Nonparametric estimation of dynamics of monotone trajectories

TL;DR

This paper introduces a nonparametric estimator for the dynamics of monotone trajectories, proving its consistency and optimal convergence rate, with applications demonstrated through simulations and real data analysis.

Contribution

It presents a novel nonparametric estimator for monotone trajectory dynamics, establishing its theoretical properties and practical effectiveness.

Findings

Estimator is consistent under regularity conditions

Achieves optimal $L^2$-loss convergence rate

Effective in real data application to growth trajectories

Abstract

We study a class of nonlinear nonparametric inverse problems. Specifically, we propose a nonparametric estimator of the dynamics of a monotonically increasing trajectory defined on a finite time interval. Under suitable regularity conditions, we prove consistency of the proposed estimator and show that in terms of $L^2$-loss, the optimal rate of convergence for the proposed estimator is the same as that for the estimation of the derivative of a trajectory. This is a new contribution to the area of nonlinear nonparametric inverse problems. We conduct a simulation study to examine the finite sample behavior of the proposed estimator and apply it to the Berkeley growth data.