论文标题
通过多元Stein方程中的傅立叶方法在任何维度中测试正态性
Testing normality in any dimension by Fourier methods in a multivariate Stein equation
论文作者
论文摘要
我们研究了一类新型的仿射不变和一致的多元正常性测试。测试基于标准$ d $ - 变量正态分布的表征，该分布通过连接到部分微分方程的初始值问题的唯一解决方案，该方程是由多元Stein方程激励的。测试标准是一个适当的加权$ l^2 $统计。我们在零假设以及在正态性的连续性和固定替代方案下得出了测试统计数据的极限分布。在固定替代方案下的限制方差以及基础替代方案相对于多元正常定律的距离距离的渐近置信区间的一致估计器。在仿真研究中，我们表明，与突出的竞争者相比，测试很强，渐近置信区间的经验覆盖率会收敛到标称水平。我们提供了一个真实的数据示例，并概述了进一步研究的主题。
We study a novel class of affine invariant and consistent tests for multivariate normality. The tests are based on a characterization of the standard $d$-variate normal distribution by means of the unique solution of an initial value problem connected to a partial differential equation, which is motivated by a multivariate Stein equation. The test criterion is a suitably weighted $L^2$-statistic. We derive the limit distribution of the test statistic under the null hypothesis as well as under contiguous and fixed alternatives to normality. A consistent estimator of the limiting variance under fixed alternatives as well as an asymptotic confidence interval of the distance of an underlying alternative with respect to the multivariate normal law is derived. In simulation studies, we show that the tests are strong in comparison with prominent competitors, and that the empirical coverage rate of the asymptotic confidence interval converges to the nominal level. We present a real data example, and we outline topics for further research.
