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Bayesian Regression Analysis With Scale Mixtures Of Normals

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  • Fernández, Carmen
  • Steel, Mark F.J.

Abstract

This paper considers a Bayesian analysis of the linear regression model under independent sampling from general scale mixtures of normals. Using a common reference prior, we investigate the validity of Bayesian inference and the existence of posterior moments of the regression and scale parameters. We find that whereas existence of the posterior distribution does not depend on the choice of the design matrix or the mixing distribution, both of them can crucially intervene in the existence of posterior moments. We identify some useful characteristics that allow for an easy verification of the existence of a wide range of moments. In addition, we provide full characterizations under sampling from finite mixtures of normals, Pearson VII, or certain modulated normal distributions. For empirical applications, a numerical implementation based on the Gibbs sampler is recommended.

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  • Fernández, Carmen & Steel, Mark F.J., 2000. "Bayesian Regression Analysis With Scale Mixtures Of Normals," Econometric Theory, Cambridge University Press, vol. 16(1), pages 80-101, February.
    • Handle: RePEc:cup:etheor:v:16:y:2000:i:01:p:80-101_16
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    1. Jung, Yeun Ji & Hobert, James P., 2014. "Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 92-100.
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    3. Saverio Ranciati & Giuliano Galimberti & Gabriele Soffritti, 2019. "Bayesian variable selection in linear regression models with non-normal errors," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(2), pages 323-358, June.
    4. Gernot Doppelhofer & Melvyn Weeks, 2011. "Robust Growth Determinants," CESifo Working Paper Series 3354, CESifo.
    5. Li, Haoxiang & Qin, Qian & Jones, Galin L., 2024. "Convergence analysis of data augmentation algorithms for Bayesian robust multivariate linear regression with incomplete data," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    6. De la Cruz, Rolando, 2008. "Bayesian non-linear regression models with skew-elliptical errors: Applications to the classification of longitudinal profiles," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 436-449, December.
    7. Ley, Eduardo & Steel, Mark F.J., 2012. "Mixtures of g-priors for Bayesian model averaging with economic applications," Journal of Econometrics, Elsevier, vol. 171(2), pages 251-266.
    8. Juarez, Miguel A. & Steel, Mark F. J., 2006. "Model-based Clustering of non-Gaussian Panel Data," MPRA Paper 880, University Library of Munich, Germany.
    9. Rubio, Francisco Javier & Steel, Mark F. J., 2014. "Bayesian modelling of skewness and kurtosis with two-piece scale and shape transformations," MPRA Paper 57102, University Library of Munich, Germany.
    10. Yongho Ko & Seungwoo Han, 2017. "A Duration Prediction Using a Material-Based Progress Management Methodology for Construction Operation Plans," Sustainability, MDPI, vol. 9(4), pages 1-12, April.
    11. Salas-Gonzalez, Diego & Kuruoglu, Ercan E. & Ruiz, Diego P., 2009. "A heavy-tailed empirical Bayes method for replicated microarray data," Computational Statistics & Data Analysis, Elsevier, vol. 53(5), pages 1535-1546, March.
    12. Miguel A. Juárez & Mark F. J. Steel, 2010. "Non‐gaussian dynamic bayesian modelling for panel data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 25(7), pages 1128-1154, November/.
    13. Jose T.A.S. Ferreira & Mark F.J. Steel, 2004. "Bayesian Multivariate Regression Analysis with a New Class of Skewed Distributions," Econometrics 0403001, University Library of Munich, Germany.
    14. Daniel R. Kowal & David S. Matteson & David Ruppert, 2019. "Functional Autoregression for Sparsely Sampled Data," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(1), pages 97-109, January.
    15. Laura Liu, 2017. "Density Forecasts in Panel Models: A semiparametric Bayesian Perspective," PIER Working Paper Archive 17-006, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 28 Apr 2017.
    16. Guodong Shan & Yiheng Hou & Baisen Liu, 2020. "Bayesian robust estimation of partially functional linear regression models using heavy-tailed distributions," Computational Statistics, Springer, vol. 35(4), pages 2077-2092, December.
    17. Heinen, Andréas & Valdesogo, Alfonso, 2020. "Spearman rank correlation of the bivariate Student t and scale mixtures of normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    18. Abanto-Valle, C.A. & Bandyopadhyay, D. & Lachos, V.H. & Enriquez, I., 2010. "Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 2883-2898, December.
    19. Fengkai Yang & Haijing Yuan, 2017. "A Non-iterative Bayesian Sampling Algorithm for Linear Regression Models with Scale Mixtures of Normal Distributions," Computational Economics, Springer;Society for Computational Economics, vol. 49(4), pages 579-597, April.
    20. Vilca, Filidor & Balakrishnan, N. & Zeller, Camila Borelli, 2014. "A robust extension of the bivariate Birnbaum–Saunders distribution and associated inference," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 418-435.
    21. Jorge A. Barahona & Yolanda M. Gómez & Emilio Gómez-Déniz & Osvaldo Venegas & Héctor W. Gómez, 2024. "Scale Mixture of Exponential Distribution with an Application," Mathematics, MDPI, vol. 12(1), pages 1-17, January.
    22. Rubio, Francisco Javier & Liseo, Brunero, 2014. "On the independence Jeffreys prior for skew-symmetric models," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 91-97.

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