Estimating a monotone trend

TL;DR

This paper investigates nonparametric estimation of a nondecreasing trend in time series data with stationary fluctuations, analyzing the asymptotic behavior of isotonic estimators and proposing modifications to address boundary issues.

Contribution

It provides new weak convergence results for isotonic estimators under minimal conditions and introduces modifications to improve boundary estimation accuracy.

Findings

Rescaled isotonic estimators converge to Chernoff's distribution.

Proposed boundary modifications reduce spiking problems.

Simulation studies confirm approximation accuracy.

Abstract

Motivated by global warming issues, we consider a time se- ries that consists of a nondecreasing trend observed with station- ary fluctuations, nonparametric estimation of the trend under monotonicity assumption is considered. The rescaled isotonic es- timators at an interior point are shown to converge to Chernoff's distribution under minimal conditions on the stationary errors. Since the isotonic estimators suffer from the spiking problem at the end point, two modifications are proposed. The estima- tion errors for both estimators of the boundary point are shown to have interesting limiting distributions. Approximation accu- racies are assessed through simulations. One highlight of our treatment is the proof of the weak convergence results which involve several recent techniques developed in the study of con- ditional central limit questions. These weak convergences can be shown to hold conditionally given the starting values.