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A NEW CHARACTERIZATION OF THE NORMAL DISTRIBUTION AND TEST FOR NORMALITY

Published online by Cambridge University Press:  19 May 2015

Anil K. Bera ,
Antonio F. Galvao ,
Liang Wang  and
Zhijie Xiao
Anil K. Bera
Affiliation:
University of Illinois
Antonio F. Galvao
Affiliation:
University of Iowa
Liang Wang
Affiliation:
University of Wisconsin-Milwaukee
Zhijie Xiao*
Affiliation:
Boston College and Shandong University
*
*Address correspondence to Zhijie Xiao, Department of Economics, Boston College, Chestnut Hill, MA 02467; e-mail: xiaoz@bc.edu.
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Abstract

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.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 5 , October 2016 , pp. 1216 - 1252
Copyright
Copyright © Cambridge University Press 2015 

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