A Pure-Tail Ordering Based on the Ratio of the Quantile Functions

Abstract

In the intuitive approach, a distribution function $F$ is said to be not more heavily tailed than $G$ if $\lim \sup_{x \rightarrow \infty} \bar{F}/\bar{G} < \infty$. An alternative is to consider the behavior of the ratio $F^{-1}(u)/G^{-1}(u)$, in a neighborhood of one. The present paper examines the relationship between these two criteria and concludes that the intuitive approach gives a more thorough comparison of distribution functions than the ratio of the quantile functions approach in the case $F$ or $G$ have tails that decrease faster than, or at, an exponential rate. If $F$ or $G$ have slowly varying tails, the intuitive approach gives a less thorough comparison of distributions. When $F$ or $G$ have polynomial tails, the approaches agree.