The Gaussian Hare and the Laplacian Tortoise

Computability of l1 vs l2 Regression Estimators

By: Stephen Portnoy and Roger Koenker

Since Gauss(1821), it has been generally accepted that l_2 methods of combining observations by minimizing sums of squared errors have significant computational advantages over earlier l_1 methods based on minimization of absolute errors advocated by Boscovich, Laplace and others. However, l_1 methods are known to have significant robustness advantages over l_2 methods in many applications, and related quantile regression methods provide a useful, complementary approach to classical least-squares estimation of statistical models. Combining recent advances in interior point methods for solving linear programs with a new statistical preprocessing approach for l_1-type problems, we show that l_1 methods can be made competitive with l_2 methods in terms of computational speed throughout the entire range of problem sizes, including those based on massive datasets.
This paper has appeared (with discussion) in Statistical Science, in the November, 1997 issue. Related software, which is written in S and ratfor is also available.

A new version of this code is available in Ox. The Ox version is quite concise and (almost) as fast as the fortran. It should be emphasized that the this new "interior point" approach to computing quantile regression estimates is still somewhat experimental. For problems of modest size and for a more developed approach to software for inference in quantile regression, the simplex approach embodied in the earlier Splus/Fortran code available ... is probably preferable. Comments on any of this are most welcome. For an alternative explanation of what is going on in the paper you could look here.
. Some details on the provenance of the above picture may be of interest: The background is taken from a wood engraving by J.J. Grandville from Fables de La Fontaine, Paris, 1838, and reprinted in AESOP Five Centuries of Illustrated Fables, selected by J.J. McKendry, Metropolitan Museum of Art: New York. The picture of Gauss is taken from a 1803 Portrait by J.C.A. Schwartz which appears as the frontispiece in Gauss: A Biographical Study, by W.K. Buhler, Springer-Verlag: New York. The picture of Laplace is taken from Cauchy: Un Mathematicien Legitimiste au XIXe Siecle, by Bruno Belhoste, Belin: Paris. No source is specified for this picture.

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