Trimmed likelihood estimators for stochastic differential equations with an application to crack growth analysis from photos

Abstract

We introduce trimmed likelihood estimators for processes given by a stochastic differential equation for which a transition density is known or can be approximated and present an algorithm to calculate them. To measure the fit of the observations to a given stochastic process, two performance measures based on the trimmed likelihood estimator are proposed. The approach is applied to crack growth data which are obtained from a series of photos by backtracking large cracks which were detected in the last photo. Such crack growth data are contaminated by several outliers caused by errors in the automatic image analysis. We show that trimming 20% of the data of a growth curve leads to good results when 100 obtained crack growth curves are fitted with the Ornstein-Uhlenbeck process and the Cox-Ingersoll-Ross processes while the fit of the Geometric Brownian Motion is significantly worse. The method is sensitive in the sense that crack curves obtained under different stress conditions provide significantly different parameter estimates.