Statistical Science

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

Stephen Portnoy and Roger Koenker

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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.

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Statist. Sci., Volume 12, Number 4 (1997), 279-300.

First available in Project Euclid: 22 August 2002

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$\ell_1$ $L_1$ least absolute deviations median regression quantiles interior point statistical preprocessing linear programming simplex method simultaneous confidence bands


Portnoy, Stephen; Koenker, Roger. The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators. Statist. Sci. 12 (1997), no. 4, 279--300. doi:10.1214/ss/1030037960.

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See also

  • Includes: Ronald A. Thisted. Comment by Ronald A. Thisted.
  • Includes: M. R. Osborne. Comment by M. R. Osborne.
  • Includes: Stephen Portnoy, Roger Koenker. Rejoinder by Stephen Portnoy and Roger Koenker.