Non-parametric Bayesian inference on bivariate extremes

Journal of the Royal Statistical Society. Series B: Statistical Methodology

Volumn 73, Issue 3, 2011, Pages 377-406

  Guillotte, Simon ^a   Perron, François ^b   Segers, Johan ^c  

^a UNIVERSITY OF PRINCE EDWARD ISLAND   (Canada)

^b UNIVERSITÉ DE MONTRÉAL   (Canada)

^c UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

irreducibility; Bayes; Bivariate extreme value distribution; Extreme conditional quantiles; Markov chain Monte Carlo methods; Metropolis within Gibbs sampling; Prediction; Rare event probabilities; Reversible jumps; Spectral measure

Indexed keywords

EID: 79955034763     PISSN: 13697412     EISSN: 14679868     Source Type: Journal    
DOI: 10.1111/j.1467-9868.2010.00770.x     Document Type: Article

References (46)

1. Aitchison, J. and Dunsmore, I. R. (1975) Statistical Prediction Analysis. Cambridge: Cambridge University Press.
2. Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J.(2004) Statistics of Extremes: Theory and Applications. Chichester: Wiley.
3. Boldi, M.-O. and Davison, A. C. (2007) A mixture model for multivariate extremes. J. R. Statist. Soc. B, 69, 217-229.
4. Chib, S. and Greenberg, E. (1995) Understanding the Metropolis-Hastings algorithm. Am. Statistn, 49, 327-335.
5. Coles, S. G. (2001) An Introduction to Statistical Modeling of Extreme Values. London: Springer.
6. Coles, S. G. and Tawn, J. A. (1991) Modelling extreme multivariate events. J. R. Statist. Soc. B, 53, 377-392.
7. Coles, S. G. and Tawn, J. A. (1994) Statistical methods for multivariate extremes: an application to structural design (with discussion). Appl. Statist., 43 1-48.
8. Coles, S. G. and Tawn, J. A. (1996) A Bayesian analysis of extreme rainfall data. Appl. Statist., 45, 463-478.
9. Coles, S. G. and Tawn, J. A. (2005) Bayesian modelling of extreme surges on the UK east coast. Phil. Trans. R. Soc. Lond. A, 363, 1387-1406.
10. de Haan, L. and Ferreira, A. (2006) Extreme Value Theory. New York: Springer.
11. de Haan, L., Neves, C. and Peng, L. (2008) Parametric tail copula estimation and model testing. J. Multiv. Anal., 99, 1260-1275.
12. de Haan, L. and Resnick, S. (1977) Limit theory for multidimensional sample extremes. Z. Wahrsch. Ver. Geb., 40, 317-337.
13. de Haan, L. and de Ronde, J. (1998) Sea and wind: multivariate extremes at work. Extremes, 1, 7-45.
14. de Haan, L. and Sinha, A. K. (1999) Estimating the probability of a rare event. Ann. Statist., 27, 732-759.
15. Einmahl, J. H. J., de Haan, L. and Li, D. (2006) Weighted approximations to tail copula processes with application to testing the bivariate extreme value condition. Ann. Statist., 34, 1987-2014.
16. Einmahl, J., de Haan, L. and Piterbarg, V. (2001) Nonparametric estimation of the spectral measure of an extreme value distribution. Ann. Statist., 29, 1401-1423.
17. Einmahl, J. H. J., Krajina, A. and Segers, J. (2008) A method of moments estimator of tail dependence. Bernoulli, 14, 1003-1026.
18. 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.
19. Fisher, R. (1929) Tests of significance in harmonic analysis. Proc. R. Soc. Lond. A, 125, 54-59.
20. Fremlin, D. H. (2003) Measure Theory, vol. 2, Broad Foundations. Colchester: Torres Fremlin.
21. Fritsch, F. N. and Butland, J. (1984) A method for constructing local monotone piecewise cubic interpolants. SIAM J. Sci. Statist. Comput., 5, 300-304.
22. Fritsch, F. N. and Carlson, R. E. (1980) Monotone piecewise cubic interpolation. SIAM J. Numer. Anal., 17, 238-246.
23. Gamerman, D. and Lopes, H. F. (2006) Stochastic simulation for Bayesian inference. In Markov Chain Monte Carlo in Practice, 2nd edn (eds W. R. Gilks, S. Richardson and D. J. Spiegelhalter ). Boca Raton: Chapman and Hall-CRC Press.
24. Green, P. J. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711-732.
25. Guillotte, S. and Perron, F. (2008) A Bayesian estimator for the dependence function of a bivariate extreme-value distribution. Can. J. Statist., 36, 383-396.
26. Hobert, J. P., Robert, C. P. and Goutis, C. (1997) Connectedness conditions for the convergence of the Gibbs sampler. Statist. Probab. Lett., 33, 235-240.
27. Joe, H., Smith, R. L. and Weissman, I. (1992) Bivariate threshold methods for extremes. J. R. Statist. Soc. B, 54, 171-183.
28. Kotz, S. and Nadarajah, S. (2000) Theory and applications. In Extreme Value Distributions. London: Imperial College Press.
29. Ledford, A. W. and Tawn, J. A. (1996) Statistics for near independence in multivariate extreme values. Biometrika, 83, 169-187.
30. Ledford, A. W. and Tawn, J. A. (1997) Modelling dependence within joint tail regions. J. R. Statist. Soc. B, 59, 475-499.
31. Marin J.-M., Mengersen, K. L. and Robert, C.(2005) Bayesian modelling and inference on mixtures of distributions. In Handbook of Statistics (eds D. Dey and C. Rao ), vol. 25. Amsterdam: Elsevier.
32. McNeil, A. (1997) Estimating the tails of loss severity distributions using extreme value theory. Astin Bull., 27, 117-137.
33. Meyn, S. and Tweedie, R. L. (2009) Markov Chains and Stochastic Stability, 2nd edn. Cambridge: Cambridge University Press.
34. Nadarajah, S., Anderson, C. W. and Tawn, J. A. (1998) Ordered multivariate extremes. J. R. Statist. Soc. B, 60, 473-496.
35. Ramos, A. and Ledford, A. (2009) A new class of models for bivariate joint tails. J. R. Statist. Soc. B, 71, 219-241.
36. Resnick, S. I. (1997) Discussion of the danish data on large fire insurance losses. Astin Bull., 27, 139-151.
37. Resnick, S. (2003) Hidden regular variation, second order regular variation and asymptotic independence. Extremes, 5, 303-336.
38. Robert, C. P. (2007) The Bayesian Choice. New York: Springer.
39. Robert, C. P. and Casella, G. (2004) Monte Carlo Statistical Methods, 2nd edn. New York: Springer.
40. Roberts, G. O. and Rosenthal, J. S. (2006) Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains. Ann. Appl. Probab., 16, 2123-2139.
41. Roberts, G. O. and Smith, A. F. M. (1994) Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms. Stoch. Process. Applic., 49, 207-216.
42. Schmidt, R. and Stadtmüller, U. (2006) Non-parametric estimation of tail dependence. Scand. J. Statist., 33, 307-335.
43. Smith, R. L. (1994) Multivariate threshold methods. In Extreme Value Theory and Applications (eds J. Galambos, J. Lechner and E. Simiu ), pp. 225-248. Dordrecht: Kluwer.
44. Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. Biometrika, 75, 397-415.
45. Tierney, L. (1994) Markov chains for exploring posterior distributions (with discussion). Ann. Statist., 22, 1701-1762.
46. Tierney, L. (1996) Introduction to general state-space Markov chain theory. In Markov Chain Monte Carlo in Practice (eds W. R. Gilks, S. Richardson and D. J. Spiegelhalter ), pp. 59-74. London: Chapman and Hall.