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### Résumé de communication

Résumé 29 :

Relative efficiency and robustness in principal component analysis
Ibazizen, Mohamed
L.M.A., Université de Poitiers

It is well known that the classical PCA is extremely sensitive to outlying observations (Huber, 1981 for example).In the statistical literature, this problem is well appreciated and a number of robust procedures have been developed. The main tool is based on a robust estimation of the covariance or correlation matrix $\Sigma$ : M estimators (Maronna, 1976; Devlin, Gnanadesikan and Kettenring, 1981), S estimators (Rousseuw and Leroy, 1987; Davies, 1987), MVE and MCD estimators (Rousseuw, Croux, Haesbroeck, 1999). A second approach is to replace the criteria of least squares by another which lead to a robust method. (Kamiya and Eguchi, 2001; Ibazizen and Dauxois, 2003). These authors proposed a new method based on a concave and sufficiently differentiable loss function $\rho$. In particular, they found the influence function of the principal component vector in an explicit form and established the robustness of the method for a suitable choice of $\rho$. The problem is that the classical estimator $\hat{\gamma}$ (non-robust) of the first eigenvector of the $\Sigma$ is the best from the point of view of efficiency when the underlying distribution is normal. The aim of this work is to combine a high efficiency with appealing robustness properties of the proposed estimator $\hat{\gamma_\star}$