Four-Moment Approximations and Fast p-Values for BHEP Tests of Multivariate Normality
Abstract
The Baringhaus--Henze--Epps--Pulley (BHEP) tests form a widely applicable class of affine invariant and consistent tests for multivariate normality. Under the null hypothesis, the test statistic converges to a weighted sum of independent chi-squared random variables. Although closed-form expressions for the first three cumulants of this limiting distribution are known for arbitrary dimension d and smoothing parameter β, an explicit expression for the fourth cumulant has so far been available only in a special univariate case. We derive the fourth cumulant for arbitrary d≥1 and β>0 and use the resulting first four moments to construct Johnson and Pearson approximations to the limiting null distribution. These yield essentially instantaneous analytic approximations to BHEP p-values, without numerical eigenvalue calculations. Using the recently obtained complete spectrum as a benchmark, we show that both four-moment approximations are highly accurate over a broad range of dimensions, smoothing parameters and upper-tail probabilities, and substantially improve on two- and three-parameter lognormal approximations. Monte Carlo results indicate good finite-sample calibration for many parameter combinations, although convergence to the limiting distribution may be slow in higher dimensions for extreme values of the smoothing parameter.
Related
- Hermite Spectra and Kernel Factorization for Gaussian-Weighted Tests of Normality
- Convergence analysis of moment-based approximations in kinetic theory
- Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers
- Strong Approximations for Multivariate Integrated Empirical Processes: Two-Sample Testing, Change-Point Detection, and Inference with Estimated Parameters
- Goodness-of-Fit Tests via Kernel Density Estimation: Univariate and Multivariate Approaches
BibTeX
@misc{ebner2026fourmoment,
title = {{Four-Moment Approximations and Fast $p$-Values for BHEP Tests of Multivariate Normality}},
author = {Bruno Ebner and Norbert Henze},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35037},
eprint = {2609.35037},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35037},
}
RIS
TY - UNPB
TI - Four-Moment Approximations and Fast p-Values for BHEP Tests of Multivariate Normality
AU - Bruno Ebner
AU - Norbert Henze
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35037
UR - https://arxiv.org/abs/2609.35037
AB - The Baringhaus--Henze--Epps--Pulley (BHEP) tests form a widely applicable class of affine invariant and consistent tests for multivariate normality. Under the null hypothesis, the test statistic converges to a weighted sum of independent chi-squared random variables. Although closed-form expressions for the first three cumulants of this limiting distribution are known for arbitrary dimension d and smoothing parameter β, an explicit expression for the fourth cumulant has so far been available only in a special univariate case. We derive the fourth cumulant for arbitrary d≥1 and β>0 and use the resulting first four moments to construct Johnson and Pearson approximations to the limiting null distribution. These yield essentially instantaneous analytic approximations to BHEP p-values, without numerical eigenvalue calculations. Using the recently obtained complete spectrum as a benchmark, we show that both four-moment approximations are highly accurate over a broad range of dimensions, smoothing parameters and upper-tail probabilities, and substantially improve on two- and three-parameter lognormal approximations. Monte Carlo results indicate good finite-sample calibration for many parameter combinations, although convergence to the limiting distribution may be slow in higher dimensions for extreme values of the smoothing parameter.
ER -