Simultaneous Inference for Monotone and Smoothly Time-Varying Functions Under Complex Temporal Dynamics

We propose a new framework for the simultaneous inference of monotone and smoothly time-varying functions under complex temporal dynamics. This will be done using 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) using 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. 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. It has also 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 used for testing the impact of sunshine duration on temperature which is believed to be increasing due to severe greenhouse effect. The validity of the proposed methods has been justified in theory as well as by extensive simulations. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.