The Edgeworth expansion for distributions of extreme values
type: Item
instance of: scholarly article
title: The Edgeworth expansion for distributions of extreme values
zbMATH Open document ID: 1001.62009
DOI: 10.1007/BF02881879
author: Changguo Jiang
author: Shihong Cheng
published in: Science in China. Series A
publication date: 2002-08-15T00:00:00Z
review text: Let \(X_{1}, X_{2}, \cdots \) be a sequence of i.i.d random variables with a common distribution \(F.\) Define \(V=(-\log^{-1}F)^{\leftarrow}\), where \(f^{\leftarrow}(t)=\inf \{ x\in(a,b) : f(x) \geq t \},\) \(\forall t\in (f(a), f(b)).\) The authors consider the rate of convergence problem of extreme values. Under the condition of second order derivative \(V''\) a second order von Mises condition (SOMC) for \(F\) is defined. L. De Haan and S.I. Resnick gave a first order Edgeworth expansion for \(\{ P(\max \{ X_{1}, \dots, X_{n} \} - b_{n})/a_{n} \leq x \}\) by choosing \(a_{n}=nV'(n)\) and \[ b_{n}= \begin{cases} nV'(n) , & \gamma\geq 0;\\ V(\infty)+nV'(n)/\gamma, & \gamma<0,\end{cases} \] and obtained the rate of uniform convergence. If the SOMC is satisfied, then \(V\) has to be a generalized regular variation of second order (GRVSO). In this paper, the authors give a complete solution for the Edgeworth expansion problem when \(V\) does not have any derivative. More precisely, they prove that the GRVSO condition is not only sufficient for the Edgeworth expansion, but also necessary.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62G32
zbMATH DE Number: 1782099
zbMATH Keywords: regular variation of second order
zbMATH Keywords: Edgeworth expansions
zbMATH Keywords: rate of uniform convergence
reviewed by: Tae Il Jeon
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02881879
OpenAlex ID: W2226247083
description: scientific article; zbMATH DE number 1782099