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Quantile Function Expansion Using Regularly Varying Functions

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  • Thomas Fung

    (Macquarie University
    University of Sydney)

  • Eugene Seneta

    (University of Sydney)

Abstract

We present a simple result that allows us to evaluate the asymptotic order of the remainder of a partial asymptotic expansion of the quantile function h(u) as u → 0+ or 1−. This is focussed on important univariate distributions when h(⋅) has no simple closed form, with a view to assessing asymptotic rate of decay to zero of tail dependence in the context of bivariate copulas. Motivation of this study is illustrated by the asymptotic behaviour of the tail dependence of Normal copula. The Normal, Skew-Normal and Gamma are used as initial examples. Finally, we discuss approximation to the lower quantile of the Variance-Gamma and Skew-Slash distributions.

Suggested Citation

  • Thomas Fung & Eugene Seneta, 2018. "Quantile Function Expansion Using Regularly Varying Functions," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1091-1103, December.
    • Handle: RePEc:spr:metcap:v:20:y:2018:i:4:d:10.1007_s11009-017-9593-0
    DOI: 10.1007/s11009-017-9593-0
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    References listed on IDEAS

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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. J. D. Beasley & S. G. Springer, 1977. "The Percentage Points of the Normal Distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 26(1), pages 118-121, March.
    3. Fung, Thomas & Seneta, Eugene, 2016. "Tail asymptotics for the bivariate skew normal," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 129-138.
    4. Anthony W. Ledford & Jonathan A. Tawn, 1997. "Modelling Dependence within Joint Tail Regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 475-499.
    5. Antonella Capitanio, 2010. "On the approximation of the tail probability of the scalar skew-normal distribution," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 299-308.
    6. Arslan, Olcay, 2008. "An alternative multivariate skew-slash distribution," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2756-2761, November.
    7. Fung, Thomas & Seneta, Eugene, 2011. "The bivariate normal copula function is regularly varying," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1670-1676, November.
    8. Paul M. Voutier, 2010. "A New Approximation to the Normal Distribution Quantile Function," Papers 1002.0567, arXiv.org, revised Feb 2010.
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    Cited by:

    1. Hu, Shuang & Peng, Zuoxiang & Nadarajah, Saralees, 2022. "Tail dependence functions of the bivariate Hüsler–Reiss model," Statistics & Probability Letters, Elsevier, vol. 180(C).
    2. Fung, Thomas & Seneta, Eugene, 2021. "Tail asymptotics for the bivariate equi-skew generalized hyperbolic distribution and its Variance-Gamma special case," Statistics & Probability Letters, Elsevier, vol. 178(C).
    3. Daouia, Abdelaati & Stupfler, Gilles & Usseglio-Carleve, Antoine, 2024. "A unified theory of extreme Expected Shortfall inference," TSE Working Papers 24-1565, Toulouse School of Economics (TSE).

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