In this article, we propose a new class of consistent tests for $p$-variate normality. These tests are based on the characterization of the standard multivariate normal distribution, that the Hessian of the corresponding cumulant generating function is identical to the $p\times p$ identity matrix and the idea of decomposing the information from the joint distribution into the dependence copula and all marginal distributions. Under the null hypothesis of multivariate normality, our proposed test statistic is independent of the unknown mean vector and covariance matrix so that the distribution-free critical value of the test can be obtained by Monte Carlo simulation. We also derive the asymptotic null distribution of proposed test statistic and establish the consistency of the test against different fixed alternatives. Last but not least, a comprehensive and extensive Monte Carlo study also illustrates that our test is a superb yet computationally convenient competitor to many well-known existing test statistics.

翻译：本文提出了一类新的$p$元正态性一致检验方法。该检验基于标准多元正态分布的特性——其累积生成函数的Hessian矩阵恒等于$p\times p$单位矩阵，以及将联合分布信息分解为相依连接函数与所有边缘分布的思想。在多元正态性原假设下，我们提出的检验统计量与未知均值向量和协方差矩阵无关，因此可通过蒙特卡洛模拟获得检验的分布无关临界值。我们还推导了所提出检验统计量的渐近零分布，并建立了针对不同固定备择假设的检验一致性。最后，全面而广泛的蒙特卡洛研究亦表明，与众多知名的现有检验统计量相比，我们的检验方法是兼具优越性与计算便捷性的有力竞争者。