The Annals of Statistics

On testing the extreme value index via the pot-method

Michael Falk

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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.

Article information

Ann. Statist., Volume 23, Number 6 (1995), 2013-2035.

First available in Project Euclid: 15 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F05: Asymptotic properties of tests
Secondary: 60G70: Extreme value theory; extremal processes 60G55: Point processes

Generalized Pareto distribution $\delta$-neighborhood peaks over threshold point process of exceedances log-likelihood ratio local asymptotic normality asymptotic sufficiency Hellinger distance


Falk, Michael. On testing the extreme value index via the pot-method. Ann. Statist. 23 (1995), no. 6, 2013--2035. doi:10.1214/aos/1034713645.

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