Central limit theory of Hotelling's T 2 statistic with dimensionality reduction

In this paper, we use dimensional reduction technique to study the central limit theory (CLT) of Hotelling’s [Formula: see text] statistic. Specifically, we use a matrix denoted by [Formula: see text], to map [Formula: see text]-dimensional sample vectors to a [Formula: see text] dimensional subspace, where [Formula: see text] or [Formula: see text]. Under the condition of [Formula: see text] as [Formula: see text], we obtain the CLT of Hotelling’s [Formula: see text] statistic. Moreover, the selection of [Formula: see text] is also considered, which demonstrates good performance in the simulation studies.