A NEW CHARACTERIZATION OF THE NORMAL DISTRIBUTION AND TEST FOR NORMALITY
研究了样本均值与分位数之间的渐近协方差函数，发现当且仅当数据来自正态分布时，该协方差在所有分位数上为常数，并基于此提出了一种新的正态性检验方法，在蒙特卡洛模拟中表现良好。
We study the asymptotic covariance function of the sample mean and quantile, and derive a new and surprising characterization of the normal distribution: the asymptotic covariance between the sample mean and quantile is constant across all quantiles, if and only if the underlying distribution is normal. This is a powerful result and facilitates statistical inference. Utilizing this result, we develop a new omnibus test for normality based on the quantile-mean covariance process. Compared to existing normality tests, the proposed testing procedure has several important attractive features. Monte Carlo evidence shows that the proposed test possesses good finite sample properties. In addition to the formal test, we suggest a graphical procedure that is easy to implement and visualize in practice. Finally, we illustrate the use of the suggested techniques with an application to stock return datasets.