Browse > Article
http://dx.doi.org/10.4134/BKMS.b150804

CONVERGENCE RATE OF EXTREMES FOR THE GENERALIZED SHORT-TAILED SYMMETRIC DISTRIBUTION  

Lin, Fuming (School of Science Sichuan University of Science & Engineering)
Peng, Zuoxiang (School of Mathematics and Statistics Southwestern University)
Yu, Kaizhi (School of Statistics Southwestern University of Finance and Economics)
Publication Information
Bulletin of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 1549-1566 More about this Journal
Abstract
Denote $M_n$ the maximum of n independent and identically distributed variables from the generalized short-tailed symmetric distribution. This paper shows the pointwise convergence rate of the distribution of $M_n$ to exp($\exp(-e^{-x})$) and the supremum-metric-based convergence rate as well.
Keywords
generalized short-tailed symmetric distribution; maximum; extreme value distribution; rate of convergence;
Citations & Related Records
연도 인용수 순위
  • Reference
1 S. Chen, C. Wang, and G. Zhang, Rates of convergence of extreme for general error distribution under power normalization, Statist. Probab. Lett. 82 (2012), no. 2, 385-395.   DOI
2 S. Cheng and C. Jiang, The Edgeworth expansion for distributions of extreme values, Sci. China Math. 44 (2001), no. 4, 427-437.   DOI
3 L. de Haan and A. Ferreira, Extreme Value Theory: An Introduction, Springer, New York, 2006.
4 L. de Haan and S. I. Resnick, Second-order regular variation and rates of convergence in extreme-value theory, Ann. Probab. 24 (1996), no. 1, 97-124.   DOI
5 P. Embrechts, C. Kluppelberg, and T. Mikosch, Modelling Extreme Events for Insurance and Finance, Springer, Berlin, 1997.
6 M. Falk, J. Hussler, and R. D. Reiss, Laws of Small Numbers: Extremes and Rare Events, Birkhauser, Switzerland, 2004.
7 B. V. Gnedenko, Sur la distribution limite du terme maximum d'une serie aleatoire, Ann. Math. 44 (1943), 423-453.   DOI
8 P. Hall, On the rate of convergence of normal extremes, J. Appl. Probab. 16 (1979), no. 2, 433-439.   DOI
9 W. J. Hall and J. A. Wellner, The rate of convergence in law of the maximum of an exponential sample, Stat. Neerl. 33 (1979), no. 3, 151-154.   DOI
10 S. Kotz and S. Nadarajah, Extreme Value Distributions: Theory and Applications, Imperial College Press, London, 2000.
11 M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer, New York, 1983.
12 X. Liao and Z. Peng, Convergence rates of limit distribution of maxima of lognormal samples, J. Math. Anal. Appl. 395 (2012), no. 2, 643-653.   DOI
13 X. Liao, Z. Peng, and S. Nadarajah, Tail properties and asymptotic expansions for the maximum of the logarithmic skew-normal distribution, J. Appl. Probab. 50 (2013), no. 3, 900-907.   DOI
14 X. Liao, Z. Peng, S. Nadarajah, and X. Wang, Rates of convergence of extremes from skew-normal samples, Statist. Probab. Lett. 84 (2014), 40-47.   DOI
15 F. Lin and Y. Jiang, A general version of the short-tailed symmetric distribution, Comm. Statist. Theory Methods 41 (2012), no. 12, 2088-2095.   DOI
16 F. Lin and Z. Peng, Tail behavior and extremes of short-tailed symmetric distribution, Comm. Statist. Theory Methods 39 (2010), no. 15, 2811-2817.   DOI
17 F. Lin, X. Zhang, Z. Peng, and Y. Jiang, On the rate of convergence of STSD extremes, Comm. Statist. Theory Methods 40 (2011), no. 10, 1795-1806.   DOI
18 Z. Peng, S. Nadarajah, and F. Lin, Convergence rate of extremes for the general error distribution, J. Appl. Probab. 47 (2010), no. 3, 668-679.   DOI
19 Z. Peng, B. Tong, and S. Nadarajah, Tail behavior of the general error distribution, Comm. Statist. Theory Methods 38 (2009), no. 11-12, 1884-1892.   DOI
20 S. I. Resnick, Extreme values, Regular Variation and Point Processes, Springer, New York, 1987.
21 M. L. Tiku and D. C. Vaughan, A family of short-tailed symmetric distributions, Technical Report, McMaster University, Canada, 1999.