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

Abstract

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 $T^{2}$ enabling the inferential analysis of large-$p$ small-$n$ normal data sets under mild assumptions.