Robust regression using repeated medians

TLDR: The repeated median algorithm as mentioned in this paper is a robustified U-statistic in which nested medians replace the single mean and maintains the high 50% breakdown value and resist the effects of outliers even when they comprise nearly half of the data.

Abstract: : The repeated median algorithm is a robustified U-statistic in which nested medians replace the single mean. Unlike many generalizations of the univariate median, repeated median estimates maintain the high 50% breakdown value and can resist the effects of outliers even when they comprise nearly half of the data. Because they are calculated directly, not iteratively, repeated median procedures can be used as starting values for iterative robust estimation methods. For bivariate linear regression with symmetric errors, repeated median estimates are unbiased and Fisher consistent, and their efficiency under Gaussian sampling can be comparable to the efficiency of the univariate median. (Author)