Computing the Least Quartile Difference Estimator in the Plane

Abstract

A common problem in linear regression is that largely aberrant values can strongly influence the results. The least quartile difference (LQD) regression estimator is highly robust, since it can resist up to almost 50% largely deviant data values without becoming extremely biased. Additionally, it shows good behavior on Gaussian data – in contrast to many other robust regression methods. However, the LQD is not widely used yet due to the high computational effort needed when using common algorithms, e.g. the subset algorithm of Rousseeuw and Leroy. For computing the LQD estimator for n data points in the plane, we propose a randomized algorithm with expected running time O(n^2 log^2 n) and an approximation algorithm with a running time of roughly O(n^2 log n). It can be expected that the practical relevance of the LQD estimator will strongly increase thereby.