Advertisement

Extremes

pp 1–47 | Cite as

Generalised least squares estimation of regularly varying space-time processes based on flexible observation schemes

  • Sven Buhl
  • Claudia Klüppelberg
Article
  • 13 Downloads

Abstract

Regularly varying stochastic processes model extreme dependence between process values at different locations and/or time points. For such stationary processes we propose a two-step parameter estimation of the extremogram, when some part of the domain of interest is fixed and another increasing. We provide conditions for consistency and asymptotic normality of the empirical extremogram centred by a pre-asymptotic version for such observation schemes. For max-stable processes with Fréchet margins we provide conditions, such that the empirical extremogram (or a bias-corrected version) centred by its true version is asymptotically normal. In a second step, for a parametric extremogram model, we fit the parameters by generalised least squares estimation and prove consistency and asymptotic normality of the estimates. We propose subsampling procedures to obtain asymptotically correct confidence intervals. Finally, we apply our results to a variety of Brown-Resnick processes. A simulation study shows that the procedure works well also for moderate sample sizes.

Keywords

Brown-Resnick process Extremogram Generalised least squares estimation Max-stable process Observation schemes Regularly varying process Semiparametric estimation Space-time process 

AMS 2000 Subject Classifications

Primary: 60F05 60G70 62F12 62G32 Secondary: 37A25 62M30 62P12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

Sven Buhl acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through the TUM International Graduate School of Science and Engineering (IGSSE).

References

  1. Asadi, P., Davison, A.C., Engelke, S.: Extremes on river networks. Ann. Appl Stat. 9(4), 2023–2050 (2015)MathSciNetCrossRefGoogle Scholar
  2. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of extremes, theory and applications. Wiley, Chichester (2004)CrossRefGoogle Scholar
  3. Blanchet, J., Davison, A.: Spatial modeling of extreme snow depth. Ann. Appl Stat. 5(3), 1699–1724 (2011)MathSciNetCrossRefGoogle Scholar
  4. Bolthausen, E.: On the central limit theorem for stationary mixing random fields. Ann. Probab. 10(4), 1047–1050 (1982)MathSciNetCrossRefGoogle Scholar
  5. Brown, B., Resnick, S.: Extreme values of independent stochastic processes. J. Appl. Probab. 14(4), 732–739 (1977)MathSciNetCrossRefGoogle Scholar
  6. Buhl, S., Klüppelberg, C.: Anisotropic Brown-Resnick space-time processes: estimation and model assessment. Extremes 19, 627–660 (2016).  https://doi.org/10.1007/s10687-016-0257-1r MathSciNetCrossRefGoogle Scholar
  7. Buhl, S., Klüppelberg, C.: Limit theory for the empirical extremogram of random fields. Stoch. Process. Appl. 128(6), 2060–2082 (2018)MathSciNetCrossRefGoogle Scholar
  8. Buhl, S., Davis, R., Klüppelberg, C., Steinkohl, C.: Semiparametric estimation for isotropic max-stable space-time processes. Bernoulli, in press, arXiv:1609.04967v3[stat.ME] (2018)
  9. Cho, Y., Davis, R., Ghosh, S.: Asymptotic properties of the spatial empirical extremogram. Scand. J Stat. 43(3), 757–773 (2016)MathSciNetCrossRefGoogle Scholar
  10. Davis, R., Mikosch, T.: The extremogram: a correlogram for extreme events. Bernoulli 15(4), 977–1009 (2009)MathSciNetCrossRefGoogle Scholar
  11. Davis, R., Klüppelberg, C., Steinkohl, C.: Max-stable processes for extremes of processes observed in space and time. J. Korean Stat. Soc. 42(3), 399–414 (2013a)Google Scholar
  12. Davis, R., Klüppelberg, C., Steinkohl, C.: Statistical inference for max-stable processes in space and time. JRSS B 75(5), 791–819 (2013b)Google Scholar
  13. Davis, R., Mikosch, T., Zhao, Y.: Measures of serial extremal dependence and their estimation. Stoch. Process. Appl. 123(7), 2575–2602 (2013c)Google Scholar
  14. Davison, A. C., Padoan, S. A., Ribatet, M.: Statistical modeling of spatial extremes. Stat. Sci. 27(2), 161–186 (2012c)Google Scholar
  15. de Fondeville, R., Davison, A.: High-dimensional peaks-over-threshold inference for the Brown-Resnick process. Biometrika 105(3), 575–592 (2018)MathSciNetCrossRefGoogle Scholar
  16. de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12(4), 1194–1204 (1984)MathSciNetCrossRefGoogle Scholar
  17. de Haan, L., Ferreira, A.: Extreme value theory: An introduction. Springer Series in Operations Research and Financial Engineering, New York (2006)CrossRefGoogle Scholar
  18. Dombry, C., Eyi-Minko, F.: Strong mixing properties of max-infinitely divisible random fields. Stoch Process. Appl. 122(11), 3790–3811 (2012)MathSciNetCrossRefGoogle Scholar
  19. Dombry, C., Engelke, S., Oesting, M.: Exact simulation of max-stable processes. Biometrika 103, 303–317 (2016)MathSciNetCrossRefGoogle Scholar
  20. Dombry, C., Genton, M. G., Huser, R., Ribatet, M.: Full likelihood inference for max-stable data. arXiv:1703.08665 (2018)
  21. Drees, H.: Bootstrapping empirical processes of cluster functionals with application to extremograms. arXiv:1511.00420v1[math.ST] (2015)
  22. Einmahl, J., Kiriliouk, A., Segers, J.: A continuous updating weighted least squares estimator of tail dependence in high dimensions. Extremes 21(2), 205–233 (2018)MathSciNetCrossRefGoogle Scholar
  23. Embrechts, P., Koch, E., Robert, C.: Space-time max-stable models with spectral separability. Adv. Appl. Probab. 48(A), 77–97 (2016)MathSciNetCrossRefGoogle Scholar
  24. Engelke, S., Malinowski, A., Kabluchko, Z., Schlather, M.: Estimation of Hüsler-Reiss distributions and Brown-Resnick processes. JRSS B 77(1), 239–265 (2015)CrossRefGoogle Scholar
  25. Fasen, V., Klüppelberg, C., Schlather, M.: High-level dependence in time series models. Extremes 13(1), 1–33 (2010)MathSciNetCrossRefGoogle Scholar
  26. Giné, E., Hahn, M. G., Vatan, P.: Max-infinitely divisible and max-stable sample continuous processes. Probab. Theory Rel. Fields 87, 139–165 (1990)MathSciNetCrossRefGoogle Scholar
  27. Hult, H., Lindskog, F.: Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274 (2005)MathSciNetCrossRefGoogle Scholar
  28. Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publications de l’Institut Mathematique (Beograd)́, 80, 121–140 (2006)MathSciNetCrossRefGoogle Scholar
  29. Huser, R.: Statistical Modeling and Inference for Spatio-Temporal Extremes. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, Lausanne (2013)Google Scholar
  30. Huser, R., Davison, A.: Composite likelihood estimation for the Brown-Resnick process. Biometrika 100(2), 511–518 (2013)MathSciNetCrossRefGoogle Scholar
  31. Huser, R., Davison, A.: Space-time modelling of extreme events. JRSS B 76(2), 439–461 (2014)MathSciNetCrossRefGoogle Scholar
  32. Huser, R., Genton, M.G.: Non-stationary dependence structures for spatial extremes. Journal of Agricultural Biological and Environmental Statistics 21(3), 470–491 (2016)MathSciNetCrossRefGoogle Scholar
  33. Ibragimov, I., Linnik, Y.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen (1971)zbMATHGoogle Scholar
  34. Kabluchko, Z., Schlather, M., de Haan, L.: Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37(5), 2042–2065 (2009)MathSciNetCrossRefGoogle Scholar
  35. Lahiri, S. N., Lee, Y., Cressie, N.: On asymptotic distribution and asymptotic efficiency of least squares estimators of spatial variogram parameters. J. Stat. Plan. Inf. 103(1), 65–85 (2002)MathSciNetCrossRefGoogle Scholar
  36. Li, B., Genton, M., Sherman, M.: On the asymptotic joint distribution of sample space-time covariance estimators. Bernoulli 14(1), 208–248 (2008)MathSciNetCrossRefGoogle Scholar
  37. Opitz, T.: Extremal t processes: Elliptical domain of attraction and a spectral representation. J. Multivar. Anal. 122, 409–413 (2013)MathSciNetCrossRefGoogle Scholar
  38. Padoan, S., Ribatet, M., Sisson, S.: Likelihood-based inference for max-stable processes. JASA 105(489), 263–277 (2009)MathSciNetCrossRefGoogle Scholar
  39. Politis, D. N., Romano, J. P., Wolf, M.: Subsampling. Springer, New York (1999)CrossRefGoogle Scholar
  40. Resnick, S. : Point processes, regular variation and weak convergence. Adv. Appl. Probab. 18(1), 66–138 (1986)MathSciNetCrossRefGoogle Scholar
  41. Resnick, S.: Heavy-tail phenomena, probabilistic and statistical modeling. Springer, New York (2007)zbMATHGoogle Scholar
  42. Schlather, M.: Randomfields, contributed package on random field simulation for R. http://cran.r-project.org/web/packages/RandomFields/
  43. Steinkohl, C.: Statistical modelling of extremes in space and time using max-stable processes. Ph.D. Thesis. Technische Universität München, München (2013)Google Scholar
  44. Thibaud, E., Aalto, J., Cooley, D. S., Davison, A. C., Heikkinen, J.: Bayesian inference for the Brown-Resnick process, with an application to extreme low temperatures. Ann. Appl. Stat. 10(4), 2303–2324 (2016)MathSciNetCrossRefGoogle Scholar
  45. Wadsworth, J., Tawn, J.: Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika 101(1), 1–15 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Mathematical SciencesTechnical University of MunichGarchingGermany

Personalised recommendations