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

Part of: Limit theorems Stochastic processes Distribution theory

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 

References

Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178.Google Scholar
Azzalini, A., Dei Cappello, T. and Kotz, S. (2003). Log-skew-normal and log-skew-t distributions as models for family income data. J. Income Distribution 11, 12–20.Google Scholar
Bolance, C., Guillen, M., Pelican, E. and Vernic, R. (2008). Skewed bivariate models and nonparametric estimation for the CTE risk measure. Insurance Math. Econom. 43, 386–393.Google Scholar
Chai, H. S. and Bailey, K. R. (2008). Use of log-skew-normal distribution in analysis of continuous data with a discrete component at zero. Statist. Medicine 27, 3643–3655.Google Scholar
Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions. Springer, New York.Google Scholar
Goldie, C. M. and Resnick, S. (1988). Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution. J. Appl. Prob. 20, 706–718.Google Scholar
Huang, C. Y. and Ku, M. S. (2010). Asymmetry effect of particle size distribution on content uniformity and over-potency risk in low-dose solid drugs. J. Pharm. Sci. 99, 4351–4362.Google Scholar
Li, X., Wu, Z., Chakravarthy, V. D. and Wu, Z. (2011). A low-complexity approximation to lognormal sum distributions via transformed log skew normal distribution. IEEE Trans. Vehicular Tech. 60, 4040–4045.Google Scholar
Liao, X., Peng, Z., Nadarajah, S. and Wang, X. (2012). Rates of convergence of extremes from skew normal samples. Preprint. Available at http://arxiv.org/abs/1212.1004v1.Google Scholar
Lin, D. G. and Stoyanov, J. (2009). The logarithmic skew-normal distributions are moment-indeterminate. J. Appl. Prob. 46, 909–916.Google Scholar
Mahmud, S., Lou, W. Y. W. and Johnston, N. W. (2010). A probit- log- skew-normal mixture model for repeated measures data with excess zeros, with application to a cohort study of paediatric respiratory symptoms. BMC Medical Res. Methodology 10, article no. 55.Google Scholar
Marchenko, Y. V. and Genton, M. G. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data. Environmetrics 21, 318–340.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Wu, Z. et al. (2009). A novel highly accurate log skew normal approximation method to lognormal sum distributions. In Proc. 2009 IEEE Wireless Communications and Networking Conf., IEEE pp. 392–397.Google Scholar