An efficient class of weighted trimmed means for linear regression models

We propose and study a class of weighted trimmed means based on the symmetric quantile functions for the location and linear regression models. A robustness comparison with the underlying distribution of a symmetric-type heavy tail is given. The weighted trimmed mean in optimal trimming under symmetric distributions is shown to have an asymptotic variance very close to the Cramér-Rao lower bound. For fixed weight setting, the weighted trimmed mean is still relatively more efficient in terms of asymptotic variance than the trimmed mean based on regression quantiles. From the parametric point of view, the computationally easy weighted trimmed mean is shown to be an efficient alternative to maximum likelihood estimation which is usually computationally difficult for most underlying distributions except the ideal case of normal ones. From the nonparametric point of view, this weighted trimmed mean is shown to be an efficient alternative robust estimator. A methodology for confidence ellipsoids and hypothesis testing based on the weighted trimmed mean is also introduced.