Improved Convergence Rates of Normal Extremes

Yijun Zhu () and Han Xiao ()
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Yijun Zhu: Rutgers University
Han Xiao: Rutgers University

A chapter in Robust and Multivariate Statistical Methods, 2023, pp 451-476 from Springer

Abstract: Abstract It is well known that the convergence of the normal extremes to the limiting Gumbel distribution is extremely slow, at the rate of ( log n ) − 1 $$(\log n)^{-1}$$ . We show that after a monotone transform, the convergence rate of the squared normal extremes can be improved to ( log n ) − 3 $$(\log n)^{-3}$$ . Simulations confirm that the convergence is much faster than existing results uniformly, especially when the sample is of moderate sizes around hundreds or thousands. More importantly, it is observed that the convergence rate at the upper tail is substantially improved, which has direct implications for hypothesis tests based on the maximum type test statistics.

Keywords: Extreme value theory; Gumbel distribution; Normal distribution; Rate of convergence (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-22687-8_21

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DOI: 10.1007/978-3-031-22687-8_21

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