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A test for normality and independence based on characteristic function

Author

Listed:
  • Wiktor Ejsmont

    (Wroclaw University of Science and Technology)

  • Bojana Milošević

    (Faculty of Mathematics)

  • Marko Obradović

    (Faculty of Mathematics)

Abstract

In this article we prove a generalization of the Ejsmont characterization (Ejsmont in Stat Probab Lett 114:1–5, 2016) of the multivariate normal distribution. Based on it, we propose a new test for independence and normality. The test uses an integral of the squared modulus of the difference between the product of empirical characteristic functions and some constant. Special attention is given to the case of testing for univariate normality in which we derive the test statistic explicitly in terms of Bessel function and explore asymptotic properties. The simulation study also includes the cases of testing for bivariate and trivariate normality and independence, as well as multivariate normality. We show the quality performance of our test in comparison to some popular powerful competitors. The practical application of the proposed normality and independence test is discussed and illustrated using a real dataset.

Suggested Citation

  • Wiktor Ejsmont & Bojana Milošević & Marko Obradović, 2023. "A test for normality and independence based on characteristic function," Statistical Papers, Springer, vol. 64(6), pages 1861-1889, December.
    • Handle: RePEc:spr:stpapr:v:64:y:2023:i:6:d:10.1007_s00362-022-01365-1
    DOI: 10.1007/s00362-022-01365-1
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    References listed on IDEAS

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    1. 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.
    2. Henze, Norbert & Wagner, Thorsten, 1997. "A New Approach to the BHEP Tests for Multivariate Normality," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 1-23, July.
    3. L. Baringhaus & N. Henze, 1988. "A consistent test for multivariate normality based on the empirical characteristic function," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 35(1), pages 339-348, December.
    4. Ejsmont, Wiktor, 2016. "A characterization of the normal distribution by the independence of a pair of random vectors," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 1-5.
    5. Bruno Ebner & Norbert Henze, 2020. "Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 845-892, December.
    6. Norbert Henze & María Dolores Jiménez-Gamero, 2019. "A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 499-521, June.
    7. Leo Breiman & Jerome H. Friedman, 1997. "Predicting Multivariate Responses in Multiple Linear Regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(1), pages 3-54.
    8. Bruno Ebner & Norbert Henze, 2020. "Rejoinder on: Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 911-913, December.
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