COMPUTATION OF HIGH BREAKDOWN NONLINEAR-REGRESSION PARAMETERS

Most algorithms for estimating high breakdown regression estimators in linear regression rely on finding the least squares fit to many p-point elemental sets, where p is the dimension of the X matrix. Such an approach is computationally infeasible in nonlinear regression. This article presents a new algorithm for computing high breakdown estimates in nonlinear regression that requires only a small number of least squares fits to p points. The algorithm is used to compute Rousseeuw's least median of squares (LMS) estimate and Yohai's MM estimate in both simulations and examples. It is also used to compute bootstrapped and Monte Carlo standard error estimates for MM estimates, which are compared with asymptotic standard errors (ASE's). Using the PROGRESS algorithm for a two-parameter nonlinear model with sample size 30 would require finding the least squares fit to 435 two-point subsets of the data. In the settings considered in this article, the proposed algorithm performs just as well with 25 as with 435 least squares fits, thus substantially reducing computation time. Using the new algorithm to compare standard error estimates for MM estimates reveals that, at least in the setting considered here, bootstrapped and to a lesser extent ASE's become increasingly unreliable as the percentage of outliers increases, In one example presented, the high breakdown estimates are useful in determining that a point appearing to be an outlier in the least squares analysis probably should not be considered an outlier. In the other example, the least squares analysis reveals no outliers and suggests that the model is inappropriate for the data. The residual plots using high breakdown estimates reveal that two points are outliers, and the model provides a reasonable fit to the remainder of the data.