Robust estimation of constant and time-varying parameters in nonlinear ordinary differential equation models

Abstract

Ordinary differential equation (ODE) models are quite popular for modelling complex dynamic processes in many scientific fields, and the parameters in these models are usually unknown, and we need to estimate them using statistical methods. When some observations are contaminated, regular estimation methods, such as nonlinear least-square estimation, will bring large bias. In this paper, robust estimations of both constant and time-varying parameters in ODE models using M-estimators are proposed, and their asymptotic properties are obtained under some mild conditions. We focus on Huber M-estimator, and also provide a method to adjust the Huber parameter automatically to the observations. The proposed method is compared to existing methods in numerical simulations and CD8 + T cell data analysis. It is demonstrated that our method gain substantial efficiency as well as robust properties.

Differential equations in data analysis

“…Cao, Huang, and Wu (2012) developed a statistically efficient estimation methodology for time‐varying coefficients. Robust estimators were developed by Hu, Qiu, and Cui (2015), while a penalized technique was presented in Li, Zhu, and Wang (2015). Hong and Lian (2012) derived asymptotic properties of two‐step methods in the context of time‐varying coefficients.…”