Location adjustment for the minimum volume ellipsoid estimator

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• 作者：Christophe Croux ; Gentiane Haesbroeck and Peter J. Rousseeuw
• 关键词：intercept adjustment ; L1 estimation ; location estimation ; location adjustment ; minimum volume ellipsoid ; robustness
• 刊名：Statistics and Computing
• 出版时间：2002
• 出版年：2002
• 期刊代码：116_09603174
• 类别：bio
• 卷：12
• 期：3
• 页码：191-200
• 数据来源：sp

摘要

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 L₁location 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 L₁ and the MVB in the multivariate setting, revealing the superiority of L₁.