# Quantile function expansion using regularly varying functions

## TL;DR

This paper develops a method to evaluate the asymptotic behavior of quantile functions for distributions lacking closed-form expressions, aiding in understanding tail dependence in bivariate copulas.

## Contribution

It introduces a simple approach to assess the asymptotic order of quantile function remainders using regularly varying functions, especially for complex univariate distributions.

## Key findings

- Provides asymptotic order evaluation for quantile function remainders.
- Applies method to Normal, Skew-Normal, and Gamma distributions.
- Discusses approximation techniques for Variance-Gamma and Skew-Slash distributions.

## 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 $u\to 0^+$ or $1^-$. This is focussed on important univariate distributions when $h(\cdot)$ 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. The Introduction motivates the study in terms of the standard Normal. 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.