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：我们提出了一个新的框架，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光滑时变函数进行同步推断。我们利用非参数单调估计量的高斯近似，并通过精心设计的自助法构造渐近正确的同步置信带（SCBs）。我们研究了两个普遍且实际但此前关注有限的情景。第一种情景是中等高维时间序列中单调光滑趋势的同步推断，所提出的算法已用于多个区域的温度曲线的联合推断。具体而言，大多数现有方法仅针对单一单调光滑趋势设计。在这种情况下，我们提出的SCB在实证上展现出最窄的宽度，同时保持置信水平。第二种情景涉及时变线性模型中单调光滑回归系数函数的同步推断。所提出的算法已用于检验日照时数对温度的影响，而温室效应假说认为该影响呈上升趋势。所提方法的有效性在理论上和广泛模拟中均得到了验证。