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

2009/03/25 by Saralees Nadarajah, Nadarajah, Saralees, Christopher S. Withers +1
Mathematics · #FOS: Computer and information sciences #Methodology (stat.ME) #stat.ME

paper · pdf · doi:10.48550/arxiv.0903.4391

arxiv created 2009/03/25 · arxiv updated 2009/12/01

Abstract

Let X_nr be the rth largest of a random sample of size n from a distribution F (x) = 1 - ∑_{i = 0}^∞ 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 - si}, 1 ≤ i ≤ k \/n^1/α for fixed \bf s = (s₁, ..., s_k), where k ≥ 1. Examples include the Cauchy, Student t, F, second extreme distributions and stable laws of index α< 1.

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