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Likelihood-free Bayesian estimation of multivariate quantile distributions


  • Drovandi, Christopher C.
  • Pettitt, Anthony N.


In this paper, we present new multivariate quantile distributions and utilise likelihood-free Bayesian algorithms for inferring the parameters. In particular, we apply a sequential Monte Carlo (SMC) algorithm that is adaptive in nature and requires very little tuning compared with other approximate Bayesian computation algorithms. Furthermore, we present a framework for the development of multivariate quantile distributions based on a copula. We consider bivariate and time series extensions of the g-and-k distribution under this framework, and develop an efficient component-wise updating scheme free of likelihood functions to be used within the SMC algorithm. In addition, we trial the set of octiles as summary statistics as well as functions of these that form robust measures of location, scale, skewness and kurtosis. We show that these modifications lead to reasonably precise inferences that are more closely comparable to computationally intensive likelihood-based inference. We apply the quantile distributions and algorithms to simulated data and an example involving daily exchange rate returns.

Suggested Citation

  • Drovandi, Christopher C. & Pettitt, Anthony N., 2011. "Likelihood-free Bayesian estimation of multivariate quantile distributions," Computational Statistics & Data Analysis, Elsevier, vol. 55(9), pages 2541-2556, September.
  • Handle: RePEc:eee:csdana:v:55:y:2011:i:9:p:2541-2556

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    References listed on IDEAS

    1. C. C. Drovandi & A. N. Pettitt, 2011. "Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation," Biometrics, The International Biometric Society, vol. 67(1), pages 225-233, March.
    2. Pierre Del Moral & Arnaud Doucet & Ajay Jasra, 2006. "Sequential Monte Carlo samplers," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 411-436.
    3. Christopher C. Drovandi & Anthony N. Pettitt & Malcolm J. Faddy, 2011. "Approximate Bayesian computation using indirect inference," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 60(3), pages 317-337, May.
    4. J. Møller & A. N. Pettitt & R. Reeves & K. K. Berthelsen, 2006. "An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants," Biometrika, Biometrika Trust, vol. 93(2), pages 451-458, June.
    5. Nicolas Chopin, 2002. "A sequential particle filter method for static models," Biometrika, Biometrika Trust, vol. 89(3), pages 539-552, August.
    6. Knut Heggland & Arnoldo Frigessi, 2004. "Estimating functions in indirect inference," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(2), pages 447-462.
    7. Mark A. Beaumont & Jean-Marie Cornuet & Jean-Michel Marin & Christian P. Robert, 2009. "Adaptive approximate Bayesian computation," Biometrika, Biometrika Trust, vol. 96(4), pages 983-990.
    8. Badrinath, S G & Chatterjee, Sangit, 1988. "On Measuring Skewness and Elongation in Common Stock Return Distributions: The Case of the Market Index," The Journal of Business, University of Chicago Press, vol. 61(4), pages 451-472, October.
    9. Matthias Fischer, 2010. "Generalized Tukey-type distributions with application to financial and teletraffic data," Statistical Papers, Springer, vol. 51(1), pages 41-56, January.
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    Cited by:

    1. Li, J. & Nott, D.J. & Fan, Y. & Sisson, S.A., 2017. "Extending approximate Bayesian computation methods to high dimensions via a Gaussian copula model," Computational Statistics & Data Analysis, Elsevier, vol. 106(C), pages 77-89.
    2. Kobayashi, Genya, 2014. "A transdimensional approximate Bayesian computation using the pseudo-marginal approach for model choice," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 167-183.
    3. Oh, Man-Suk & Park, Eun Sug & So, Beong-Soo, 2016. "Bayesian variable selection in binary quantile regression," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 177-181.
    4. Ajay Jasra, 2015. "Approximate Bayesian Computation for a Class of Time Series Models," International Statistical Review, International Statistical Institute, vol. 83(3), pages 405-435, December.
    5. Ji, Yonggang & Lin, Nan & Zhang, Baoxue, 2012. "Model selection in binary and tobit quantile regression using the Gibbs sampler," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 827-839.
    6. Menéndez, P. & Fan, Y. & Garthwaite, P.H. & Sisson, S.A., 2014. "Simultaneous adjustment of bias and coverage probabilities for confidence intervals," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 35-44.


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