Tests for the Gaussianity of a time series with application to financial time series

从多元正态检验的角度出发，提出了一种检验平稳时间序列过程的高斯性的新方法。其基本思想是首先标准化p个变量，然后用p-dim的矩结构--厄米特多项式来变换它们。基于变换后的变量进行多元正态检验。检验统计量服从x^2分布的渐近分布。这些结果被应用于通过考虑滞后变量p-dim变量来检验单变量时间序列的高斯性。将该方法应用于股票收益率的检验，剔除了正态分布的高斯性，并处理了曲线族的多元正态检验问题，得到了局部最优不变检验及其分布。找到了MarDia检验是局部最佳不变量的曲线族。

In a view of testing multivariate normality, a new testing procedure for testing the Gaussianity of stationary time series process is proposed. The basic idea is that first standardizing p variates and then transforming them by Hermitian polynomials, the moment structure of the p-dim. multivariate normality is tested based on the transformed variates. The test statistic is shown to be asymptotically distributed as x^2-distribution. These results are applied to testing the Gaussianity of a univariate time series by considering lagged variates p-dim variates. The method is applied to returns of stocks and the Gaussianity is rejected.Also the problem of testing multivariate normality against a curved family is treated and the locally best invariant test and its distribution are derived. A curved family for which the Mardia test is locally best invariant is found.