Abstract
In this short note, under power normalization we establish the higher-order expansions of probability density function and cumulative distribution function of maximum from logarithmic general error distribution.
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References
Barakat HM, Nigm EM, EI-Adll ME (2010) Comparison between the rates of convergence of extremes under linear and under power normalization. Stat Pap 51:149–164
de Haan L, Resnick S (1996) Second order regular variation and rates of convergence in extreme value theory. Ann Probab 24:97–124
Hall P (1979) On the rate of convergence of normal extremes. J Appl Probab 16:433–439
Hall WJ, Wellner JA (1979) The rate of convergence in law of the maximum of an exponential sample. Stat Neerl 33:151–154
Liao X, Peng Z (2012) Convergence rates of limit distribution of maxima of lognormal samples. J Math Anal Appl 395:643–653
Liao X, Peng Z, Nadarajah S (2014a) Tail behavior and limit distribution of maximum of logarithmic general error distribution. Commun Stat 43:5276–5289
Liao X, Peng Z, Nadarajah S, Wang X (2014b) Rates of convergence of extremes from skew-normal samples. Stat Probab Lett 84:40–47
Mohan NR, Ravi S (1993) Max domains of attraction of univariate and multivariate p-max stable laws. Theory Probab Appl 37:632–643
Nair KA (1981) Asymptotic distribution and moments of normal extremes. Ann Probab 9:150–153
Nasri-Roudsari D (1999) Limit distribution of generalized order statistics under power normalization. Commun Stat 28:1379–1389
Pancheva E (1985) Limit theorems for extreme order statistics under nonlinear normalization. Lect Notes Math 115:284–309
Peng Z, Nadarajah S, Lin F (2010) Convergence rate of extremes for the general error distribution. J Appl Probab 47:668–679
Peng Z, Shuai Y, Nadarajah S (2013) On convergence of extremes under power normalization. Extremes 16:285–301
Resnick SI (1987) Extreme values, regular variation, and point processes. Springer, New York
Sreehari M (2009) General max-stable laws. Extremes 12:187–200
Subramanya UR (1994) On max domain of attraction of univariate p-max stable laws. Stat Probab Lett 19:271–279
Acknowledgments
We would like to appreciate two referees for reading the paper and making helpful comments which greatly improved the paper. This work was supported by the National Natural Science Foundation of China (11171275), the Natural Science Foundation Project of CQ (cstc2012jjA00029), the Fundamental Research Funds for the Central Universities (XDJK2013C021, XDJK2014D020) and the Doctoral Grant of Southwest University (SWU113012).
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Yang, G., Li, T. Expansions on extremes from logarithmic general error distribution under power normalization. Stat Papers 57, 781–793 (2016). https://doi.org/10.1007/s00362-015-0679-x
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DOI: https://doi.org/10.1007/s00362-015-0679-x
Keywords
- Logarithmic general error distribution
- Higher-order expansion
- Power normalization
- \(p\)-max stable laws
- Maximum