We propose a new framework for the simultaneous inference of monotone smooth 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 which have received limited attention. 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. The second scenario involves simultaneous inference of monotone smooth regression coefficient functions in time-varying 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 Greenhouse effect hypothesis. The validity of the proposed methods has been justified theoretically as well as extensive simulations.

翻译：我们提出一种新框架，用于在复杂时间动态下对单调平滑时变函数进行联合推断，该方法利用单调重排与非参数估计技术。通过高斯近似对非参数单调估计量进行逼近，并借助精心设计的自助法构造渐近精确的联合置信带（SCBs）。我们研究了两个具有实际意义但鲜受关注的场景：其一为中等高维时间序列中单调平滑趋势的联合推断，所提算法已应用于多区域温度曲线的联合分析。具体而言，现有方法多针对单一单调平滑趋势设计，而在本例中，我们提出的SCB在保持置信水平的同时，其宽度在现有方法中经验性地达到最窄。第二个场景涉及时变线性模型中单调平滑回归系数函数的联合推断。该算法已用于检验日照时长对温度的影响——这被认为符合温室效应假说中温度上升的预期。所提方法的有效性已通过理论证明及大量数值模拟得到验证。