Expansions for Quantiles and Multivariate Moments of Extremes for Distributions of Pareto Type

Abstract

Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F(x)=1-{\sum }_{i=0}^{\infty }{c}_i{x}^{-\alpha -i\beta }$ for $\alpha >0$ and $\beta >0$. An inversion theorem is proved and used to derive an expansion for the quantile $F^{-1}(u)$ and powers of it. From this an expansion in powers of $(n^{-1},n^{-\beta /\alpha })$ is given for the multivariate moments of the extremes $\{X_{n,n-s_i},1\le i\le k\}/n^{1/\alpha }$ for fixed $\mathbf{s}=(s_1,...,s_k)$, where $k\ge 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $\alpha <1$.