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A New Characterization Of The Normal Distribution And Test For Normality

Author

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  • Bera, Anil K.
  • Galvao, Antonio F.
  • Wang, Liang
  • Xiao, Zhijie

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.

Suggested Citation

  • Bera, Anil K. & Galvao, Antonio F. & Wang, Liang & Xiao, Zhijie, 2016. "A New Characterization Of The Normal Distribution And Test For Normality," Econometric Theory, Cambridge University Press, vol. 32(5), pages 1216-1252, October.
    • Handle: RePEc:cup:etheor:v:32:y:2016:i:05:p:1216-1252_00
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    Cited by:

    1. Dong, Kangyin & Dong, Xiucheng & Ren, Xiaohang, 2020. "Can expanding natural gas infrastructure mitigate CO2 emissions? Analysis of heterogeneous and mediation effects for China," Energy Economics, Elsevier, vol. 90(C).
    2. Steffen Betsch & Bruno Ebner, 2020. "Testing normality via a distributional fixed point property in the Stein characterization," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 105-138, March.
    3. Panagiotis Palaios & Evangelia Papapetrou, 2022. "Oil prices, labour market adjustment and dynamic quantile connectedness analysis: evidence from Greece during the crisis," Journal of Economic Structures, Springer;Pan-Pacific Association of Input-Output Studies (PAPAIOS), vol. 11(1), pages 1-21, December.
    4. Sarantis Lolos & Panagiotis Palaios & Evangelia Papapetrou, 2023. "Tourism-led growth asymmetries in Greece: evidence from quantile regression analysis," Portuguese Economic Journal, Springer;Instituto Superior de Economia e Gestao, vol. 22(1), pages 125-148, January.
    5. L. Ndwandwe & J. S. Allison & L. Santana & I. J. H. Visagie, 2023. "Testing for the Pareto type I distribution: a comparative study," METRON, Springer;Sapienza Università di Roma, vol. 81(2), pages 215-256, August.
    6. Marcel, Bräutigam & Marie, Kratz, 2018. "On the Dependence between Quantiles and Dispersion Estimators," ESSEC Working Papers WP1807, ESSEC Research Center, ESSEC Business School.
    7. Tauchmann, Harald, 2019. "Fixed-effects estimation of the linear discrete-time hazard model: An adjusted first-differences estimator," FAU Discussion Papers in Economics 09/2019, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    8. Ron Mittelhammer & George Judge & Miguel Henry, 2022. "An Entropy-Based Approach for Nonparametrically Testing Simple Probability Distribution Hypotheses," Econometrics, MDPI, vol. 10(1), pages 1-19, January.
    9. Gabriel Montes Rojas & Andrés Sebastián Mena, 2020. "Density estimation using bootstrap quantile variance and quantile-mean covariance," Documentos de trabajo del Instituto Interdisciplinario de Economía Política IIEP (UBA-CONICET) 2020-50, Universidad de Buenos Aires, Facultad de Ciencias Económicas, Instituto Interdisciplinario de Economía Política IIEP (UBA-CONICET).
    10. Ya. Yu. Nikitin, 2018. "Local exact Bahadur efficiencies of two scale-free tests of normality based on a recent characterization," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(6), pages 609-618, August.
    11. Marcel Bräutigam & Marie Kratz, 2018. "On the Dependence between Quantiles and Dispersion Estimators," Working Papers hal-02296832, HAL.

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