Yves Romain

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.

Tensor products and statistics

Abstract A dictionary between operator-based and matrix-based languages in multivariate statistical analysis is proposed. Then this formulary is applied to asymptotic factorial analyses, especially for giving asymptotic covariance matrices and operators in an explicit form. Finally, we present the mathematical foundations on which are based the functional tools, i.e. tensor products of linear spaces, of vectors, and of operators.

Reduced principal component analysis

We define three kinds of Reduced Principal Component Analysis and study their asymptotic behaviours

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.

On the integral with respect to the tensor product of two random measures

A Fubini-type formula for the integral with respect to the tensor product of two random measures is established in an intrinsic way. This permits one to consider a convolution product. The results are applied to a stationary continuous random function (which may be multiplicatively written with two stationary components) and to principal component analysis in the frequency domain.

Recent Results on Random and Spectral Measures with Some Applications in Statistics

In this talk we de ne and study rst the convolution product of two spectral measures and secondly the tensor and convolution products of random measures. Then we propose some applications in stationary processes statistics. 13.