Electronic Journal of Statistics

Testing for a generalized Pareto process

Stefan Aulbach and Michael Falk

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We investigate two models for the following setup: We consider a stochastic process $\boldsymbol{X}\in C[0,1]$ whose distribution belongs to a parametric family indexed by $\vartheta\in\Theta\subset \mathbb{R} $. In case $\vartheta=0$, $\boldsymbol{X}$ is a generalized Pareto process. Based on $n$ independent copies $\boldsymbol{X}^{(1)},\dots,\boldsymbol{X}^{(n)}$ of $\boldsymbol{X}$, we establish local asymptotic normality (LAN) of the point process of exceedances among $\boldsymbol{X}^{(1)},\dots,\boldsymbol{X}^{(n)}$ above an increasing threshold line in each model.

The corresponding central sequences provide asymptotically optimal sequences of tests for testing $H_{0}:\vartheta=0$ against a sequence of alternatives $H_{n}:\vartheta=\vartheta_{n}$ converging to zero as $n$ increases. In one model, with an underlying exponential family, the central sequence is provided by the number of exceedances only, whereas in the other one the exceedances themselves contribute, too. However it turns out that, in both cases, the test statistics also depend on some additional and usually unknown model parameters.

We, therefore, consider an omnibus test statistic sequence as well and compute its asymptotic relative efficiency with respect to the optimal test sequence.

Article information

Electron. J. Statist., Volume 6 (2012), 1779-1802.

First available in Project Euclid: 4 October 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F05: Asymptotic properties of tests
Secondary: 60G70: Extreme value theory; extremal processes 62G32: Statistics of extreme values; tail inference

Functional extreme value theory generalized Pareto process max-stable process point process of exceedances local asymptotic normality central sequence asymptotic optimal test sequence asymptotic relative efficiency


Aulbach, Stefan; Falk, Michael. Testing for a generalized Pareto process. Electron. J. Statist. 6 (2012), 1779--1802. doi:10.1214/12-EJS728. https://projecteuclid.org/euclid.ejs/1349355602

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