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December 1995 On testing the extreme value index via the pot-method
Michael Falk
Ann. Statist. 23(6): 2013-2035 (December 1995). DOI: 10.1214/aos/1034713645


Consider an iid sample $Y_1, \dots, Y_n$ of random variables with common distribution function F, whose upper tail belongs to a neighborhood of the upper tail of a generalized Pareto distribution $H_{\beta}, \beta \epsilon \mathbb{R}$. We investigate the testing problem $\beta = \beta_0$ against a sequence $\beta = \beta_n$ of contiguous 0 n alternatives, based on the point processes $N_n$ of the exceedances among $Y_i$ over a sequence of thresholds $t_n$. It turns out that the (random) number of exceedances $\tau (n)$ over $t_n$ is the central sequence for the log-likelihood ratio $d \mathsf{L}_{\beta_n} (N_n)/ d \mathsf{L}_{\beta_0} (N_n)$, yielding its local asymptotic normality (LAN). This result implies in particular that $\tau (n)$ carries asymptotically all the information about the underlying parameter $\beta$, which is contained in $N_n$. We establish sharp bounds for the rate at which $\tau (n)$ becomes asymptotically sufficient, which show, however, that this is quite a poor rate. These results remain true if we add an unknown scale parameter.


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Michael Falk. "On testing the extreme value index via the pot-method." Ann. Statist. 23 (6) 2013 - 2035, December 1995.


Published: December 1995
First available in Project Euclid: 15 October 2002

zbMATH: 0856.62027
MathSciNet: MR1389863
Digital Object Identifier: 10.1214/aos/1034713645

Primary: 62F05
Secondary: 60G55, 60G70

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 6 • December 1995
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