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Bayesian inference in spherical linear models: robustness and conjugate analysis

Author

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  • Arellano-Valle, R.B.
  • del Pino, G.
  • Iglesias, P.

Abstract

The early work of Zellner on the multivariate Student-t linear model has been extended to Bayesian inference for linear models with dependent non-normal error terms, particularly through various papers by Osiewalski, Steel and coworkers. This article provides a full Bayesian analysis for a spherical linear model. The density generator of the spherical distribution is here allowed to depend both on the precision parameter [phi] and on the regression coefficients [beta]. Another distinctive aspect of this paper is that proper priors for the precision parameter are discussed. The normal-chi-squared family of prior distributions is extended to a new class, which allows the posterior analysis to be carried out analytically. On the other hand, a direct joint modelling of the data vector and of the parameters leads to conjugate distributions for the regression and the precision parameters, both individually and jointly. It is shown that some model specifications lead to Bayes estimators that do not depend on the choice of the density generator, in agreement with previous results obtained in the literature under different assumptions. Finally, the distribution theory developed to tackle the main problem is useful on its own right.

Suggested Citation

  • Arellano-Valle, R.B. & del Pino, G. & Iglesias, P., 2006. "Bayesian inference in spherical linear models: robustness and conjugate analysis," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 179-197, January.
  • Handle: RePEc:eee:jmvana:v:97:y:2006:i:1:p:179-197
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    References listed on IDEAS

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    1. Arellano-Valle, Reinaldo B., 2001. "On some characterizations of spherical distributions," Statistics & Probability Letters, Elsevier, vol. 54(3), pages 227-232, October.
    2. Arellano-Valle, Reinaldo B. & Bolfarine, Heleno, 1995. "On some characterizations of the t-distribution," Statistics & Probability Letters, Elsevier, vol. 25(1), pages 79-85, October.
    3. Osiewalski, Jacek & Steel, Mark F. J., 1993. "Robust bayesian inference in elliptical regression models," Journal of Econometrics, Elsevier, vol. 57(1-3), pages 345-363.
    4. Osiewalski, Jacek & Steel, Mark F. J., 1993. "Bayesian marginal equivalence of elliptical regression models," Journal of Econometrics, Elsevier, vol. 59(3), pages 391-403, October.
    5. Ng, Vee Ming, 2002. "Robust Bayesian Inference for Seemingly Unrelated Regressions with Elliptical Errors," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 409-414, November.
    6. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
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    Cited by:

    1. Vidal, Ignacio & Arellano-Valle, Reinaldo B., 2010. "Bayesian inference for dependent elliptical measurement error models," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2587-2597, November.
    2. Casanova, María P. & Iglesias, Pilar & Bolfarine, Heleno & Salinas, Victor H. & Peña, Alexis, 2010. "Semiparametric Bayesian measurement error modeling," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 512-524, March.
    3. Hosseini, Reshad & Sra, Suvrit & Theis, Lucas & Bethge, Matthias, 2016. "Inference and mixture modeling with the Elliptical Gamma Distribution," Computational Statistics & Data Analysis, Elsevier, vol. 101(C), pages 29-43.

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