  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.
