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2012 Testing for a generalized Pareto process
Stefan Aulbach, Michael Falk
Electron. J. Statist. 6: 1779-1802 (2012). DOI: 10.1214/12-EJS728

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.

Citation

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Stefan Aulbach. Michael Falk. "Testing for a generalized Pareto process." Electron. J. Statist. 6 1779 - 1802, 2012. https://doi.org/10.1214/12-EJS728

Information

Published: 2012
First available in Project Euclid: 4 October 2012

zbMATH: 1295.62017
MathSciNet: MR2988464
Digital Object Identifier: 10.1214/12-EJS728

Subjects:
Primary: 62F05
Secondary: 60G70, 62G32

Rights: Copyright © 2012 The Institute of Mathematical Statistics and the Bernoulli Society

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