When is Hotelling's $T^2$ asymptotically $F$-distributed?

Abstract

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 $\mathbf{X}\in {\mathbb{R}}^{p\times n}$ with a fixed, possibly heavy-tailed entry law $\xi$. Under symmetry and ${\mathrm{limsup}}_{n\to \infty }p/n<1$, 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 $\mathbf{P}={\mathbf{I}}_n-{\mathbf{X}}^{\top }(\mathbf{X}{\mathbf{X}}^{\top }{)}^{-1}\mathbf{X}$. The central quantity $R=1^{\top }P1$ measures the geometric alignment between the averaging direction and the random row space of $\mathbf{X}$. For symmetric finite-variance entries, the sufficient approximation extends to every sequence $1\le p<n$, 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 ${\lim }_{x\to \infty }{x}^{5/2}\mathbb{P}(|\xi |>x)=0$ when $p/n\to \gamma \in (0,1)$, 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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