A New Characterization of the Normal Distribution and Test for Normality

48 Pages Posted: 3 Apr 2015

See all articles by Anil Bera

Anil Bera

University of Illinois

Antonio F. Galvao

Michigan State University

Liang Wang

University of Wisconsin - Milwaukee

Zhijie Xiao

Boston College - Department of Finance and Department of Economics

Date Written: April 1, 2015

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.

Keywords: Asymptotic covariance, characterization of distribution, normality, quantile process, mean, testing

Suggested Citation

Bera, Anil and Galvao, Antonio F. and Wang, Liang and Xiao, Zhijie, A New Characterization of the Normal Distribution and Test for Normality (April 1, 2015). Available at SSRN: https://ssrn.com/abstract=2588812 or http://dx.doi.org/10.2139/ssrn.2588812

Anil Bera

University of Illinois ( email )

601 E John St
Champaign, IL 61820
United States

Antonio F. Galvao (Contact Author)

Michigan State University ( email )

486 W. Circle Drive 110 Marshall-Adams Hall
East Lansing, MI 48824
United States

Liang Wang

University of Wisconsin - Milwaukee ( email )

Bolton Hall 802
3210 N. Maryland Ave.
Milwaukee, WI 53211
United States

Zhijie Xiao

Boston College - Department of Finance and Department of Economics ( email )

United States

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