Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T05:32:37.031Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Limit theorems Distribution theory Stochastic processes

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Balakrishnan, N. and Stepanov, A. (2005). A note on the number of observations near an order statistic. J. Statist. Planning Infer. 134, 1–14.Google Scholar
Box, G. E. P. and Tiao, G. C. (1962). A further look at robustness via Bayes's theorem. Biometrika 49, 419–432.CrossRefGoogle Scholar
Box, G. E. P. and Tiao, G. C. (1964). A note on criterion robustness and inference robustness. Biometrika 51, 169–173.CrossRefGoogle Scholar
Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MA.Google Scholar
Do, M. N. and Vetterli, M. (2002). Wavelet-based texture retrievel using generalized Gaussian density and Kullback–Leibler distance. IEEE Trans. Image Process. 11, 146–158.CrossRefGoogle Scholar
Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. John Wiley, New York.Google Scholar
Hu, Z. and Su, C. (2003). Limit theorems for the number and sum of near-maxima for medium tails. Statist. Prob. Lett. 63, 229–237.CrossRefGoogle Scholar
Li, Y. (1999). A note on the number of records near the maximum. Statist. Prob. Lett. 43, 153–158.Google Scholar
Mohan, N. R. and Ravi, S. (1993). Max domains of attraction of univariate and multivariate p-max stable laws. Theory Prob. Appl. 37, 632–643.CrossRefGoogle Scholar
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370.CrossRefGoogle Scholar
Osiewalski, J. and Steel, M. F. J. (1993). Robust Bayesian inference in lq-spherical models. Biometrika 80, 456–460.Google Scholar
Pakes, A. G. (2000). The number and sum of near-maxima for thin-tailed populations. Adv. Appl. Prob. 32, 1100–1116.Google Scholar
Pakes, A. G. (2004). Criteria for convergence of the number of near maxima for long tails. Extremes 7, 123–134.Google Scholar
Pakes, A. G. and Steutel, F. W. (1997). On the number of records near the maximum. Austral. J. Statist. 39, 179–192.Google Scholar
Peng, Z., Nadarajah, S. and Lin, F. (2010). Convergence rate of extremes for the general error distribution. J. Appl. Prob. 47, 668–679.Google Scholar
Peng, Z., Tong, B. and Nadarajah, S. (2009). Tail behavior of the general error distribution. Commun. Statist. Theory Meth. 38, 1884–1892.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Subbotin, M. (1923). On the law of frequency of errors. Mat. Sb. 31, 296–301.Google Scholar
Swamy, P. A. V. B. and Mehta, J. S. (1977). Robustness of Theil's mixed regression estimators. Canad. J. Statist. 5, 93–109.CrossRefGoogle Scholar
Tiao, G. C. and Lund, D. R. (1970). The use of OLUMV estimators in inference robustness studies of the location parameters of a class of symmetric distributions. J. Amer. Statist. Assoc. 65, 370–386.Google Scholar
West, M. (1984). Outlier models and prior distributions in Bayesian linear regression. J. R. Statist. Soc. B 46, 431–439.Google Scholar