Open Access
March, 1992 A Pure-Tail Ordering Based on the Ratio of the Quantile Functions
Javier Rojo
Ann. Statist. 20(1): 570-579 (March, 1992). DOI: 10.1214/aos/1176348541


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.


Download Citation

Javier Rojo. "A Pure-Tail Ordering Based on the Ratio of the Quantile Functions." Ann. Statist. 20 (1) 570 - 579, March, 1992.


Published: March, 1992
First available in Project Euclid: 12 April 2007

zbMATH: 0745.62011
MathSciNet: MR1150363
Digital Object Identifier: 10.1214/aos/1176348541

Primary: 62E10
Secondary: 60E05 , 62B15

Keywords: density-quantile functions , exponential tails , polynomial tails , quantile functions , scale-invariant tails , swiftly varying tails , Tail-orderings

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 1 • March, 1992
Back to Top