Inference for the mean of large p small n data: A finite-sample high-dimensional generalization of Hotelling’s theorem

We provide a generalization of Hotelling’s Theorem that enables inference (i) for the mean vector of a multivariate normal population and (ii) for the comparison of the mean vectors of two multivariate normal populations, when the number p of components is larger than the number n of sample units and the (common) covariance matrix is unknown. In particular, we extend some recent results presented in the literature by finding the (finite-n) p-asymptotic distribution of the Generalized Hotelling’s T² enabling the inferential analysis of large-p small-n normal data sets under mild assumptions.