EFFICIENT ROBUST ESTIMATION OF NONLINEAR REGRESSION PARAMETERS by Arnold

The least median of squares estimator (Rousseeuw, 1984)1 of linear regression parameters is a high breakdown estimator, meaning that, unlike the least squares estimator, it performs reasonably well when up to 50% outliers are present in a data set. Unfortunately, it lacks efficiency under normal errors. This disadvantage can be overcome by using the least median of squares estimator as a starting value for an MM-estimator (Yohai, 1987)2 that retains the high breakdown point of the least median of squares estimator, but has high efficiency, like the least square estimator, when the errors are normally distributed. In this dissertation I prove the consistency of the least median of squares estimator in nonlinear regression, thus showing that it can be used as a starting value for an MM-estimator of nonlinear regression parameters. I then present a new definition of the breakdown point that can be used in nonlinear regression, and show that the least median of squares estimator is a high breakdown nonlinear regression estimator. A computer algorithm for computing these estimates is presented, then the performance of these estimators and the least squares estimators are then compared using simulations and examples. 1Rousseeuw, P.J. (1984) "Least Median of Squares Regression," Journal of the A merican Statistical Association 79:871-880. 2Yohai, V.J. (1987) "High Breakdown Point and High Efficiency Robust Estimates for Regression," Annals of Statistics 15(2):642-656.