Electronic Journal of Statistics

Testing for a generalized Pareto process

Stefan Aulbach and Michael Falk

Full-text: Open access

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

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

Keywords
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

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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References

  • Aulbach, S., Bayer, V. and Falk, M. (2012). A multivariate piecing-together approach with an application to operational loss data., Bernoulli 18 455-475. 10.3150/10-BEJ343
  • Aulbach, S. and Falk, M. (2012). Local asymptotic normality in $\delta$-neighborhoods of standard generalized Pareto processes., J. Statist. Plann. Inference 142 1339-1347. 10.1016/j.jspi.2011.12.011
  • Aulbach, S., Falk, M. and Hofmann, M. (2012a). The multivariate piecing-together approach revisited., J. Multivariate Anal. 110 161-170. 10.1016/j.jmva.2012.02.002
  • Aulbach, S., Falk, M. and Hofmann, M. (2012b). On max-stable processes and the functional $D$-norm., Extremes. to appear. 10.1007/s10687-012-0160-3
  • Balkema, A. A. and de Haan, L. (1974). Residual life time at great age., Ann. Probab. 2 792-804. 10.1214/aop/1176996548
  • Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004)., Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. Wiley, Chichester, UK. 10.1002/0470012382
  • Buishand, T. A., de Haan, L. and Zhou, C. (2008). On spatial extremes: With application to a rainfall problem., Ann. Appl. Stat. 2 624-642. 10.1214/08-AOAS159
  • de Haan, L. and Ferreira, A. (2006)., Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York. See http://people.few.eur.nl/ldehaan/EVTbook.correction.pdf and http://home.isa.utl.pt/~anafh/corrections.pdf for corrections and extensions.
  • de Haan, L. and Pereira, T. T. (2006). Spatial extremes: Models for the stationary case., Ann. Statist. 34 146-168. 10.1214/009053605000000886
  • Falk, M. (1998). Local asymptotic normality of truncated empirical processes., Ann. Statist. 26 692-718. 10.1214/aos/1028144855
  • Falk, M., Hüsler, J. and Reiss, R.-D. (2010)., Laws of Small Numbers: Extremes and Rare Events, 3rd ed. Birkhäuser, Basel.
  • Falk, M. and Liese, F. (1998). LAN of thinned empirical processes with an application to fuzzy set density estimation., Extremes 1 323-349. 10.1023/A:1009981817526
  • Falk, M. and Michel, R. (2009). Testing for a multivariate generalized Pareto distribution., Extremes 12 33-51. 10.1007/s10687-008-0067-1
  • Ferreira, A. and de Haan, L. (2012). The generalized Pareto process; with application. Technical Report. arXiv:1203.2551v1, [math.PR].
  • Pfanzagl, J. (1994)., Parametric Statistical Theory. De Gruyter, Berlin.
  • Pickands, J. III (1975). Statistical inference using extreme order statistics., Ann. Statist. 3 119-131. 10.1214/aos/1176343003
  • Reiss, R.-D. (1993)., A Course on Point Processes. Springer, New York.
  • Rootzén, H. and Tajvidi, N. (2006). Multivariate generalized Pareto distributions., Bernoulli 12 917-930. 10.3150/bj/1161614952
  • Tajvidi, N. (1996). Characterisation and Some Statistical Aspects of Univariate and Multivariate Generalised Pareto Distributions. PhD thesis, Chalmers University of Technology, Gothenburg., http://www.maths.lth.se/matstat/staff/nader/fullpub.html