Exponential bounds for regularized Hotelling’s T2 statistic in high dimension

We obtain exponential inequalities for regularized Hotelling’s $T_n^2$ statistics, that take into account the potential high dimensional aspects of the problem. We explore the finite sample properties of the tail of these statistics by deriving exponential bounds for symmetric distributions and also for general distributions under weak moment assumptions (we never assume exponential moments). For this, we use a penalized estimator of the covariance matrix and propose an optimal choice for the penalty coefficient.