Quantile function approximation using regularly varying functions

We address asymptotic approximation $y(u)$ to the quantile function $h(u)$ as $u\to 0^+$ or $1^-,$ in particular for important distributions when $h(\cdot)$ has no simple closed form. The performance of such an approximation may be judged by how well it cancels the cumulative distribution function $g(\cdot)$; or by how close it is to $h(u)$. We establish a result linking the two performance criteria by using regularly varying functions. For applications, several initial terms of an asymptotic expansion of $h(u)$ plus the order of the lower order remainder term, as $ u \to 0^+$ or $1^-$ can be obtained. This is tantamount to assessing the order of difference: $h(u) - y(u),$ which our approach enables us to do. The Normal, Skew-Normal and Gamma are used as examples. Finally, we discuss approximation to the lower quantile of the Variance-Gamma and Skew-Slash distributions.