The Annals of Applied Statistics

Extreme value modelling of water-related insurance claims

Christian Rohrbeck, Emma F. Eastoe, Arnoldo Frigessi, and Jonathan A. Tawn

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Abstract

This paper considers the dependence between weather events, for example, rainfall or snow-melt, and the number of water-related property insurance claims. Weather events which cause severe damages are of general interest; decision makers want to take efficient actions against them while the insurance companies want to set adequate premiums. The modelling is challenging since the underlying dynamics vary across geographical regions due to differences in topology, construction designs and climate. We develop new methodology to improve the existing models which fail to model high numbers of claims. The statistical framework is based on both mixture and extremal mixture modelling, with the latter being based on a discretized generalized Pareto distribution. Furthermore, we propose a temporal clustering algorithm and derive new covariates which lead to a better understanding of the association between claims and weather events. The modelling of the claims, conditional on the locally observed weather events, both fit the marginal distributions well and capture the spatial dependence between locations. Our methodology is applied to three cities across Norway to demonstrate its benefits.

Article information

Source
Ann. Appl. Stat. Volume 12, Number 1 (2018), 246-282.

Dates
Received: April 2017
Revised: June 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1520564472

Digital Object Identifier
doi:10.1214/17-AOAS1081

Keywords
Extremal dependence extremal mixture insurance claims mixture modelling Poisson hurdle model spatio-temporal modelling

Citation

Rohrbeck, Christian; Eastoe, Emma F.; Frigessi, Arnoldo; Tawn, Jonathan A. Extreme value modelling of water-related insurance claims. Ann. Appl. Stat. 12 (2018), no. 1, 246--282. doi:10.1214/17-AOAS1081. https://projecteuclid.org/euclid.aoas/1520564472


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References

  • Anderson, C. W. (1970). Extreme value theory for a class of discrete distributions with applications to some stochastic processes. J. Appl. Probab. 7 99–113.
  • Anderson, C. W. (1980). Local limit theorems for the maxima of discrete random variables. Math. Proc. Cambridge Philos. Soc. 88 161–165.
  • Anderson, C. W., Coles, S. G. and Hüsler, J. (1997). Maxima of Poisson-like variables and related triangular arrays. Ann. Appl. Probab. 7 953–971.
  • Behrens, C. N., Lopes, H. F. and Gamerman, D. (2004). Bayesian analysis of extreme events with threshold estimation. Stat. Model. 4 227–244.
  • Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. J. Roy. Statist. Soc. Ser. B 36 192–236.
  • Besag, J., York, J. and Mollié, A. (1991). Bayesian image restoration, with two applications in spatial statistics. Ann. Inst. Statist. Math. 43 1–59.
  • Boldi, M.-O. and Davison, A. C. (2007). A mixture model for multivariate extremes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 69 217–229.
  • Bottolo, L., Consonni, G., Dellaportas, P. and Lijoi, A. (2003). Bayesian analysis of extreme values by mixture modeling. Extremes 6 25–47.
  • Botzen, W. J. and van den Bergh, J. C. (2008). Insurance against climate change and flooding in the Netherlands: Present, future, and comparison with other countries. Risk Anal. 28 413–426.
  • Botzen, W. J. and van den Bergh, J. C. (2012). Risk attitudes to low-probability climate change risks: WTP for flood insurance. J. Econ. Behav. Organ. 82 151–166.
  • Brockwell, A. E. (2007). Universal residuals: A multivariate transformation. Statist. Probab. Lett. 77 1473–1478.
  • Brooks, S. P. and Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Statist. 7 434–455.
  • Buddana, A. and Kozubowski, T. J. (2014). Discrete Pareto distributions. Econ. Qual. Control 29 143–156.
  • Carreau, J. and Bengio, Y. (2009). A hybrid Pareto model for asymmetric fat-tailed data: The univariate case. Extremes 12 53–76.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London.
  • Coles, S., Heffernan, J. and Tawn, J. A. (1999). Dependence measures for extreme value analyses. Extremes 2 339–365.
  • Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. Roy. Statist. Soc. Ser. B 53 377–392.
  • Coles, S. G. and Tawn, J. A. (1996). Modelling extremes of the areal rainfall process. J. Roy. Statist. Soc. Ser. B 58 329–347.
  • Davison, A. C. and Smith, R. L. (1990). Models for exceedances over high thresholds. J. Roy. Statist. Soc. Ser. B 52 393–442.
  • de Carvalho, M. and Davison, A. C. (2014). Spectral density ratio models for multivariate extremes. J. Amer. Statist. Assoc. 109 764–776.
  • Department for Environment, Food and Rural Affairs (2004). Review of UK climate change indicators. Available from http://www.ecn.ac.uk/iccuk/.
  • Eastoe, E. F. and Tawn, J. A. (2009). Modelling non-stationary extremes with application to surface level ozone. J. R. Stat. Soc. Ser. C. Appl. Stat. 58 25–45.
  • Einmahl, J. H. J., de Haan, L. and Sinha, A. K. (1997). Estimating the spectral measure of an extreme value distribution. Stochastic Process. Appl. 70 143–171.
  • Einmahl, J. H. J. and Segers, J. (2009). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Statist. 37 2953–2989.
  • Frees, E. W. and Valdez, E. A. (2008). Hierarchical insurance claims modeling. J. Amer. Statist. Assoc. 103 1457–1469.
  • Frigessi, A., Haug, O. and Rue, H. (2002). A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes 5 219–235.
  • Hanson, T. E., de Carvalho, M. and Chen, Y. (2017). Bernstein polynomial angular densities of multivariate extreme value distributions. Statist. Probab. Lett. 128 60–66.
  • Haug, O., Dimakos, X. K., Vårdal, J. F., Aldrin, M. and Meze-Hausken, E. (2011). Future building water loss projections posed by climate change. Scand. Actuar. J. 2011 1–20.
  • Hitz, A. (2017). Modelling of Extremes. Unpublished Oxford University PhD thesis.
  • Holmes, J. and Moriarty, W. (1999). Application of the generalized Pareto distribution to extreme value analysis in wind engineering. J. Wind Eng. Ind. Aerodyn. 83 1–10.
  • Jenkins, G. J., Perry, M. C. and Prior, M. J. (2008). The climate of the United Kingdom and recent trends. Met Office Hadley Centre, Exeter, UK.
  • Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012). Loss Models: From Data to Decisions, 4th ed. Wiley, Hoboken, NJ.
  • Knabb, R. D., Rhome, J. R. and Brown, D. P. (2005). Tropical Cyclone Report: Hurricane Katrina: 23–30 August 2005. Technical report, National Hurricane Center.
  • Kubilay, A., Derome, D., Blocken, B. and Carmeliet, J. (2013). CFD simulation and validation of wind-driven rain on a building facade with an Eulerian multiphase model. Build. Environ. 61 69–81.
  • Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
  • Li, Y., Cai, W. and Campbell, E. (2005). Statistical modeling of extreme rainfall in southwest Western Australia. J. Climate 18 852–863.
  • MacDonald, A., Scarrott, C. J., Lee, D., Darlow, B., Reale, M. and Russell, G. (2011). A flexible extreme value mixture model. Comput. Statist. Data Anal. 55 2137–2157.
  • Mills, E. (2005). Insurance industry in a climate of change. Science 309 1040–1044.
  • Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed. Springer, New York.
  • Pickands, J. III (1971). The two-dimensional Poisson process and extremal processes. J. Appl. Probab. 8 745–756.
  • Prieto, F., Gómez-Déniz, E. and Sarabia, J. M. (2014). Modelling road accident blackspots data with the discrete generalized Pareto distribution. Accident Anal. Prev. 71 38–49.
  • Resnick, S. I. (2013). Extreme Values, Regular Variation and Point Processes. Springer, New York.
  • Russell, B. T., Cooley, D. S., Porter, W. C., Reich, B. J. and Heald, C. L. (2016). Data mining to investigate the meteorological drivers for extreme ground level ozone events. Ann. Appl. Stat. 10 1673–1698.
  • Sanders, C. H. and Phillipson, M. C. (2003). UK adaption strategy and technical measures: The impacts of climate change on buildings. Build. Res. Inf. 31 210–221.
  • Scheel, I., Ferkingstad, E., Frigessi, A., Haug, O., Hinnerichsen, M. and Meze-Hausken, E. (2013). A Bayesian hierarchical model with spatial variable selection: The effect of weather on insurance claims. J. R. Stat. Soc. Ser. C. Appl. Stat. 62 85–100.
  • Schuster, S. S., Blong, R. J. and McAneney, K. J. (2006). Relationship between radar-derived hail kinetic energy and damage to insured buildings for severe hailstorms in Eastern Australia. Atmos. Res. 81 215–235.
  • Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464.
  • Shimura, T. (2012). Discretization of distributions in the maximum domain of attraction. Extremes 15 299–317.
  • Smith, J. (1985). Diagnostic checks of non-standard time series models. J. Forecast. 4 283–291.
  • Smith, R. L. and Goodman, D. J. (2000). Bayesian risk analysis. In Extremes and Integrated Risk Management (P. Embrechts, ed.) 235–251. Risk Books, London.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B. Stat. Methodol. 64 583–639.
  • Yip, K. C. and Yau, K. K. (2005). On modeling claim frequency data in general insurance with extra zeros. Insurance Math. Econom. 36 153–163.