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Representing Koziol’s Kurtoses

In: Mathematical and Statistical Methods for Actuarial Sciences and Finance

Author

Listed:
  • Nicola Loperfido

    (Università Degli Studi di Urbino Carlo Bo)

Abstract

The Koziol’s kurtosis of a random vector is the sum of its squared fourth moments. Similarly, the Koziol’s excess kurtosis of a random vector is the sum of its fourth standardized cumulants. Koziol’s kurtoses provide some insight into several features of the underlying distributions, as for example tails and modes. Moreover, they are invariant with respect to one-to-one affine transformations. We prove that Koziol’s kurtosis is a simple analytical function of the cokurtosis matrix and the covariance matrix. A similar result holds for Koziol’s excess kurtosis. Derivations of both results use fourth moments and cumulants of linearly transformed random vectors, whose properties already appeared in the literature but are formally proved in this paper for the first time, to the best of our knowledge. Applications to financial econometrics and portfolio selection are briefly discussed.

Suggested Citation

  • Nicola Loperfido, 2021. "Representing Koziol’s Kurtoses," Springer Books, in: Marco Corazza & Manfred Gilli & Cira Perna & Claudio Pizzi & Marilena Sibillo (ed.), Mathematical and Statistical Methods for Actuarial Sciences and Finance, pages 323-328, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-78965-7_47
    DOI: 10.1007/978-3-030-78965-7_47
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