The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators

Portnoy, Stephen ; Koenker, Roger

Statist. Sci., Tome 12 (1997) no. 1, p. 279-300 / Harvested from Project Euclid

Résumé

Since the time of Gauss, it has been generally accepted that $\ell_2$-methods of combining observations by minimizing sums of squared errors have significant computational advantages over earlier $\ell_1$-methods based on minimization of absolute errors advocated by Boscovich, Laplace and others. However, $\ell_1$-methods are known to have significant robustness advantages over $\ell_2$-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 $\ell_1$-type problems, we obtain a 10- to 100-fold improvement in computational speeds over current (simplex-based) $\ell_1$-algorithms in large problems, demonstrating that $\ell_1$-methods can be made competitive with $\ell_2$-methods in terms of computational speed throughout the entire range of problem sizes. Formal complexity results suggest that $\ell_1$-regression can be made faster than least-squares regression for n sufficiently large and p modest.

Détail

Publié le : 1997-11-14
Classification:  $\ell_1$,  $L_1$,  least absolute deviations,  median,  regression quantiles,  interior point,  statistical preprocessing,  linear programming,  simplex method,  simultaneous confidence bands