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Generalised least squares estimation of regularly varying space-time processes based on flexible observation schemes. (English) Zbl 1422.60030

Summary: 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.

MSC:

60F05 Central limit and other weak theorems
60G70 Extreme value theory; extremal stochastic processes
62F12 Asymptotic properties of parametric estimators
62G32 Statistics of extreme values; tail inference
62M30 Inference from spatial processes
62P12 Applications of statistics to environmental and related topics

Software:

RandomFields
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References:

[1] Asadi, P., Davison, A.C., Engelke, S.: Extremes on river networks. Ann. Appl Stat. 9(4), 2023-2050 (2015) · Zbl 1397.62482 · doi:10.1214/15-AOAS863
[2] Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of extremes, theory and applications. Wiley, Chichester (2004) · Zbl 1070.62036 · doi:10.1002/0470012382
[3] Blanchet, J., Davison, A.: Spatial modeling of extreme snow depth. Ann. Appl Stat. 5(3), 1699-1724 (2011) · Zbl 1228.62154 · doi:10.1214/11-AOAS464
[4] Bolthausen, E.: On the central limit theorem for stationary mixing random fields. Ann. Probab. 10(4), 1047-1050 (1982) · Zbl 0496.60020 · doi:10.1214/aop/1176993726
[5] Brown, B., Resnick, S.: Extreme values of independent stochastic processes. J. Appl. Probab. 14(4), 732-739 (1977) · Zbl 0384.60055 · doi:10.2307/3213346
[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 · Zbl 1357.62279 · doi:10.1007/s10687-016-0257-1
[7] Buhl, S., Klüppelberg, C.: Limit theory for the empirical extremogram of random fields. Stoch. Process. Appl. 128(6), 2060-2082 (2018) · doi:10.1016/j.spa.2017.08.018
[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) · Zbl 1434.62183
[9] Cho, Y., Davis, R., Ghosh, S.: Asymptotic properties of the spatial empirical extremogram. Scand. J Stat. 43(3), 757-773 (2016) · Zbl 1468.62284 · doi:10.1111/sjos.12202
[10] Davis, R., Mikosch, T.: The extremogram: a correlogram for extreme events. Bernoulli 15(4), 977-1009 (2009) · Zbl 1200.62104 · doi:10.3150/09-BEJ213
[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) · Zbl 1294.62118
[12] Davis, R., Klüppelberg, C., Steinkohl, C.: Statistical inference for max-stable processes in space and time. JRSS B 75(5), 791-819 (2013b) · Zbl 1411.60071
[13] Davis, R., Mikosch, T., Zhao, Y.: Measures of serial extremal dependence and their estimation. Stoch. Process. Appl. 123(7), 2575-2602 (2013c) · Zbl 1294.60076
[14] Davison, A. C., Padoan, S. A., Ribatet, M.: Statistical modeling of spatial extremes. Stat. Sci. 27(2), 161-186 (2012c) · Zbl 1330.86021
[15] de Fondeville, R., Davison, A.: High-dimensional peaks-over-threshold inference for the Brown-Resnick process. Biometrika 105(3), 575-592 (2018) · Zbl 1499.62158 · doi:10.1093/biomet/asy026
[16] de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12(4), 1194-1204 (1984) · Zbl 0597.60050 · doi:10.1214/aop/1176993148
[17] de Haan, L., Ferreira, A.: Extreme value theory: An introduction. Springer Series in Operations Research and Financial Engineering, New York (2006) · Zbl 1101.62002 · doi:10.1007/0-387-34471-3
[18] Dombry, C., Eyi-Minko, F.: Strong mixing properties of max-infinitely divisible random fields. Stoch Process. Appl. 122(11), 3790-3811 (2012) · Zbl 1260.60101 · doi:10.1016/j.spa.2012.06.013
[19] Dombry, C., Engelke, S., Oesting, M.: Exact simulation of max-stable processes. Biometrika 103, 303-317 (2016) · Zbl 1499.62343 · doi:10.1093/biomet/asw008
[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) · Zbl 1402.62088 · doi:10.1007/s10687-017-0303-7
[23] Embrechts, P., Koch, E., Robert, C.: Space-time max-stable models with spectral separability. Adv. Appl. Probab. 48(A), 77-97 (2016) · Zbl 1426.60058 · doi:10.1017/apr.2016.43
[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) · Zbl 1414.60038 · doi:10.1111/rssb.12074
[25] Fasen, V., Klüppelberg, C., Schlather, M.: High-level dependence in time series models. Extremes 13(1), 1-33 (2010) · Zbl 1226.60079 · doi:10.1007/s10687-009-0084-8
[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) · Zbl 0688.60031 · doi:10.1007/BF01198427
[27] Hult, H., Lindskog, F.: Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249-274 (2005) · Zbl 1070.60046 · doi:10.1016/j.spa.2004.09.003
[28] Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publications de l’Institut Mathematique (Beograd)́, 80, 121-140 (2006) · Zbl 1164.28005 · doi:10.2298/PIM0694121H
[29] Huser, R.: Statistical Modeling and Inference for Spatio-Temporal Extremes. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, Lausanne (2013)
[30] Huser, R., Davison, A.: Composite likelihood estimation for the Brown-Resnick process. Biometrika 100(2), 511-518 (2013) · Zbl 1452.62702 · doi:10.1093/biomet/ass089
[31] Huser, R., Davison, A.: Space-time modelling of extreme events. JRSS B 76(2), 439-461 (2014) · Zbl 07555457 · doi:10.1111/rssb.12035
[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) · Zbl 1347.62246 · doi:10.1007/s13253-016-0247-4
[33] Ibragimov, I., Linnik, Y.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen (1971) · Zbl 0219.60027
[34] Kabluchko, Z., Schlather, M., de Haan, L.: Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37(5), 2042-2065 (2009) · Zbl 1208.60051 · doi:10.1214/09-AOP455
[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) · Zbl 0989.62049 · doi:10.1016/S0378-3758(01)00198-7
[36] Li, B., Genton, M., Sherman, M.: On the asymptotic joint distribution of sample space-time covariance estimators. Bernoulli 14(1), 208-248 (2008) · Zbl 1155.62010 · doi:10.3150/07-BEJ6196
[37] Opitz, T.: Extremal t processes: Elliptical domain of attraction and a spectral representation. J. Multivar. Anal. 122, 409-413 (2013) · Zbl 1282.60054 · doi:10.1016/j.jmva.2013.08.008
[38] Padoan, S., Ribatet, M., Sisson, S.: Likelihood-based inference for max-stable processes. JASA 105(489), 263-277 (2009) · Zbl 1397.62172 · doi:10.1198/jasa.2009.tm08577
[39] Politis, D. N., Romano, J. P., Wolf, M.: Subsampling. Springer, New York (1999) · Zbl 0931.62035 · doi:10.1007/978-1-4612-1554-7
[40] Resnick, S. : Point processes, regular variation and weak convergence. Adv. Appl. Probab. 18(1), 66-138 (1986) · Zbl 0597.60048 · doi:10.2307/1427239
[41] Resnick, S.: Heavy-tail phenomena, probabilistic and statistical modeling. Springer, New York (2007) · Zbl 1152.62029
[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) · Zbl 1294.62118
[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) · Zbl 1454.62462 · doi:10.1214/16-AOAS980
[45] Wadsworth, J., Tawn, J.: Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika 101(1), 1-15 (2014) · Zbl 1400.62104 · doi:10.1093/biomet/ast042
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