The Annals of Applied Statistics

A two-step approach to model precipitation extremes in California based on max-stable and marginal point processes

Hongwei Shang, Jun Yan, and Xuebin Zhang

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In modeling spatial extremes, the dependence structure is classically inferred by assuming that block maxima derive from max-stable processes. Weather stations provide daily records rather than just block maxima. The point process approach for univariate extreme value analysis, which uses more historical data and is preferred by some practitioners, does not adapt easily to the spatial setting. We propose a two-step approach with a composite likelihood that utilizes site-wise daily records in addition to block maxima. The procedure separates the estimation of marginal parameters and dependence parameters into two steps. The first step estimates the marginal parameters with an independence likelihood from the point process approach using daily records. Given the marginal parameter estimates, the second step estimates the dependence parameters with a pairwise likelihood using block maxima. In a simulation study, the two-step approach was found to be more efficient than the pairwise likelihood approach using only block maxima. The method was applied to study the effect of El Niño-Southern Oscillation on extreme precipitation in California with maximum daily winter precipitation from 35 sites over 55 years. Using site-specific generalized extreme value models, the two-step approach led to more sites detected with the El Niño effect, narrower confidence intervals for return levels and tighter confidence regions for risk measures of jointly defined events.

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Ann. Appl. Stat., Volume 9, Number 1 (2015), 452-473.

First available in Project Euclid: 28 April 2015

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Composite likelihood estimating function extreme value analysis risk analysis spatial dependence


Shang, Hongwei; Yan, Jun; Zhang, Xuebin. A two-step approach to model precipitation extremes in California based on max-stable and marginal point processes. Ann. Appl. Stat. 9 (2015), no. 1, 452--473. doi:10.1214/14-AOAS804.

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  • Aulbach, S. and Falk, M. (2012). Testing for a generalized Pareto process. Electron. J. Stat. 6 1779–1802.
  • Bacro, J.-N. and Gaetan, C. (2012). A review on spatial extreme modelling. In Advances and Challenges in Space–Time Modelling of Natural Events (E. Porcu, J. M. Montero and M. Schlather, eds.). Lecture Notes in Statistics 207 103–124. Springer, Berlin.
  • Bacro, J.-N. and Gaetan, C. (2014). Estimation of spatial max-stable models using threshold exceedances. Stat. Comput. 24 651–662.
  • Balkema, A. A. and de Haan, L. (1974). Residual life time at great age. Ann. Probab. 2 792–804.
  • Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2003). Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC, Boca Raton, FL.
  • Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Ann. Statist. 29 1165–1188.
  • Blanchet, J. and Davison, A. C. (2011). Spatial modeling of extreme snow depth. Ann. Appl. Stat. 5 1699–1725.
  • 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.
  • Capéraà, P., Fougères, A.-L. and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84 567–577.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London.
  • Cooley, D., Naveau, P. and Poncet, P. (2006). Variograms for spatial max-stable random fields. In Dependence in Probability and Statistics. Lecture Notes in Statist. 187 373–390. Springer, New York.
  • Cooley, D., Nychka, D. and Naveau, P. (2007). Bayesian spatial modeling of extreme precipitation return levels. J. Amer. Statist. Assoc. 102 824–840.
  • Davison, A. C. and Gholamrezaee, M. M. (2012). Geostatistics of extremes. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 581–608.
  • Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Series in Statistical and Probabilistic Mathematics 1. Cambridge Univ. Press, Cambridge.
  • Davison, A. C., Padoan, S. A. and Ribatet, M. (2012). Statistical modeling of spatial extremes. Statist. Sci. 27 161–186.
  • de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab. 12 1194–1204.
  • de Haan, L. and Pereira, T. T. (2006). Spatial extremes: Models for the stationary case. Ann. Statist. 34 146–168.
  • Falk, M. and Guillou, A. (2008). Peaks-over-threshold stability of multivariate generalized Pareto distributions. J. Multivariate Anal. 99 715–734.
  • Falk, M., Hüsler, J. and Reiss, R.-D. (2010). Laws of Small Numbers: Extremes and Rare Events. Springer, Basel.
  • Falk, M. and Michel, R. (2009). Testing for a multivariate generalized Pareto distribution. Extremes 12 33–51.
  • Fawcett, L. and Walshaw, D. (2007). Improved estimation for temporally clustered extremes. Environmetrics 18 173–188.
  • Fawcett, L. and Walshaw, D. (2012). Estimating return levels from serially dependent extremes. Environmetrics 23 272–283.
  • Ferreira, A. and de Haan, L. (2014). The generalized Pareto process; with a view towards application and simulation. Bernoulli 20 1717–1737.
  • Ferreira, A. and de Haan, L. (2015). On the block maxima method in extreme value theory: PWM estimators. Ann. Statist. 43 276–298.
  • Genest, C. and Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. Ann. Statist. 37 2990–3022.
  • Genton, M. G., Ma, Y. and Sang, H. (2011). On the likelihood function of Gaussian max-stable processes. Biometrika 98 481–488.
  • Gershunov, A. and Barnett, T. P. (1998). ENSO influence on intraseasonal extreme rainfall and temperature frequencies in the contiguous United States: Observations and model results. Journal of Climate 11 1575–1586.
  • Godambe, V. P., eds. (1991). Estimating Functions. Oxford Statistical Science Series 7. Oxford Univ. Press, New York.
  • Gudendorf, G. and Segers, J. (2010). Extreme-value copulas. In Proceedings of the Workshop on Copula Theory and Its Applications (P. Jaworski, F. Durante, W. K. Härdle and T. Rychlik, eds.) 127–146. Springer, Berlin.
  • Guillou, A. and Hall, P. (2001). A diagnostic for selecting the threshold in extreme value analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 293–305.
  • Hosking, J. R. M., Wallis, J. R. and Wood, E. F. (1985). Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27 251–261.
  • Huser, R. and Davison, A. C. (2013). Composite likelihood estimation for the Brown–Resnick process. Biometrika 100 511–518.
  • Huser, R. and Davison, A. C. (2014). Space–time modelling of extreme events. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 439–461.
  • Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models. J. Multivariate Anal. 94 401–419.
  • Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37 2042–2065.
  • Katz, R. W., Parlange, M. B. and Naveau, P. (2002). Statistics of extremes in hydrology. Advances in Water Resources 25 1287–1304.
  • Kauermann, G. and Carroll, R. J. (2001). A note on the efficiency of sandwich covariance matrix estimation. J. Amer. Statist. Assoc. 96 1387–1396.
  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
  • Mancl, L. A. and DeRouen, T. A. (2001). A covariance estimator for GEE with improved small-sample properties. Biometrics 57 126–134.
  • Northrop, P. J. and Jonathan, P. (2011). Threshold modelling of spatially dependent nonstationary extremes with application to hurricane-induced wave heights. Environmetrics 22 799–809.
  • Opitz, T. (2013). Extremal $t$ processes: Elliptical domain of attraction and a spectral representation. J. Multivariate Anal. 122 409–413.
  • Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. J. Amer. Statist. Assoc. 105 263–277.
  • Pickands, J. III (1971). The two-dimensional Poisson process and extremal processes. J. Appl. Probab. 8 745–756.
  • Pickands, J. III (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119–131.
  • Ribatet, M. (2013). SpatialExtremes: Modelling spatial extremes. R package version 2.0-0.
  • Rootzén, H. and Tajvidi, N. (2006). Multivariate generalized Pareto distributions. Bernoulli 12 917–930.
  • Ropelewski, C. F. and Halpert, M. S. (1986). North American precipitation and temperature patterns associated with the El Niño/Southern Oscillation (ENSO). Monthly Weather Review 114 2352–2362.
  • Ropelewski, C. F. and Halpert, M. S. (1996). Quantifying southern oscillation–precipitation relationships. Journal of Climate 9 1043–1059.
  • Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5 33–44.
  • Shang, H., Yan, J. and Zhang, X. (2011). El Niño–Southern oscillation influence on winter maximum daily precipitation in California in a spatial model. Water Resources Research 47 W11507.
  • Shang, H., Yan, J. and Zhang, X. (2015). Supplement to “A two-step approach to model precipitation extremes in California based on max-stable and marginal point processes.” DOI:10.1214/14-AOAS804SUPP.
  • Smith, R. L. (1989). Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone. Statist. Sci. 4 367–377.
  • Smith, R. L. (1990). Max-stable processes and spatial extremes. Univ. Surrey. Unpublished manuscript.
  • Smith, R. L. (1991). Regional estimation from spatially dependent data. Preprint.
  • Takeuchi, K. (1976). Distribution of informational statistics and a criterion of model fitting. Suri-Kagaku (Mathematical Sciences) 153 12–18.
  • Tanaka, S. and Takara, K. (2002). A study on threshold selection in POT analysis of extreme floods. In The Extremes of the Extremes: Extraordinary Floods 271 (A. Snorasson, H. Finnsdottir and M. Moss, eds.) 299–304. IAHS Publication, Oxford.
  • Thibaud, E. and Opitz, T. (2013). Efficient inference and simulation for elliptical Pareto processes. Available at arXiv:1401.0168.
  • Thompson, P., Cai, Y., Reeve, D. and Stander, J. (2009). Automated threshold selection methods for extreme wave analysis. Coastal Engineering 56 1013–1021.
  • Varin, C. (2008). On composite marginal likelihoods. AStA Adv. Stat. Anal. 92 1–28.
  • Varin, C. and Vidoni, P. (2005). A note on composite likelihood inference and model selection. Biometrika 92 519–528.
  • Wadsworth, J. L. and Tawn, J. A. (2012). Dependence modelling for spatial extremes. Biometrika 99 253–272.
  • Wadsworth, J. L. and Tawn, J. A. (2014). Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika 101 1–15.
  • Zhang, X., Wang, J., Zwiers, F. W. and Groisman, P. Y. (2010). The influence of large scale climate variability on winter maximum daily precipitation over North America. Journal of Climate 23 2902–2915.
  • Zhao, Y. and Joe, H. (2005). Composite likelihood estimation in multivariate data analysis. Canad. J. Statist. 33 335–356.

Supplemental materials

  • Additional simulation results and data analysis.: We provide a sandwich variance estimator, additional tables summarizing the simulation study and additional figures in analyzing the California precipitation data.