Location adjustment for the minimum volume ellipsoid estimator

in Statistics and Computing (2002), 12(3), 191-200

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 ... [more ▼]

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 both L-1 and the MVB in the multivariate setting, revealing the superiority of L-1. [less ▲]

A note on finite-sample efficiencies of estimators for the minimum volume ellipsoid

in Journal of Statistical Computation & Simulation (2002), 72(7), 585-596

Among the most well known estimators of multivariate location and scatter is the Minimum Volume Ellipsoid (MVE). Many algorithms have been proposed to compute it. Most of these attempt merely to ... [more ▼]

Among the most well known estimators of multivariate location and scatter is the Minimum Volume Ellipsoid (MVE). Many algorithms have been proposed to compute it. Most of these attempt merely to approximate as close as possible the exact MVE, but some of them led to the definition of new estimators which maintain the properties of robustness and affine equivariance that make the MVE so attractive. Rousseeuw and van Zomeren (1990) used the (p+1)- subset estimator which was modified by Croux and Haesbroeck (1997) to give rise to the averaged (p+1)- subset estimator . This note shows by means of simulations that the averaged (p+1)-subset estimator outperforms the exact estimator as far as finite-sample efficiency is concerned. We also present a new robust estimator for the MVE, closely related to the averaged (p+1)-subset estimator, but yielding a natural ranking of the data. [less ▲]

Maxbias curves of robust scale estimators based on subranges

in Metrika (2001), 53(2), 101-122

A maxbias curve is a powerful tool to describe the robustness of an estimator. It is an asymptotic concept which tells how much an estimator can change due to a given fraction of contamination. In this ... [more ▼]

A maxbias curve is a powerful tool to describe the robustness of an estimator. It is an asymptotic concept which tells how much an estimator can change due to a given fraction of contamination. In this paper, maxbias curves are computed for some univariate scale estimators based on subranges: trimmed standard deviations, interquantile ranges and the univariate Minimum Volume Ellipsoid (MVE) and Minimum Covariance Determinant (MCD) scale estimators. These estimators are intuitively appealing and easy to calculate. Since the bias behavior of scale estimators may differ depending on the type of contamination (outliers or inliers), expressions for both explosion and implosion maxbias curves are given. On the basis of robustness and efficiency arguments, the MCD scale estimator with 25% breakdown point can be recommended for practical use. [less ▲]

Formation mathématique par la résolution de problèmes

Book published by De Boeck Université (2000)

Entre manuel scolaire et traité d'heuristique, illustré de centaines d'exemples variés, significatifs et souvent inédits, cet ouvrage présente une démarche originale de résolution de problèmes ... [more ▼]

Entre manuel scolaire et traité d'heuristique, illustré de centaines d'exemples variés, significatifs et souvent inédits, cet ouvrage présente une démarche originale de résolution de problèmes mathématiques. [less ▲]

Estimateurs Robustes pour les Composantes Principales

in Proceedings des XXXII Journées de Statistique (2000)

La formation quantitative des économistes à la lumière de l'évolution des rapports entre les mathématiques et l'économie

in Histoire et épistémologie dans l'éducation mathématique: de la maternelle à l'université, Proceedings I (1999)

Dans cet article, on s'efforce de décrire, de comprendre et de justifier l'accroissement progressif des mathématiques dans la formation des économistes; à cet effet, on analyse l'évolution temporelle de ... [more ▼]

Dans cet article, on s'efforce de décrire, de comprendre et de justifier l'accroissement progressif des mathématiques dans la formation des économistes; à cet effet, on analyse l'évolution temporelle de ces deux disciplines. Une attention particulière est portée sur l'enseignement de la finance. [less ▲]

Modèles mathématiques en finance

Book - Presses Ferrer (1998)

L'ouvrage reprend de manière progressive les principaux résultats théoriques nécessaires à une bonne compréhension des produits financiers usuels, en allant de l'intérêt simple aux modèles stochastiques ... [more ▼]

L'ouvrage reprend de manière progressive les principaux résultats théoriques nécessaires à une bonne compréhension des produits financiers usuels, en allant de l'intérêt simple aux modèles stochastiques et chaotiques. [less ▲]

Le chaos

in Bair, Jacques; Haesbroeck, Gentiane; Justens, Daniel (Eds.) et al Modèles mathématiques en finance: de l'incohérence à l'incertitude et au chaos (1998)

Monotonous stability for neutral fixed points

in Bulletin of the Belgian Mathematical Society Simon Stevin (1997), 4(5), 639-646

We give subtle, simple and precise results about the convergence or the divergence of the sequence (x(n)), where x(j) = f(x(j-1)) for every integer j, when the initial element x(0) is in the neighbourhood ... [more ▼]

We give subtle, simple and precise results about the convergence or the divergence of the sequence (x(n)), where x(j) = f(x(j-1)) for every integer j, when the initial element x(0) is in the neighbourhood of a neutral fixed point, i.e. a point x* such that f(x*) = x* with \f'(x*)\ = 1 (where f is a C-infinity function defined on a subset of R). [less ▲]

La dérivée schwarzienne

in Mathématique et Pédagogie (1996), 108

Dans cette note est introduit le concept de dérivée schwarzienne pour une fonction réelle univariée. Diverses propriétés et applications sont également proposées.