Abstract
Estimating multivariate location and scatter with both affine equivariance and positive breakdown has always been difficult. A well-known estimator which satisfies both properties is the Minimum Volume Ellipsoid Estimator (MVE). Computing the exact MVE is often not feasible, so one usually resorts to an approximate algorithm. In the regression setup, algorithms for positive-breakdown estimators like Least Median of Squares typically recompute the intercept at each step, to improve the result. This approach is called intercept adjustment. In this paper we show that a similar technique, called location adjustment, can be applied to the MVE. For this purpose we use the Minimum Volume Ball (MVB), in order to lower the MVE objective function. An exact algorithm for calculating the MVB is presented. As an alternative to MVB location adjustment we propose L 1 location adjustment, which does not necessarily lower the MVE objective function but yields more efficient estimates for the location part. Simulations compare the two types of location adjustment. We also obtain the maxbias curves of L 1 and the MVB in the multivariate setting, revealing the superiority of L 1.
Similar content being viewed by others
References
Cook R.D., Hawkins D.M., and Weisberg S. 1993. Exact iterative computation of the robust multivariate minimum volume ellipsoid estimator. Statistics and Probability Letters 16: 213-218.
Hawkins D.M. 1993a. The Feasible Set Algorithm for least median of squares regression. Computational Statistics and Data Analysis 16: 81-101.
Hawkins D.M. 1993b. A Feasible Solution Algorithm for the minimum volume ellipsoid estimator in multivariate data. Computational Statistics 8: 95-107.
He X. and Simpson D.G. 1993. Lower bounds for contamination bias: Globally minimax versus locally linear estimation. The Annals of Statistics 21: 314-337.
Hössjer O. and Croux C. 1995. Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter. Nonparametric Statistics 4: 293-308.
Lopuhaä H.P. 1992. Highly efficient estimators of multivariate location with high breakdown point. The Annals of Statistics 20: 398-413.
Lopuhaä H.P. and Rousseeuw P.J. 1991. Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. The Annals of Statistics 19: 229-248.
Maronna R.A. and Yohai V.J. 1995. The behavior of the Stahel-Donoho robust multivariate estimator. Journal of the American Statistical Association 90: 330-341.
Martin R.D., Yohai V.J., and Zamar R.H. 1989. Min-max bias robust regression. The Annals of Statistics 17: 1608-1630.
Rousseeuw P.J. 1984. Least median of squares regression. Journal of the American Statistical Association 79: 871-880.
Rousseeuw P.J. 1985. Multivariate estimation with high breakdown point. In: Grossmann W., Pflug G., Vincze I., and Wertz W. (Eds.), Mathematical Statistics and Applications, Vol. B, Reidel, Dordrecht, pp. 283-297.
Rousseeuw P.J. and Leroy A.M. 1987. Robust Regression and Outlier Detection. John Wiley, New York.
Rousseeuw P.J. and van Zomeren B.C. 1990. Unmasking multivariate outliers and leverage points. Journal of the American Statistical Association 85: 633-639.
Titterington D.M. 1975. Optimal design: Some geometrical aspects of D-optimality. Biometrika 62: 313-320.
Woodruff D.L. and Rocke D.M. 1993. Heuristic search algorithms for the minimum volume ellipsoid. Journal of Computational and Graphical Statistics 2: 69-95.
Rights and permissions
About this article
Cite this article
Croux, C., Haesbroeck, G. & Rousseeuw, P.J. Location adjustment for the minimum volume ellipsoid estimator. Statistics and Computing 12, 191–200 (2002). https://doi.org/10.1023/A:1020713207683
Issue Date:
DOI: https://doi.org/10.1023/A:1020713207683