Sharp moderate deviations and uniform Berry--Esseen bounds for Hotelling's T2-statistic via Stein's method
Abstract
Let X1,…,Xn be independent random vectors in ℝd with common mean μn, where d≥2 is a fixed integer. Let X̅n=n−1∑i=1nXi denote the sample mean and let Σ̂n=1/n∑i=1n (Xi−X̅n) (Xi−X̅n)⊤ denote the sample covariance matrix. Assume that the average covariance matrix is positive definite and that Σ̂n is nonsingular almost surely. For testing H0:μn=μ0, the Hotelling statistic is T2H,n =n(X̅n−μ0)⊤ Σ̂n−1 (X̅n−μ0). Under H0, we establish finite-sample Cramér-type moderate deviation results and uniform Berry--Esseen bounds under finite pth moments for 2<p≤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 sup0≤ x≤ bn | (ℙ(T2H,n≥ x2))/(ℙ(χd2≥ x2))−1 | = O ((1+bn)p n−(p−2)/2) for every nonnegative sequence bn=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.
Related
- Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains
- Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes
- Moderate deviations and laws of the iterated logarithm for geometric functionals in the sparse regime
- Moderate deviations for the Maki--Thompson rumour model
- Normal approximation, moderate and large deviations for the elephant random walk with stops and random step sizes
BibTeX
@misc{liu2026sharp,
title = {{Sharp moderate deviations and uniform Berry--Esseen bounds for Hotelling's $T^2$-statistic via Stein's method}},
author = {Song-Hao Liu and Qi-Man Shao and Hao Shi and Jing-Yu Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08074},
eprint = {2611.08074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08074},
}
RIS
TY - UNPB
TI - Sharp moderate deviations and uniform Berry--Esseen bounds for Hotelling's T^2-statistic via Stein's method
AU - Song-Hao Liu
AU - Qi-Man Shao
AU - Hao Shi
AU - Jing-Yu Xu
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08074
UR - https://arxiv.org/abs/2611.08074
AB - Let X_1,…,X_n be independent random vectors in ℝ^d with common mean μ_n, where d≥2 is a fixed integer. Let X̅_n=n^−1∑_i=1^nX_i denote the sample mean and let Σ̂_n=1/n∑_i=1^n (X_i−X̅_n) (X_i−X̅_n)^⊤ denote the sample covariance matrix. Assume that the average covariance matrix is positive definite and that Σ̂_n is nonsingular almost surely. For testing H_0:μ_n=μ_0, the Hotelling statistic is T^2_H,n =n(X̅_n−μ_0)^⊤ Σ̂_n^−1 (X̅_n−μ_0). Under H_0, we establish finite-sample Cramér-type moderate deviation results and uniform Berry--Esseen bounds under finite pth moments for 2<p≤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≤ x≤ b_n | (ℙ(T^2_H,n≥ x^2))/(ℙ(χ_d^2≥ x^2))−1 | = O ((1+b_n)^p n^−(p−2)/2) 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.
ER -