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Characteristic function and moment generating function of multivariate folded normal distribution

Author

Listed:
  • Matej Benko

    (Brno University of Technology)

  • Zuzana Hübnerová

    (Brno University of Technology)

  • Viktor Witkovský

    (Slovak Academy of Sciences)

Abstract

In this study, we derive the characteristic function of the multivariate folded normal distribution, a distribution that arises when the magnitudes—but not the signs—of a normally distributed random vector are of interest. The folded normal distribution is widely applicable across various fields. Thus, obtaining an analytical expression for its characteristic function is pivotal in understanding its fundamental properties. Moreover, this allows one to facilitate numerical evaluations of complex distributions involving linear combinations of absolute values of dependent normal variables. The derivation is based on a novel expression of the moment generating function, formulated using the cumulative distribution function of the multivariate normal distribution. To validate our findings, we present two examples using our MATLAB implementation. We compare the characteristic function for the sum of the absolute values of elements of a multivariate normal vector with the simulated empirical counterpart. Additionally, we derive the second mixed moment of the bivariate folded normal distribution from the moment generating function, demonstrating its agreement with known theoretical expressions.

Suggested Citation

  • Matej Benko & Zuzana Hübnerová & Viktor Witkovský, 2025. "Characteristic function and moment generating function of multivariate folded normal distribution," Statistical Papers, Springer, vol. 66(4), pages 1-23, June.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:4:d:10.1007_s00362-025-01711-z
    DOI: 10.1007/s00362-025-01711-z
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    References listed on IDEAS

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    1. Psarakis, Stelios & Panaretos, John, 2001. "On Some Bivariate Extensions of the Folded Normal and the Folded-T Distributions," MPRA Paper 6383, University Library of Munich, Germany.
    2. Michail Tsagris & Christina Beneki & Hossein Hassani, 2014. "On the Folded Normal Distribution," Mathematics, MDPI, vol. 2(1), pages 1-17, February.
    3. Kourtis, Apostolos, 2014. "On the distribution and estimation of trading costs," Journal of Empirical Finance, Elsevier, vol. 28(C), pages 104-117.
    4. Kim, Hyoung-Moon & Genton, Marc G., 2011. "Characteristic functions of scale mixtures of multivariate skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 102(7), pages 1105-1117, August.
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