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

Abstract

Let $\mathbf X_1,\ldots,\mathbf X_n$ be independent random vectors in $\mathbb R^d$ with common mean $\boldsymbolμ_n$, where $d\ge2$ 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 $\widehatΣ_n$ is nonsingular almost surely. For testing $H_0:\boldsymbolμ_n=\boldsymbolμ_0$, 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 $2<p\le3$. 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.