## Bayesian Analysis

### Bayesian inference for shape mixtures of skewed distributions, with application to regression analysis

#### Abstract

We introduce a class of shape mixtures of skewed distributions and study some of its main properties. We discuss a Bayesian interpretation and some invariance results of the proposed class. We develop a Bayesian analysis of the skew-normal, skew-generalized-normal, skew-normal-t and skew-t-normal linear regression models under some special prior specifications for the model parameters. In particular, we show that the full posterior of the skew-normal regression model parameters is proper under an arbitrary proper prior for the shape parameter and noninformative prior for the other parameters. We implement a convenient hierarchical representation in order to obtain the corresponding posterior analysis. We illustrate our approach with an application to a real dataset on characteristics of Australian male athletes.

#### Article information

Source
Bayesian Anal. Volume 3, Number 3 (2008), 513-539.

Dates
First available in Project Euclid: 22 June 2012

https://projecteuclid.org/euclid.ba/1340370436

Digital Object Identifier
doi:10.1214/08-BA320

Mathematical Reviews number (MathSciNet)
MR2434401

Zentralblatt MATH identifier
1330.62242

#### Citation

Arellano-Valle, Reinaldo B.; Castro, Luis M.; Genton, Marc G.; Gómez, Héctor W. Bayesian inference for shape mixtures of skewed distributions, with application to regression analysis. Bayesian Anal. 3 (2008), no. 3, 513--539. doi:10.1214/08-BA320. https://projecteuclid.org/euclid.ba/1340370436

#### References

• Arellano-Valle, R. B. and Azzalini, A. (2006). "On the unification of families of skew-normal distributions." Scandinavian Journal of Statistics, 33: 561–574.
• Arellano-Valle, R. B., Bolfarine, H., and Lachos, V. H. (2007). "Bayesian inference for skew-normal linear mixed models." Journal of Applied Statistics, 34: 663–682.
• Arellano-Valle, R. B., Branco, M. D., and Genton, M. G. (2006). "A unified view on skewed distributions arising from selections." The Canadian Journal of Statistics, 34: 581–601.
• Arellano-Valle, R. B., del Pino, G., and Iglesias, P. (2006). "Bayesian inference in spherical linear models: robustness and conjugate analysis." Journal of Multivariate Analysis, 97: 179–197.
• Arellano-Valle, R. B., del Pino, G., and San Martín, E. (2002). "Definition and probabilistic properties of skew-distributions." Statistics and Probability Letters, 58: 111–121.
• Arellano-Valle, R. B. and Genton, M. G. (2005). "On fundamental skewed distributions." Journal of Multivariate Analysis, 96: 93–116.
• Arellano-Valle, R. B., Gómez, H. W., and Quintana, F. A. (2004). "A new class of skew-normal distributions." Communications in Statistics, Theory and Methods, 33: 1465–1480.
• Azzalini, A. (1985). "A class of distributions which includes the normal ones." Scandinavian Journal of Statistics, 12: 171–178.
• –- (2005). "The skew-normal distribution and related multivariate families. With discussion by Marc G. Genton and a rejoinder by the author." Scandinavian Journal of Statistics, 32: 159–200.
• Azzalini, A. and Capitanio, A. (2003). "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew $t$ distribution." Journal of the Royal Statistical Society Series B, 65: 367–389.
• Azzalini, A. and Dalla Valle, A. (1996). "The multivariate skew-normal distribution." Biometrika, 83: 715–726.
• Azzalini, A. and Genton, M. G. (2008). "Robust likelihood methods based on the skew-$t$ and related distributions." International Statistical Review, 76: 106–129.
• Bernardo, J. and Smith, A. (2000). Bayesian Theory. New York: Wiley, second edition.
• Branscum, A. and Hanson, T. (2008). "Bayesian nonparametric meta-analysis using Polya tree mixture models." Biometrics, in press.
• Chen, M.-H., Shao, Q.-M., and Ibrahim, J. G. (2000). Monte Carlo Methods in Bayesian Computation. New York: Springer-Verlag.
• Cook, R. D. and Weisberg, S. (1994). An Introduction to Regression Graphics. New York: Wiley.
• Ellison, B. E. (1964). "Two theorems for inferences about the normal distribution with applications in acceptance sampling." Journal of the American Statistical Association, 59: 89–95.
• Geisser, S. and Eddy, W. (1979). "A predictive approach to model selection." Journal of the American Statistical Association, 74: 153–160.
• Gelfand, A. E. and Dey, D. K. (1994). "Bayesian model choice: asymptotics and exact calculations." Journal of the Royal Statistical Society Series B, 56: 501–514.
• Genton, M. G. (2004). Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality. Boca Raton, Florida: Chapman & Hall / CRC. Edited Volume.
• Genton, M. G. and Loperfido, N. (2005). "Generalized skew-elliptical distributions and their quadratic forms." Annals of the Institute of Statistical Mathematics, 57: 389–401.
• Ghosh, P., Branco, M. D., and Chakraborty, H. (2007). "Bivariate random effect model using skew normal distribution with application to HIV-RNA." Statistics in Medicine, 26: 1255–1267.
• Ghosh, P. and Gönen, M. (2008). "Bayesian modeling of multivariate average bioequivalence." Statistics in Medicine, 27: 2402–2419.
• Gómez, H. W., Venegas, O., and Bolfarine, H. (2007). "Skew-symmetric distributions generated by the distribution function of the normal distribution." Environmetrics, 18: 395–407.
• Ma, Y. and Genton, M. G. (2004). "A flexible class of skew-symmetric distributions." Scandinavian Journal of Statistics, 31: 459–468.
• Ma, Y., Genton, M. G., and Davidian, M. (2004). "Linear mixed effects models with flexible generalized skew-elliptical random effects." In Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality, 339–358. Boca Raton, Florida: Genton, M. G., Ed., Chapman & Hall / CRC.
• Nadarajah, S. and Kotz, S. (2003). "Skewed distributions generated by the normal kernel." Statistics and Probability Letters, 65: 269–277.
• Sahu, S. K., Dey, D. K., and Branco, M. D. (2003). "A new class of multivariate skew distributions with application to Bayesian regression models." The Canadian Journal of Statistics, 31: 129–150.
• Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and van der Linde, A. (2002). "Bayesian measures of model complexity and fit (with discussion)." Journal of the Royal Statistical Society Series B, 64: 583–640.
• Wang, J., Boyer, J., and Genton, M. G. (2004). "A skew-symmetric representation of multivariate distributions." Statistica Sinica, 14: 1259–1270.