Abstract The linear regression model requires robust estimation of parameters, if the measured data are contaminated by outlying measurements (outliers). While a number of robust estimators (i.e. resistant to outliers) have been proposed, this paper is focused on estimating the variance of the random regression errors. We particularly focus on the least weighted squares estimator, for which we review its properties and propose new weighting schemes together with corresponding estimates for the variance of disturbances. An illustrative example revealing the idea of the estimator to down-weight individual measurements is presented. Further, two numerical simulations presented here allow to compare various estimators. They verify the theoretical results for the least weighted squares to be meaningful. MM-estimators turn out to yield the best results in the simulations in terms of both accuracy and precision. The least weighted squares (with suitable weights) remain only slightly behind in terms of the mean square error and are able to outperform the much more popular least trimmed squares estimator, especially for smaller sample sizes.

中文翻译：

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关于（高度）鲁棒回归中误差方差的鲁棒估计

摘要 如果测量数据受到异常测量（异常值）的污染，线性回归模型需要对参数进行稳健估计。虽然已经提出了许多稳健的估计量（即抵抗异常值），但本文的重点是估计随机回归误差的方差。我们特别关注最小加权平方估计器，为此我们回顾了它的属性并提出了新的加权方案以及对干扰方差的相应估计。给出了一个说明性示例，该示例揭示了估算器对单个测量进行加权的想法。此外，这里介绍的两个数值模拟允许比较各种估计量。他们验证了最小加权平方的理论结果是有意义的。结果证明 MM 估计器在模拟中在准确性和精度方面都产生了最好的结果。最小加权平方（具有合适的权重）在均方误差方面仅略微落后，并且能够胜过更流行的最小修整平方估计器，尤其是对于较小的样本量。