Alireza AHMADABADİ, Gholamhossein GHOLAMİ, Burcu HUDAVERDİ

Bayesian estimation of bivariate Pickands dependence function

Anahtar Kelimeler:

Bivariate extreme value copula, Pickands dependence function, Bayesian method, reversible jump MCMC

Bayesian estimation of bivariate Pickands dependence function

In the present study, Bayesian method of estimating the Pickands dependence function of bivariate extreme-value copulas is proposed. Initially, cubic B-spline regression is used to model the dependence function. Then, the estimator of Pickands dependence function is obtained by the Bayesian approach. Through the estimation process, the prior and the posterior distributions of the parameter vectors are provided. The posterior sampling algorithm is presented in order to approximate the posterior distribution. We give a simulation study to measure and compare the performance of the proposed Bayesian estimator of the Pickands dependence function. A real data example is also illustrated.

Keywords:

Bivariate extreme value copula, Pickands dependence function, Bayesian method, reversible jump MCMC,

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[1] A. Ahmadabadi and B. Hudaverdi Ucer, Bivariate nonparametric estimation of the Pickands dependence function using Bernstein copula with kernel regression approach, Comput. Statist. 32 (4), 1515–1532, 2017.
[2] C.B. Barber, D.P. Dobkin and H.T. Huhdanpaa, The Quickhull algorithm for convex hulls, ACM Trans. Math. Software 22 (4), 469–483, 1996.
[3] B. Bergahus, A. Bücher and H. Dette, Minimum distance estimation of Pickands dependence function for multivariate distributions, Working Paper, 2012.
[4] S.P. Brooks, Bayesian computation: a statistical revolution, Philos. Trans. Royal Soc. A 361 (1813), 2681–2697, 2003.
[5] A. Bücher, H. Dette and S. Volgushev, New estimators of the Pickands dependence function and a test for extreme-value dependence, Ann. Statist. 39 (4), 1963–2006, 2011.
[6] P. Capéraà, A.-L.Fougeres and C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (4), 567–577, 1997.
[7] E. Cormiér, C. Genest and J.G. Neslehova, Using B-splines for nonparametric inference on bivariate extreme-value copulas, Extremes 17 (4), 633–659, 2014.
[8] P. Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 (5), 429–439, 1991.
[9] C. Genest and J.Segers, Rank-based inference for bivariate extreme-value copulas, Ann. Statist. 37 (5B), 2990–3022, 2009.
[10] G. Gholami, On the Bayesian change-point problem in regression analysis, J. Stat. Theory Appl. 9 (1), 9–27, 2010.
[11] P. Green, Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrika 82 (4), 711–732, 1995.
[12] G. Gudendorf and J. Segers, Nonparametric estimation of an extreme-value copula in arbitrary dimensions, J. Multivariate Anal. 102 (1), 37–47, 2011.
[13] S. Guillotte and F. Perron, A Bayesian estimator for the dependence function of a bivariate extremevalue distribution, Canad. J. Statist. 36 (3), 83-396, 2008.
[14] E.W. Frees and E.A. Valdez, Understanding relationships using copulas, N. Am. Actuar. J. 2 (1), 1–25, 1998.
[15] P. Hall and N. Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (5), 835–844, 2000.
[16] G. Marcon, S.A. Padoan, P. Naveau, P. Muliere and J. Segers, Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials, J. Statist. Plann. Inference 183, 1-17, 2017.
[17] J. Pickands, Multivariate extreme value distribution, in: Proceedings of the 43rd Session of the International Statistical Institute, Buenos Aires, Brazil, 859–878, 1981.
[18] J. Segers, Non-parametric inference for bivariate extreme-value copulas, in: M. Ahsanullah and S. Kirmani (ed.) Extreme Value Distributions, Nova Science Publishers, 181–203, 1985.
[19] A. Zellner, On assessing prior distributions and Bayesian regression analysis with g prior distributions, in: P. Goel and A. Zellner (ed.) Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti. Studies in Bayesian Econometrics. Elsevier, New York, 233-243, 1986.
[20] D. Zhang, M.T Wells and L. Peng, Nonparametric estimation of the dependence function for a multivariate extreme value distribution, J. Multivariate Anal. 99, 577–588, 2008.