## Journal of Applied Probability

### On the asymptotic behaviour of extremes and near maxima of random observations from the general error distributions

#### Abstract

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.

#### Article information

Source
J. Appl. Probab., Volume 51, Number 2 (2014), 528-541.

Dates
First available in Project Euclid: 12 June 2014

https://projecteuclid.org/euclid.jap/1402578641

Digital Object Identifier
doi:10.1239/jap/1402578641

Mathematical Reviews number (MathSciNet)
MR3217783

Zentralblatt MATH identifier
1305.60018

#### Citation

Vasudeva, R.; Kumari, J. Vasantha; Ravi, S. On the asymptotic behaviour of extremes and near maxima of random observations from the general error distributions. J. Appl. Probab. 51 (2014), no. 2, 528--541. doi:10.1239/jap/1402578641. https://projecteuclid.org/euclid.jap/1402578641

#### 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.
• Box, G. E. P. and Tiao, G. C. (1962). A further look at robustness via Bayes's theorem. Biometrika 49, 419–432.
• Box, G. E. P. and Tiao, G. C. (1964). A note on criterion robustness and inference robustness. Biometrika 51, 169–173.
• Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MA.
• 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.
• Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. John Wiley, New York.
• 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.
• Li, Y. (1999). A note on the number of records near the maximum. Statist. Prob. Lett. 43, 153–158.
• 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.
• Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370.
• Osiewalski, J. and Steel, M. F. J. (1993). Robust Bayesian inference in $l_q$-spherical models. Biometrika 80, 456–460.
• Pakes, A. G. (2000). The number and sum of near-maxima for thin-tailed populations. Adv. Appl. Prob. 32, 1100–1116.
• Pakes, A. G. (2004). Criteria for convergence of the number of near maxima for long tails. Extremes 7, 123–134.
• Pakes, A. G. and Steutel, F. W. (1997). On the number of records near the maximum. Austral. J. Statist. 39, 179–192.
• Peng, Z., Nadarajah, S. and Lin, F. (2010). Convergence rate of extremes for the general error distribution. J. Appl. Prob. 47, 668–679.
• Peng, Z., Tong, B. and Nadarajah, S. (2009). Tail behavior of the general error distribution. Commun. Statist. Theory Meth. 38, 1884–1892.
• Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
• Subbotin, M. (1923). On the law of frequency of errors. Mat. Sb. 31, 296–301.
• Swamy, P. A. V. B. and Mehta, J. S. (1977). Robustness of Theil's mixed regression estimators. Canad. J. Statist. 5, 93–109.
• 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.
• West, M. (1984). Outlier models and prior distributions in Bayesian linear regression. J. R. Statist. Soc. B 46, 431–439. \endharvreferences