Least Median of Squares Regression

TLDR

Classical least squares regression minimizes the sum of squared residuals, while many authors have developed robust alternatives by replacing the square with other functions such as the absolute value. The article introduces a new approach that replaces the sum of squared residuals with their median. This method replaces the sum with the median of squared residuals and generalizes to multivariate location, orthogonal regression, and hypothesis testing in linear models. The estimator tolerates up to about 50 % contamination and, in simple regression, is equivalent to locating the narrowest strip covering half the observations.

Abstract

Abstract Classical least squares regression consists of minimizing the sum of the squared residuals. Many authors have produced more robust versions of this estimator by replacing the square by something else, such as the absolute value. In this article a different approach is introduced in which the sum is replaced by the median of the squared residuals. The resulting estimator can resist the effect of nearly 50% of contamination in the data. In the special case of simple regression, it corresponds to finding the narrowest strip covering half of the observations. Generalizations are possible to multivariate location, orthogonal regression, and hypothesis testing in linear models.