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Tail Properties and Asymptotic Expansions for the Maximum of the Logarithmic Skew-Normal Distribution

Part of: Distribution theory Stochastic processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Xin Liao ,
Zuoxiang Peng  and
Saralees Nadarajah
Xin Liao*
Affiliation:
Southwest University
Zuoxiang Peng*
Affiliation:
Southwest University
Saralees Nadarajah*
Affiliation:
University of Manchester
*
∗ Postal address: School of Mathematics and Statistics, Southwest University, 400715 Chongqing, China.
∗ Postal address: School of Mathematics and Statistics, Southwest University, 400715 Chongqing, China.
∗∗∗ Postal address: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address: saralees.nadarajah@manchester.ac.uk
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Abstract

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We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

Keywords

Extreme value distributionlogarithmic skew-normal distributionmaximumpointwise convergence ratesubexponentiality

MSC classification

Primary: 62E20: Asymptotic distribution theory 60G70: Extreme value theory; extremal processes
Secondary: 60F15: Strong theorems 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 900 - 907
Copyright
© Applied Probability Trust 

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