Some New Estimation Methods for Weighted Regression When There Are Possible Outliers

The problem of estimating the variance parameter robustly in a heteroscedatic linear model is considered. The situation where the variance is a function of the explanatory variables is treated. To estimate the variance robustly in this case, it is necessary to guard against the influence of outliers in the design as well as outliers in the response. By analogy with the homoscedastic regression case, two estimators are proposed which do this. Their performance is evaluated on a number of data sets. We had considerable success with estimators that bound the "self-influence", that is, the influence an observation has on its own fitted value. We conjecture that in other situations, for example, homoscedastic regression, bounding the self-influence will lead the estimators with good robustness properties.