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arXiv · 0903.4391

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

Saralees Nadarajah·Christopher S. Withers
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Abstract

Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-α- i β}$ for $α> 0$ and $β> 0$. 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 $(n^{-1}, n^{-β/α})$ is given for the multivariate moments of the extremes $\{X_{n, n - s_i}, 1 \leq i \leq k \}/n^{1/α}$ for fixed ${\bf s} = (s_1, ..., s_k)$, where $k \geq 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $α< 1$.

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Saralees Nadarajah, Christopher S. Withers. 2009-03-25. Expansions for Quantiles and Multivariate Moments of Extremes for Distributions of Pareto Type. https://arxiv.org/abs/0903.4391

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