When is Hotelling's T2 asymptotically F-distributed?
Abstract
For Gaussian observations, a deterministic multiple of Hotelling's T2 statistic has an exact F distribution. This paper investigates when this approximation remains valid for an independent and identically distributed (i.i.d.) data matrix X∈ ℝp× n with a fixed, possibly heavy-tailed entry law ξ. Under symmetry and limsupn→ ∞ p/n<1, we prove that approximation by the F distribution in Kolmogorov distance is equivalent to belonging to the domain of attraction of the normal law. Both conditions are equivalent to relative concentration of the diagonal entries of the null-space projection P=In−X⊤(XX⊤)−1X. The central quantity R=1⊤P1 measures the geometric alignment between the averaging direction and the random row space of X. For symmetric finite-variance entries, the sufficient approximation extends to every sequence 1≤ p<n, including fixed nullity n−p. As a consequence of the F approximation, Hotelling's T2 statistic exhibits different asymptotic distributions in different regimes, including the normal, chi-square, and inverse-chi-square limits depending on p and n. Without symmetry, we establish a central limit theorem under the weak-5/2 tail condition limx→∞x5/2ℙ(|ξ|>x)=0 when p/n→ γ∈ (0,1), and obtain the F approximation as a consequence. Centered standardized Pareto laws show that the exponent 5/2 is sharp for this universal absolute-tail sufficient condition, although balanced-tail nonsymmetric laws can retain the approximation at the boundary, which reveals that the asymmetry of the tails matters for the F approximation.
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BibTeX
@misc{li2026when,
title = {{When is Hotelling's $T^2$ asymptotically $F$-distributed?}},
author = {Yanpeng Li and Guangming Pan and Wang Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12903},
eprint = {2610.12903},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12903},
}
RIS
TY - UNPB
TI - When is Hotelling's T^2 asymptotically F-distributed?
AU - Yanpeng Li
AU - Guangming Pan
AU - Wang Zhou
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12903
UR - https://arxiv.org/abs/2610.12903
AB - For Gaussian observations, a deterministic multiple of Hotelling's T^2 statistic has an exact F distribution. This paper investigates when this approximation remains valid for an independent and identically distributed (i.i.d.) data matrix X∈ ℝ^p× n with a fixed, possibly heavy-tailed entry law ξ. Under symmetry and limsup_n→ ∞ p/n<1, we prove that approximation by the F distribution in Kolmogorov distance is equivalent to belonging to the domain of attraction of the normal law. Both conditions are equivalent to relative concentration of the diagonal entries of the null-space projection P=I_n−X^⊤(XX^⊤)^−1X. The central quantity R=1^⊤P1 measures the geometric alignment between the averaging direction and the random row space of X. For symmetric finite-variance entries, the sufficient approximation extends to every sequence 1≤ p<n, including fixed nullity n−p. As a consequence of the F approximation, Hotelling's T^2 statistic exhibits different asymptotic distributions in different regimes, including the normal, chi-square, and inverse-chi-square limits depending on p and n. Without symmetry, we establish a central limit theorem under the weak-5/2 tail condition lim_x→∞x^5/2ℙ(|ξ|>x)=0 when p/n→ γ∈ (0,1), and obtain the F approximation as a consequence. Centered standardized Pareto laws show that the exponent 5/2 is sharp for this universal absolute-tail sufficient condition, although balanced-tail nonsymmetric laws can retain the approximation at the boundary, which reveals that the asymmetry of the tails matters for the F approximation.
ER -