ALGORITHMS AND COMPLEXITY FOR LEAST MEDIAN OF SQUARES REGRESSION

Given n points left brace (x//i,Y//i) right brace in the plane we study the problem of calculating the least median of squares regression line. This involves the study of the function f( alpha , beta ) equals median( vertical y//i minus ( alpha plus beta x//i) vertical ); it is piecewise linear and can have a quadratic number of local minima. Several algorithms that locate a minimizer of f are presented. The best of these has time complexity O(n**3) in the worst case. Our most practical algorithm appears to be one which has worst case behavior of O(n**3 log(n)), but we provide a probabilistic speed-up of this algorithm which appears to have expected time complexity of O((n log(nn))**2).