A mathematical programming approach for improving the robustness of least sum of absolute deviations regression

  1. Avi Giloni1,*,
  2. Bhaskar Sengupta2,†,*,
  3. Jeffrey S. Simonoff3,*

Article first published online: 27 FEB 2006

DOI: 10.1002/nav.20139

Issue

Naval Research Logistics (NRL)

Naval Research Logistics (NRL)

Volume 53, Issue 4, pages 261–271, June 2006

Additional Information

How to Cite

Giloni, A., Sengupta, B. and Simonoff, J. S. (2006), A mathematical programming approach for improving the robustness of least sum of absolute deviations regression. Naval Research Logistics, 53: 261–271. doi: 10.1002/nav.20139

Author Information

  1. 1

    Sy Syms School of Business, Yeshiva University, 500 W 185 St, BH-428, New York, NY 10033, USA

  2. 2

    ExxonMobil Research and Engineering, 1545 Route 22 East, Annandale, NJ 08801, USA

  3. 3

    Leonard N. Stern School of Business, New York University, 44 West 4th Street, New York, NY 10012, USA

  1. †

    This work was done when the author was at Yeshiva University.

*Sy Syms School of Business, Yeshiva University, 500 W 185 St, BH-428, New York, NY 10033, USA

  1. †

    This work was done when the author was at Yeshiva University.

Publication History

  1. Issue published online: 27 MAR 2006
  2. Article first published online: 27 FEB 2006
  3. Manuscript Accepted: 30 SEP 2005
  4. Manuscript Revised: 22 MAY 2005
  5. Manuscript Received: 26 JUL 2004

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Keywords:

  • algorithms;
  • breakdown point;
  • knapsack problem;
  • nonlinear mixed integer programming;
  • robust regression

Abstract

This paper discusses a novel application of mathematical programming techniques to a regression problem. While least squares regression techniques have been used for a long time, it is known that their robustness properties are not desirable. Specifically, the estimators are known to be too sensitive to data contamination. In this paper we examine regressions based on Least-sum of Absolute Deviations (LAD) and show that the robustness of the estimator can be improved significantly through a judicious choice of weights. The problem of finding optimum weights is formulated as a nonlinear mixed integer program, which is too difficult to solve exactly in general. We demonstrate that our problem is equivalent to a mathematical program with a single functional constraint resembling the knapsack problem and then solve it for a special case. We then generalize this solution to general regression designs. Furthermore, we provide an efficient algorithm to solve the general nonlinear, mixed integer programming problem when the number of predictors is small. We show the efficacy of the weighted LAD estimator using numerical examples. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006

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