PARALLEL ALGORITHMS FOR LEAST MEDIAN OF SQUARES REGRESSION

Recently much effort has been devoted to devising robust regression procedures to which the least median of squares (LMS) regression proposed by Rousseeuw (Journal of the American Statistical Association 79, 1984) belongs. LMS's breakdown point achieves the highest possible value of 50%. It may not be possible to write down closed form expressions for the estimates of LMS and the computations involved are not trivial. Efforts have been made to look for better performance algorithms. Based on the algorithm of Rousseeuw, Steele and Steiger (Discrete Applied Mathematics 14, 1986) provided a lemma that characterized the computation as a discrete optimization problem with worst time complexity of O(n3). Souvaine and Steele (Journal of the American Statistical Association 82, 1987) developed a sweep-line algorithm with time complexity of O(n2log n). This paper reports the use of parallel computing techniques to speedup the computation of LMS. The Rousseeuw's algorithm and the T-sweep algorithm, an algorithm based on Tukey's idea, have been successfully parallelized, but parallel sweep-line algorithm and its refinement may not work at all in the sense that its performance is untolerantly low due to severe load imbalancing.