## Electronic Journal of Statistics

### Testing for a generalized Pareto process

#### Abstract

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

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

Dates
First available in Project Euclid: 4 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1349355602

Digital Object Identifier
doi:10.1214/12-EJS728

Mathematical Reviews number (MathSciNet)
MR2988464

Zentralblatt MATH identifier
1295.62017

#### Citation

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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