Sharp moderate deviations and uniform Berry--Esseen bounds for Hotelling's $T^2$-statistic via Stein's method

Abstract

Let ${\mathbf{X}}_1,\dots ,{\mathbf{X}}_n$ be independent random vectors in ${\mathbb{R}}^d$ with common mean ${\mathbf{\mu }}_n$, where $d\ge 2$ is a fixed integer. Let ${\bar{\mathbf{X}}}_n=n^{-1}{\sum }_{i=1}^n{\mathbf{X}}_i$ denote the sample mean and let \[ \widehatΣ_n=\frac1n\sum_{i=1}^n (\mathbf X_i-\bar{\mathbf X}_n) (\mathbf X_i-\bar{\mathbf X}_n)^\top \] denote the sample covariance matrix. Assume that the average covariance matrix is positive definite and that ${\hat{\Sigma }}_n$ is nonsingular almost surely. For testing $H_0:{\mathbf{\mu }}_n={\mathbf{\mu }}_0$, the Hotelling statistic is \[ T^2_{H,n} =n(\bar{\mathbf X}_n-\boldsymbolμ_0)^\top \widehatΣ_n^{-1} (\bar{\mathbf X}_n-\boldsymbolμ_0). \] Under $H_0$, we establish finite-sample Cramér-type moderate deviation results and uniform Berry--Esseen bounds under finite $p$th moments for $2<p\le 3$. The moderate deviation error order is optimal over independent triangular arrays, as shown by explicit constructions. For identically distributed observations with a fixed distribution, we obtain \[ \sup_{0\le x\le b_n} \left| \frac{\mathbb P(T^2_{H,n}\ge x^2)} {\mathbb P(χ_d^2\ge x^2)}-1 \right| = O\!\left((1+b_n)^p n^{-(p-2)/2}\right) \] for every nonnegative sequence $b_n=o(n^{(p-2)/(2p)})$. The uniform Berry--Esseen bound is $O(n^{-(p-2)/2})$. The proof uses a radial Stein identity and a recursive error estimate. We apply the results to chi-squared calibration in simultaneous testing.

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