Electronic Journal of Statistics

The spectrogram: A threshold-based inferential tool for extremes of stochastic processes

Thomas Opitz, Jean-Noël Bacro, and Pierre Ribereau

Full-text: Open access

Abstract

We extend the statistical toolbox for inferring the extreme value behavior of stochastic processes by defining the spectrogram. It is based on a pseudo-polar representation of bivariate data and represents a collection of spectral measures that characterize the bivariate extremal dependence structure in terms of a univariate distribution. Based on threshold exceedances in the original event data, estimation of the spectrogram is flexible and efficient. The virtues of the spectrogram for exploratory studies and for parametric inference are highlighted. We propose a variance reduction technique that can be applied to the estimation of summary statistics like the extremogram. Parametric inference based on distance measures between the empirical and the model spectrogram, as for instance pairwise likelihoods or least squares distances, is developed. Simulated data illustrate gains in parametric estimation efficiency compared to a standard estimation approach. An application to precipitation data collected in the Cévennes region in France shows the practical utility of the introduced notions.

Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 842-868.

Dates
First available in Project Euclid: 20 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1429536044

Digital Object Identifier
doi:10.1214/15-EJS1021

Mathematical Reviews number (MathSciNet)
MR3338665

Zentralblatt MATH identifier
1321.62051

Keywords
Max-stable processes pairwise inference spatial extremes spectral measure variance reduction

Citation

Opitz, Thomas; Bacro, Jean-Noël; Ribereau, Pierre. The spectrogram: A threshold-based inferential tool for extremes of stochastic processes. Electron. J. Statist. 9 (2015), no. 1, 842--868. doi:10.1214/15-EJS1021. https://projecteuclid.org/euclid.ejs/1429536044


Export citation

References

  • [1] Amemiya, T. (1985)., Advanced Econometrics. Harvard University Press, Cambridge, MA.
  • [2] Aulbach, S. and Falk, M. (2012). Testing for a generalized Pareto process., Electron. J. Stat. 6 1779–1802.
  • [3] Bacro, J. N. and Gaetan, C. (2014). Estimation of spatial max-stable models using threshold exceedances., Stat. Comput. 24 651–662.
  • [4] Balkema, A. A. and Resnick, S. I. (1977). Max-infinite divisibility., J. Appl. Probab. 14 309–319.
  • [5] Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004)., Statistics of Extremes: Theory and Applications. Wiley.
  • [6] Brown, B. M. and Resnick, S. I. (1977). Extreme values of independent stochastic processes., J. Appl. Probab. 732–739.
  • [7] 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.
  • [8] Capéraà, P. and Fougères, A. L. (2000). Estimation of a bivariate extreme value distribution., Extremes 3 311–329.
  • [9] Carlstein, E., Do, K. A., Hall, P., Hesterberg, T. and Künsch, H. R. (1998). Matched-block bootstrap for dependent data., Bernoulli 305–328.
  • [10] Cho, Y., Davis, R. A. and Ghosh, S. (2014). Asymptotic properties of the empirical spatial extremogram., arXiv:1408.0412.
  • [11] Coles, S. G. (2001)., An Introduction to Statistical Modeling of Extreme Values. Springer.
  • [12] Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events., J. R. Stat. Soc. B 53 377–392.
  • [13] Davis, R. A., Klüppelberg, C. and Steinkohl, C. (2013). Max-stable processes for modeling extremes observed in space and time., J. Korean Stat. Society 42 399–414.
  • [14] Davis, R. A. and Mikosch, T. (2009). The extremogram: A correlogram for extreme events., Bernoulli 15 977–1009.
  • [15] Davison, A. C. and Gholamrezaee, M. M. (2012). Geostatistics of extremes., Proc. R. Stat. Soc. A 468 581–608.
  • [16] Davison, A. C. and Hinkley, D. V. (1997)., Bootstrap Methods and Their Application. Cambridge Univ. Press, Cambridge, UK.
  • [17] Davison, A. C., Padoan, S. and Ribatet, M. (2012). Statistical modelling of spatial extremes., Stat. Sci. 27 161–186.
  • [18] de Carvalho, M., Oumow, B., Segers, J. and Warchol, M. (2013). A Euclidean likelihood estimator for bivariate tail dependence., Commun. Stat. (Theory and Methods) 42 1176–1192.
  • [19] de Haan, L. (1984). A spectral representation for max-stable processes., Ann. Probab. 12 1194–1204.
  • [20] Dombry, C. and Ribatet, M. (2015). Functional regular variations, Pareto processes and peaks over threshold., Stat. Interface 8 9– 17.
  • [21] Einmahl, J. H. J., de Haan, L. and Sinha, A. K. (1997). Estimating the spectral measure of an extreme value distribution., Stoch. Processes Appl. 70 143–171.
  • [22] Einmahl, J. H. J., de Haan, L. and Piterbarg, V. I. (2001). Nonparametric estimation of the spectral measure of an extreme value distribution., Ann. Stat. 29 1401–1423.
  • [23] Einmahl, J. H. J. and Segers, J. (2009). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution., Ann. Stat. 37 2953–2989.
  • [24] Engelke, S., Malinowski, A., Kabluchko, Z. and Schlather, M. (2015). Estimation of Huesler-Reiss distributions and Brown-Resnick processes., J. R. Statist. Soc. B 77 239–265.
  • [25] Falk, M. and Michel, R. (2009). Testing for a multivariate generalized Pareto distribution., Extremes 12 33–51.
  • [26] Ferreira, A. andde Haan, L. (2014). The generalized Pareto process., Bernoulli 20 1717–1737.
  • [27] Fiebig, U., Strokorb, K. and Schlather, M. (2014). The realization problem for tail correlation functions., arXiv:1405.6876.
  • [28] Huser, R. and Davison, A. C. (2014). Space-time modelling of extreme events., J. R. Stat. Soc. B 76 439–461.
  • [29] Kabluchko, Z., Schlather, M. and De Haan, L. (2009). Stationary max-stable fields associated to negative definite functions., Ann. Probab. 37 2042–2065.
  • [30] Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values., Biometrika 83 169.
  • [31] Mandelbrot, B. B. and van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications., Soc. Ind. Appl. Math. Rev. 10 422–437.
  • [32] Opitz, T. (2013a). Extremal t processes: Elliptical domain of attraction and a spectral representation., J. Multivar. Anal. 122 409–413.
  • [33] Opitz, T. (2013b). Extrêmes multivariés et spatiaux: Approches spectrales et modèles elliptiques. PhD Thesis, 146, pages.
  • [34] Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes., J. Am. Stat. Assoc. 105 263–277.
  • [35] Pickands, J. (1981). Multivariate extreme value theory., Bull. Int. Stat. Inst. 1 859–878.
  • [36] Resnick, S. I. (1987)., Extreme Values, Regular Variation and Point Processes. Springer, Berlin.
  • [37] Resnick, S. I. (2007)., Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, Berlin.
  • [38] Robert, C. P. and Casella, G. (2004)., Monte Carlo Statistical Methods, 2nd ed. Springer, Berlin.
  • [39] Sang, H. and Genton, M. (2014). Tapered composite likelihood for spatial max-stable models., Spatial Statistics 8 86–103.
  • [40] Schabenberger, O. and Gotway, C. A. (2004)., Statistical Methods for Spatial Data Analysis. Chapman & Hall/CRC, Boca Raton.
  • [41] Schlather, M. (2002). Models for stationary max-stable random fields., Extremes 5 33–44.
  • [42] Schlather, M. and Tawn, J. A. (2003). A dependence measure for multivariate and spatial extreme values: Properties and inference., Biometrika 90 139–156.
  • [43] Shao, J. and Tu, D. (1995)., The Jackknife and Bootstrap. Springer, Berlin.
  • [44] Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished, manuscript.
  • [45] Strokorb, K., Ballani, F. and Schlather, M. (2014). Systematic co-occurrence of tail correlation functions among max-stable processes., arXiv:1402.4632.
  • [46] Strokorb, K., Ballani, F. and Schlather, M. (2015). Tail correlation functions of max-stable processes. To appear in, Extremes.
  • [47] R Core Team (2013). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria.
  • [48] Thibaud, E. and Opitz, T. (2014). Efficient inference and simulation for elliptical Pareto processes., Submitted.
  • [49] Varin, C., Reid, N. and Firth, D. (2011). An overview of composite likelihood methods., Statistica Sinica 21 5–42.