### 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.

Language | English |
---|---|

Pages | 1091-1103 |

Number of pages | 13 |

Journal | Methodology and Computing in Applied Probability |

Volume | 20 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

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*Methodology and Computing in Applied Probability*,

*20*(4), 1091-1103. https://doi.org/10.1007/s11009-017-9593-0

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*Methodology and Computing in Applied Probability*, vol. 20, no. 4, pp. 1091-1103. https://doi.org/10.1007/s11009-017-9593-0

**Quantile function expansion using regularly varying functions.** / Fung, Thomas; Seneta, Eugene.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Quantile function expansion using regularly varying functions

AU - Fung,Thomas

AU - Seneta,Eugene

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Asymptotic expansion

KW - Asymptotic tail dependence

KW - Quantile function

KW - Regularly varying functions

KW - Skew-Slash distribution

KW - Variance-Gamma distribution

UR - http://www.scopus.com/inward/record.url?scp=85056226245&partnerID=8YFLogxK

U2 - 10.1007/s11009-017-9593-0

DO - 10.1007/s11009-017-9593-0

M3 - Article

VL - 20

SP - 1091

EP - 1103

JO - Methodology and Computing in Applied Probability

T2 - Methodology and Computing in Applied Probability

JF - Methodology and Computing in Applied Probability

SN - 1387-5841

IS - 4

ER -