Statistical Applications of Random Matrix Theory: Comparison of Two Populations

During the last twenty years, Random matrix theory (RMT) has produced numerous results that allow a better understanding of large random matrices. These advances have enabled interesting applications in the domain of communication. Although this theory can contribute to many other domains such as brain imaging or genetic research, its has been rarely applied. The main barrier to the adoption of RMT may be the lack of concrete statistical results from probabilistic Random matrix theory. Indeed, direct generalisation of classical multivariate theory to high dimensional assumptions is often difficult and the proposed procedures often assume strong hypotheses on the data matrix such as normality or overly restrictive independence conditions on the data.

This thesis proposes a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors corresponding to the number of observed variables. Although the existing theory builds a very good intuition of the behaviour of these matrices, it does not provide enough results to build a satisfactory test for both the power and the robustness. Hence, inspired by spike models, we define the residual spikes and prove many theorems describing the behaviour of many statistics using eigenvectors and eigenvalues in very general cases. For example in the two central theorems of this thesis, the Invariant Angle Theorem and the Invariant Dot Product Theorem.

Using numerous generalisations of the theory, this thesis finally proposes a description of the behaviour of a statistic under a null hypothesis. This statistic allows the user to test the equality of two populations, but also other null hypotheses such as the independence of two sets of variables. Finally, the robustness of the procedure is demonstrated for different classes of models and criteria for evaluating robustness are proposed to the reader.

Therefore, the major contribution of this thesis is to propose a methodology both easy to apply and having good properties. Secondly, a large number of theoretical results are demonstrated and could be easily used to build other applications.

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