Electronic Journal of Statistics

Almost sure convergence of extreme order statistics

Zuoxiang Peng, Jiaona Li, and Saralees Nadarajah

Full-text: Open access

Abstract

Let Mn(k) denote the kth largest maximum of a sample (X1,X2,,Xn) from parent X with continuous distribution. Assume there exist normalizing constants an>0, bnℝ and a nondegenerate distribution G such that $a_{n}^{-1}(M_{n}^{(1)}-b_{n})\stackrel{w}{\to}G$. Then for fixed kℕ, the almost sure convergence of \begin{eqnarray*}\quad \qquad \frac{1}{D_{N}}\sum_{n=k}^{N}d_{n}\mathbb{I}\{M_{n}^{(1)}\,{\le}\,a_{n}x_{1}\,{+}\,b_{n},M_{n}^{(2)}\,{\le}\,a_{n}x_{2}\,{+}\,b_{n},\ldots,M_{n}^{(k)}\le a_{n}x_{k}\,{+}\,b_{n}\}\end{eqnarray*} is derived if the positive weight sequence (dn) with DN=n=1Ndn satisfies conditions provided by Hörmann. Some practical issues of this result are also discussed.

Article information

Source
Electron. J. Statist. Volume 3 (2009), 546-556.

Dates
First available in Project Euclid: 17 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1245243980

Digital Object Identifier
doi:10.1214/08-EJS303

Mathematical Reviews number (MathSciNet)
MR2519532

Zentralblatt MATH identifier
1326.62110

Subjects
Primary: 62F15: Bayesian inference
Secondary: 60G70: Extreme value theory; extremal processes 60F15: Strong theorems

Keywords
Almost sure convergence order statistics

Citation

Peng, Zuoxiang; Li, Jiaona; Nadarajah, Saralees. Almost sure convergence of extreme order statistics. Electron. J. Statist. 3 (2009), 546--556. doi:10.1214/08-EJS303. https://projecteuclid.org/euclid.ejs/1245243980.


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