Jacques Dauxois

Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference

From the results of convergence by sampling in linear principal component analysis (of a random function in a separable Hilbert space), the limiting distribution is given for the principal values and the principal factors. These results can be explicitly written in the normal case. Some applications to statistical inference are investigated.

CANONICAL ANALYSIS OF TWO EUCLIDEAN SUBSPACES AND ITS APPLICATIONS

Abstract We develop a canonical analysis both without constraints and under constraints for subspaces of Euclidean space. We also look into a canonical analysis of operators. We then apply the definitions and some of the results to the field of probability and statistics.

Testing for the Lack of a Linear Relationship

Measures of association (between two random vectors) which are suitable symmetric nondecreasing functions of canonical correlations are studied. Limiting distributions of an estimator for such a measure are obtained under lack of relationship, or in case the random vectors are correlated. This general study allows us to describe as particular cases most of the classical measures based on canonical correlations, and so, to obtain their asymptotic theory in a unified framework. Finally, a test of lack of linear relationship deriving from these measures is proposed.

Linear relative canonical analysis of Euclidean random variables, asymptotic study and some applications

We introduce the Linear Relative Canonical Analysis (LRCA) of Euclidean random variables. Then similar properties than for usual linear Canonical Analysis are obtained. Furthermore, we develop an asymptotic study of LRCA and apply the obtained results to tests for lack of relative linear association, dimensionality and invariance.

A robust principal component analysis

This work is concerned with robustness in Principal Component Analysis (PCA). The approach, which we adopt here, is to replace the criterion of least squares by another criterion based on a convex and sufficiently differentiable loss function ρ. Using this criterion we propose a robust estimate of the location vector and introduce an orthogonality with respect to (w.r.t.) ρ in order to define the different steps of a PCA. The influence functions of a vector mean and principal vectors are developed in order to provide method for obtaining a robust PCA. The practical procedure is based on an alternative-steps algorithm.