多元正态性BHEP检验的四矩近似与快速$p$值计算
Four-Moment Approximations and Fast $p$-Values for BHEP Tests of Multivariate Normality
- Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文推导了BHEP多元正态性检验极限分布的第四累积量，并基于前四阶矩构造Johnson和Pearson近似，实现了无需特征值计算的快速精确$p$值计算。
AI中文摘要：
Baringhaus--Henze--Epps--Pulley (BHEP) 检验构成了一类广泛适用的仿射不变且一致的多元正态性检验。在原假设下，检验统计量收敛到独立卡方随机变量的加权和。尽管对于任意维度 $d$ 和光滑参数 $\beta$，该极限分布的前三累积量已有闭式表达式，但第四累积量的显式表达式此前仅在特殊的单变量情形下可得。我们推导了任意 $d\geq1$ 且 $\beta>0$ 时的第四累积量，并利用所得的前四阶矩构造了极限零分布的 Johnson 和 Pearson 近似。这些近似无需数值特征值计算即可得到 BHEP $p$ 值的几乎瞬时解析近似。以最近获得的完整谱作为基准，我们表明两种四矩近似在广泛的维度、光滑参数和上尾概率范围内高度精确，并显著优于二参数和三参数对数正态近似。蒙特卡洛结果表明，对于许多参数组合，有限样本校准良好，尽管在高维且光滑参数取极端值时，向极限分布的收敛可能较慢。
英文摘要：
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 $d\geq1$ 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.
