Hi all, I've been trying to understand the breakdown point of M-Estimators in linear regression however there are a couple of issues that continue to confuse me. I would be grateful if any of you could clarify these issues or point me to a reference that may help me.
Lets begin with the quadratic estimator - it has a breakdown point of zero because a single outlier (erroneous response/output variable) or leverage point (erroneous explanatory/input/factor space variable) can skew the estimate by an arbitrary amount. The root cause of this skew is the large residual caused by the outlier/leverage point. Now, redescending M-Estimators are designed to limit the effect of large residuals. If a residual is over a certain magnitude it is down weighted. Because the residuals of both outliers and leverage points are down weighted in the same way, you would think that M-Estimators would be resistant to both types of error. But the literature says that this isn't the case. M-Estimators are said to have a zero breakdown point of zero because a single outlier can cause an arbitrarily large skew in the estimate. But why? Any residual over the cutoff point will have small effect on the summation. What is the breakdown point of an M-Estimator if only outliers are considered? Can 50% contamination be tolerated? Why are leverage points considered to have a stronger skewing effect than outliers? Is it because in a linear model (e.g. y=mx+c), the explanatory variable x (i.e. the leverage point) is multiplied by the parameter m? If this is the case, then I return to my previous argument, the leverage point may have a larger residual due to the multiplication, but the redescending M-Estimator should limit the effect of this residual. In the literature that I have read, it is often said that M-Estimators have a breakdown point of 1/(p+1) where p is the number of parameters. Is this result correct, or is it limited to the GM-Estimators that are supposed to limit the effect of leverage points by weighting residuals. You help is much appreciated. Julian . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
