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Title: Exponential bounds for regularized Hotelling statistics in high dimension Authors:  Emmanuelle Gautherat - University of Reims (France) [presenting]
Patrice Bertail - university of Paris-Ouest-Nanterre-La Defense (France)
El Mehdi Issouani - university of Paris-Ouest-Nanterre-La defense (France)
Abstract: In many applications (for instance, in genomics or natural language processes), the dimension of the parameter of interest $q$ is large in comparison to the sample size $n$ and sometimes increasing with $n$. Consider, for instance, the problem of estimating or testing a mean of variables in $\mathbb{R}^q$, with $q>n$; in that case, the empirical covariance matrix is not full rank and does not even converge to the true one when $n \to \infty$ so that the usual ``studentized statistics'' or Hotteling $T^2$ tests are no longer valid. It is thus important to construct estimators and testing procedures which take into account the high dimensional aspects of the problem. One relevant proposition which has been developed in the statistical literature is to use a penalized estimator of the covariance matrix, which is invertible and to use this matrix in tests. In that spirit, some authors have obtained asymptotically valid penalized Hotteling $T^2$ tests for the mean in the Gaussian case for high dimension framework, when $n$ and $q=q(n)$ goes to $\infty$ at some specific rate. The purpose of that work is to further explore the finite sample properties of such tests by deriving the exponential bound of some correctly penalized Hotteling $T^2$.