A pure-tail ordering based on the ratio of the quantile functions. (English) Zbl 0745.62011

Summary: In the intuitive approach, a distribution function \(F\) is said to be not more heavily tailed than \(G\) if \(\lim \sup_{x\to \infty} \bar F/\bar G < \infty\). 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.


62E10 Characterization and structure theory of statistical distributions
62E15 Exact distribution theory in statistics
62B15 Theory of statistical experiments
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