Simultaneous inference for monotone and smoothly time varying functions under complex temporal dynamics

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.