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Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes

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  • OLE EILER BARNDORFF‐NIELSEN
  • ROBERT STELZER

Abstract

. Expressions for (absolute) moments of generalized hyperbolic and normal inverse Gaussian (NIG) laws are given in terms of moments of the corresponding symmetric laws. For the (absolute) moments centred at the location parameter μ explicit expressions as series containing Bessel functions are provided. Furthermore, the derivatives of the logarithms of absolute μ‐centred moments with respect to the logarithm of time are calculated explicitly for NIG Lévy processes. Computer implementation of the formulae obtained is briefly discussed. Finally, some further insight into the apparent scaling behaviour of NIG Lévy processes is gained.

Suggested Citation

  • Ole Eiler Barndorff‐Nielsen & Robert Stelzer, 2005. "Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 617-637, December.
    • Handle: RePEc:bla:scjsta:v:32:y:2005:i:4:p:617-637
    DOI: 10.1111/j.1467-9469.2005.00466.x
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    1. Mencia, Javier F. & Sentana, Enrique, 2004. "Estimation and testing of dynamic models with generalised hyperbolic innovations," LSE Research Online Documents on Economics 24742, London School of Economics and Political Science, LSE Library.
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    1. Ignatieva, Katja & Landsman, Zinoviy, 2015. "Estimating the tails of loss severity via conditional risk measures for the family of symmetric generalised hyperbolic distributions," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 172-186.
    2. Andreas Behr, Ulrich Pötter, "undated". "Downward Wage Rigidity in Europe: A New Flexible Parametric Approach and Empirical Results," Working Papers 201159, Institute of Spatial and Housing Economics, Munster Universitary.
    3. Katja Ignatieva & Natalia Ponomareva, 2017. "Commodity currencies and commodity prices: modelling static and time-varying dependence," Applied Economics, Taylor & Francis Journals, vol. 49(15), pages 1491-1512, March.
    4. Luo, Min & Kontosakos, Vasileios E. & Pantelous, Athanasios A. & Zhou, Jian, 2019. "Cryptocurrencies: Dust in the wind?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1063-1079.
    5. Arturo Ramos, 2017. "Are the log-growth rates of city sizes distributed normally? Empirical evidence for the USA," Empirical Economics, Springer, vol. 53(3), pages 1109-1123, November.
    6. Ignatieva, Katja & Landsman, Zinoviy, 2019. "Conditional tail risk measures for the skewed generalised hyperbolic family," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 98-114.
    7. Johannes Muhle-Karbe & Marcel Nutz, 2010. "Small-Time Asymptotics of Option Prices and First Absolute Moments," Papers 1006.2294, arXiv.org, revised Jun 2011.
    8. Wan-Lun Wang & Ahad Jamalizadeh & Tsung-I Lin, 2020. "Finite mixtures of multivariate scale-shape mixtures of skew-normal distributions," Statistical Papers, Springer, vol. 61(6), pages 2643-2670, December.
    9. Ramos, Arturo, 2015. "Are the log-growth rates of city sizes normally distributed? Empirical evidence for the US," MPRA Paper 65584, University Library of Munich, Germany.
    10. Alexeev Vitali & Ignatieva Katja & Liyanage Thusitha, 2021. "Dependence Modelling in Insurance via Copulas with Skewed Generalised Hyperbolic Marginals," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 25(2), pages 1-20, April.
    11. Gwo Dong Lin & Chin-Yuan Hu, 2021. "Formulas of Absolute Moments," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 476-495, February.
    12. Puente-Ajovin, Miguel & Ramos, Arturo, 2015. "An improvement over the normal distribution for log-growth rates of city sizes: Empirical evidence for France, Germany, Italy and Spain," MPRA Paper 67471, University Library of Munich, Germany.
    13. Andreas Behr & Ulrich Pötter, 2010. "Downward Wage Rigidity in Europe: A New Flexible Parametric Approach and Empirical Results," German Economic Review, Verein für Socialpolitik, vol. 11, pages 169-187, May.
    14. Roman V. Ivanov, 2023. "The Semi-Hyperbolic Distribution and Its Applications," Stats, MDPI, vol. 6(4), pages 1-21, October.
    15. Philipp M. Möller, 2018. "Drawdown Measures And Return Moments," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(07), pages 1-42, November.
    16. Behr, Andreas & Pötter, Ulrich, 2005. "Downward wage rigidity in Europe: A new flexible parametric approach and empirical results," Beiträge zur angewandten Wirtschaftsforschung 14, University of Münster, Center of Applied Economic Research Münster (CAWM).
    17. David Scott & Diethelm Würtz & Christine Dong & Thanh Tran, 2011. "Moments of the generalized hyperbolic distribution," Computational Statistics, Springer, vol. 26(3), pages 459-476, September.
    18. Andreas Behr & Ulrich Pötter, 2010. "Downward Wage Rigidity in Europe: A New Flexible Parametric Approach and Empirical Results," German Economic Review, Verein für Socialpolitik, vol. 11(2), pages 169-187, May.

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