Efficient randomized algorithms for the repeated median line estimator

The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Recently there has been a great deal of interests is robust estimators, because of their lack of sensitivity to outlying data points. The basic measure of the robustness of an estimator is its breakdown point, intuitively, the fraction (up to 50% percent) of outlying data points that can corrupt the estimator. One problem with robust estimators is that achieving high breakdown points (near 50%) has proved to be computationally demanding. In this paper we present the best known theoretical algorithm and the first practical subquadratic algorithm for computing a 50% breakdown point estimator: the Siegel, or repeated median, estimator. We first present an O(n log n) randomized expected time algorithm, where n is the number of given points. This algorithm relies, however, on sophisticated data structures. We also present a very simple O(n log² n) randomized algorithm for this problem, which uses no complex data structures. We provide empirical evidence that, for many realistic input distributions, the running time of this second algorithm is actually O(n log n) expected time.