Bayesian Analysis

Approximate Bayesian Inference in Semiparametric Copula Models

Clara Grazian and Brunero Liseo

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We describe a simple method for making inference on a functional of a multivariate distribution, based on its copula representation. We make use of an approximate Bayesian Monte Carlo algorithm, where the proposed values of the functional of interest are weighted in terms of their Bayesian exponentially tilted empirical likelihood. This method is particularly useful when the “true” likelihood function associated with the working model is too costly to evaluate or when the working model is only partially specified.

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Bayesian Anal. Volume 12, Number 4 (2017), 991-1016.

First available in Project Euclid: 8 November 2017

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multivariate dependence Bayesian exponentially tilted empirical likelihood Spearman’s ρ tail dependence coefficients partially specified models

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Grazian, Clara; Liseo, Brunero. Approximate Bayesian Inference in Semiparametric Copula Models. Bayesian Anal. 12 (2017), no. 4, 991--1016. doi:10.1214/17-BA1080.

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  • Allingham, D., King, R. A., and Mengersen, K. L. (2009). “Bayesian estimation of quantile distributions.”Statistics and Computing, 19(2): 189–201.
  • Ané, T. and Kharoubi, C. (2003). “Dependence structure and risk measure.”The Journal of Business, 76(3): 411–438.
  • Ardia, D. and Hoogerheide, L. (2010). “Bayesian Estimation of the GARCH(1,1) Model with Student-t Innovations.”The R Journal, 2(12): 41–47.
  • Borkowf, C. B. (2002). “Computing the nonnull asymptotic variance and the asymptotic relative efficiency of Spearman’s rank correlation.”Computational Statistics & Data Analysis, 39(3): 271–286. URL
  • Burda, M. and Prokhorov, A. (2014). “Copula based factorization in Bayesian multivariate infinite mixture models.”Journal of Multivariate Analysis, 127: 200–213.
  • Cherubini, U., Luciano, E., and Vecchiato, W. (2004).Copula methods in finance. Wiley Finance Series. John Wiley & Sons, Ltd., Chichester. URL
  • Choroś, B., Ibragimov, R., and Permiakova, E. (2010). “Copula estimation.” InCopula theory and its applications, 77–91. Springer.
  • Craiu, V. R. and Sabeti, A. (2012). “In mixed company: Bayesian inference for bivariate conditional copula models with discrete and continuous outcomes.”Journal of Multivariate Analysis, 110: 106–120. URL
  • De Luca, G. and Rivieccio, G. (2012). “Multivariate tail dependence coefficients for archimedean copulae.” InAdvanced Statistical Methods for the Analysis of Large Data-Sets, 287–296. Springer.
  • Di Bernardino, E. and Rullière, D. (2016). “On tail dependence coefficients of transformed multivariate Archimedean copulas.”Fuzzy Sets and Systems, 284: 89–112. URL
  • Drovandi, C. C. and Pettitt, A. N. (2011). “Likelihood-free Bayesian estimation of multivariate quantile distributions.”Computational Statistics & Data Analysis, 55(9): 2541–2556.
  • Fermanian, J.-D., Radulović, D., and Wegkamp, M. (2004). “Weak convergence of empirical copula processes.”Bernoulli, 10(5): 847–860. URL
  • Frahm, G., Junker, M., and Schmidt, R. (2005). “Estimating the tail-dependence coefficient: properties and pitfalls.”Insurance. Mathematics & Economics, 37(1): 80–100. URL
  • Genest, C. and Favre, A. (2007). “Everything you always wanted to know about copula modeling but were afraid to ask.”Journal of Hydrologic Engineering, 347–368.
  • Genest, C., Ghoudi, K., and Rivest, L.-P. (1995). “A semiparametric estimation procedure of dependence parameters in multivariate families of distributions.”Biometrika, 82(3): 543–552. URL
  • Geweke, J. (1993). “Bayesian Treatment of the Independent Student- t Linear Model.”Journal of Applied Econometrics, 8(S): S19–40.
  • Grazian, C. and Liseo, B. (2017). “Approximate Bayesian Inference in Semiparametric Copula Models – Supplementary Material”Bayesian Analysis.
  • Gruber, L. and Czado, C. (2015). “Sequential Bayesian model selection of regular vine copulas.”Bayesian Analysis, 10(4): 937–963.
  • Hjort, N. L., Holmes, C., Mueller, P., and Walker, S. G. (eds.) (2010).Bayesian Nonparametrics:. Cambridge. URL
  • Huynh, V., Kreinovich, V., and Sriboonchitta, S. (2014). “Modeling dependence in econometrics.”Advances in Intelligent Systems and Computing, 251.
  • Joe, H. (1990). “Multivariate concordance.”Journal of Multivariate Analysis, 35(1): 12–30. URL
  • Joe, H. (2005). “Asymptotic efficiency of the two-stage estimation method for copula-based models.”Journal of Multivariate Analysis, 94(2): 401–419.
  • Joe, H. (2015).Dependence modeling with copulas, volume 134 ofMonographs on Statistics and Applied Probability. CRC Press, Boca Raton, FL.
  • Joe, H., Smith, R. L., and Weissman, I. (1992). “Bivariate threshold methods for extremes.”Journal of the Royal Statistical Society. Series B. Methodological, 54(1): 171–183. URL<171:BTMFE>2.0.CO;2-B&origin=MSN
  • Kalli, M., Griffin, J. E., and Walker, S. G. (2011). “Slice sampling mixture models.”Statistics and Computing, 21(1): 93–105.
  • Kauermann, G., Schellhase, C., and Ruppert, D. (2013). “Flexible Copula Density Estimation with Penalized Hierarchical B-splines.”Scandinavian Journal of Statistics, 40(4): 685–705.
  • Lancaster, T. and Jae Jun, S. (2010). “Bayesian quantile regression methods.”Journal of Applied Econometrics, 25(2): 287–307.
  • McAleer, M., Caporin, M., et al. (2011). “Ranking multivariate GARCH models by problem dimension: An empirical evaluation.” Technical report, Kyoto University, Institute of Economic Research.
  • Mengersen, K. L., Pudlo, P., and Robert, C. P. (2013). “Bayesian computation via empirical likelihood.”Proceedings of the National Academy of Sciences, 110(4): 1321–1326.
  • Min, A. and Czado, C. (2010). “Bayesian inference for multivariate copulas using pair-copula constructions.”Journal of Financial Econometrics, 8(4): 511–546.
  • Nieto-Barajas, L. E. and Contreras-Cristán, A. (2014). “A Bayesian nonparametric approach for time series clustering.”Bayesian Analysis, 9(1): 147–170.
  • Oh, D. H. and Patton, A. J. (2013). “Simulated method of moments estimation for copula-based multivariate models.”Journal of the American Statistical Association, 108(502): 689–700.
  • Owen, A. B. (2001).Empirical likelihood. CRC press.
  • Pitt, M., Chan, D., and Kohn, R. (2006). “Efficient Bayesian inference for Gaussian copula regression models.”Biometrika, 93(3): 537–554.
  • Rayner, G. D. and MacGillivray, H. L. (2002). “Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions.”Statistics and Computing, 12(1): 57–75.
  • Rubin, D. B. et al. (1988). “Using the SIR algorithm to simulate posterior distributions.”Bayesian Statistics, 3(1): 395–402.
  • Salazar, Y. and Ng, W. L. (2015). “Nonparametric estimation of general multivariate tail dependence and applications to financial time series.”Statistical Methods & Applications, 24(1): 121–158.
  • Schefzik, R., Thorarinsdottir, T., and Gneiting, T. (2013). “Uncertainty quantification in complex simulation models using ensemble copula coupling.”Statistical Science, 28(4): 616–640. URL
  • Schennach, S. M. (2005). “Bayesian exponentially tilted empirical likelihood.”Biometrika, 92(1): 31–46. URL
  • Schmid, F. and Schmidt, R. (2006). “Bootstrapping Spearman’s multivariate rho.” InCOMPSTAT, Proceedings in Computational Statistics, 759–766. Springer.
  • Schmid, F. and Schmidt, R. (2007). “Multivariate extensions of Spearman’s rho and related statistics.”Statistics & Probability Letters, 77(4): 407–416.
  • Schmidt, R. and Stadtmüller, U. (2006). “Non-parametric Estimation of Tail Dependence.”Scandinavian Journal of Statistics, 33(2): 307–335.
  • Schölzel, C. and Friederichs, P. (2008). “Multivariate non-normally distributed random variables in climate research – Introduction to the copula approach.”Nonlinear Processes in Geophysics, 15(5): 761–772. URL
  • Sibuya, M. (1959). “Bivariate extreme statistics, I.”Annals of the Institute of Statistical Mathematics, 11(2): 195–210.
  • Sklar, M. (2010). “Fonctions de répartition a $n$ dimensions et leurs marges.”Annales de L’ISUP, 54(1–2): 3–6. With an introduction by Denis Bosq.
  • Smith, M. and Khaled, M. (2012). “Estimation of copula models with discrete margins via Bayesian data augmentation.”Journal of the American Statistical Association, 107(497): 290–303.
  • Smith, M. S. (2013). “Bayesian approaches to copula modelling.” InBayesian theory and applications, 336–358. Oxford Univ. Press, Oxford. URL
  • Wu, J., Wang, X., and Walker, S. G. (2014). “Bayesian Nonparametric Inference for a Multivariate Copula Function.”Methodology and Computing in Applied Probability, 16(3): 747–763.
  • Yang, Y. and He, X. (2012). “Bayesian empirical likelihood for quantile regression.”The Annals of Statistics, 40(2): 1102–1131.
  • Zhu, W., Marin, J. M., and Leisen, F. (2016). “A bootstrap likelihood approach to Bayesian computation.”Australian & New Zealand Journal of Statistics, 58(2): 227–244. URL

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