上及大量仿真中均得到验证。