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

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

2009-03-25

arxiv.org/abs/0903.4391v1

Statistics

Methodology

Scientific paper

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 $\{X_{n, n - s_i}, 1 \leq i \leq k \}/n^{1/\alpha}$ 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 $\alpha < 1$.

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