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On the Asymptotic Behaviour of Extremes and Near Maxima of Random Observations from the General Error Distributions

Part of: Limit theorems Stochastic processes Distribution theory

Published online by Cambridge University Press:  19 February 2016

R. Vasudeva ,
J. Vasantha Kumari  and
S. Ravi
R. Vasudeva*
Affiliation:
University of Mysore
J. Vasantha Kumari*
Affiliation:
University of Mysore
S. Ravi*
Affiliation:
University of Mysore
*
∗ Postal address: Department of Studies in Statistics, University of Mysore, Mysore 570006, India.
∗ Postal address: Department of Studies in Statistics, University of Mysore, Mysore 570006, India.
∗ Postal address: Department of Studies in Statistics, University of Mysore, Mysore 570006, India.
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Abstract

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As the name suggests, the family of general error distributions has been used to model nonnormal errors in a variety of situations. In this article we show that the asymptotic distribution of linearly normalized partial maxima of random observations from the general error distributions is Gumbel when the parameter of these distributions lies in the interval (0, 1). Our result fills a gap in the literature. We also establish the corresponding density convergence, obtain an asymptotic distribution of the partial maxima under power normalization, and state and prove a strong law. We also study the asymptotic behaviour of observations near the partial maxima and the sum of such observations.

Keywords

Extremesgeneral error distributionGumbel distributionstrong law for partial maximanear maximapower normalization

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F15: Strong theorems 60G70: Extreme value theory; extremal processes 62E20: Asymptotic distribution theory
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 528 - 541
Copyright
© Applied Probability Trust 

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