Guy Martial Nkiet

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.

Inference for the invariance of canonical analysis under linear transformations

Using a recent result about the invariance problem in linear canonical analysis (LCA), we introduce a criterion by means of which one can see if this invariance holds when the related random vectors are transformed by linear maps. Then, the estimation of this criterion is considered as well as the problem of testing for invariance of LCA. Particularly, a new test for additional information in canonical analysis is proposed and simulations are used to gain understanding of the finite sample performance of this test and to compare it with the likelihood ratio test.

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.

Direct variable selection for discrimination among several groups

We propose a criterion for variable selection in discriminant analysis. This criterion permits to arrange the variables in decreasing order of adequacy for discrimination, so that the variable selection problem reduces to that of the estimation of suitable permutation and dimensionality. Then, estimators for these parameters are proposed and the resulting method for selecting variables is shown to be consistent. In a simulation study, we compute proportions of correct classification after variable selection in order to gain understanding of the performance of our proposal and to compare it to existing methods.