Bayesian Analysis

Objective Prior for the Number of Degrees of Freedom of a t Distribution

Abstract

In this paper, we construct an objective prior for the degrees of freedom of a $t$ distribution, when the parameter is taken to be discrete. This parameter is typically problematic to estimate and a problem in objective Bayesian inference since improper priors lead to improper posteriors, whilst proper priors may dominate the data likelihood. We find an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. Truncating the prior on the degrees of freedom is necessary, as the $t$ distribution, above a certain number of degrees of freedom, becomes the normal distribution. The defined prior is tested in simulation scenarios, including linear regression with $t$-distributed errors, and on real data: the daily returns of the closing Dow Jones index over a period of 98 days.

Article information

Source
Bayesian Anal., Volume 9, Number 1 (2014), 197-220.

Dates
First available in Project Euclid: 24 February 2014

https://projecteuclid.org/euclid.ba/1393251776

Digital Object Identifier
doi:10.1214/13-BA854

Mathematical Reviews number (MathSciNet)
MR3188305

Zentralblatt MATH identifier
1327.62168

Citation

Villa, Cristiano; Walker, Stephen G. Objective Prior for the Number of Degrees of Freedom of a t Distribution. Bayesian Anal. 9 (2014), no. 1, 197--220. doi:10.1214/13-BA854. https://projecteuclid.org/euclid.ba/1393251776

References

• Anscombe, F. J. (1967). “Topics in the investigation of linear relations fitted by the method of least squares.” Journal of the Royal Statistical Society, Series B, 29(1): 1–52.
• Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag.
• Berger, J. O., Bernardo, J. M., and Sun, D. (2012). “Objective priors for discrete parameter spaces.” Journal of the American Statistical Association, 107(498): 636–648.
• Berk, R. H. (1966). “Limiting behaviour of posterior distributions when the model is incorrect.” Annals of Mathematical Statistics, 37: 51–58.
• Blyth, S. (1994). “Local divergence and association.” Biometrika, 81(3): 579–584.
• Brown, P. J. and Walker, S. G. (2012). “Bayesian priors from loss matching.” International Statistical Review, 80(1): 60–82.
• Chu, J. T. (1956). “Errors in normal approximations to the $y$, $\tau$, and similar types of distribution.” Annals of Mathematical Statistics, 27: 780–789.
• Dmochowski, J. (1996). “Intrinsic priors via Kullback–Leibler geometry.” In Bayesian Statistics 5, 543–549. London: Oxford University Press.
• Fabozzi, F. J., Focardi, S. M., Höchstötter, M., and Rachev, S. T. (2010). Probability and Statistics for Finance. Hoboken, New Jersey: John Wiley & Sons, Inc.
• Fernandez, C. and Steel, M. F. (1999). “Multivariate Student-$t$ regression models: pitfalls and inference.” Biometrika, 86(1): 153–167.
• Fonseca, T. C. O., Ferreira, M. A. R., and Migon, H. S. (2008). “Objective Bayesian analysis for the Student-$t$ regression model.” Biometrika, 95(2): 325–333.
• Geweke, J. (1993). “Bayesian treatment of the independent Student-$t$ linear model.” Journal of Applied Econometrics, 8: S19–S40.
• Jacquier, E., Polson, N. G., and Rossi, P. E. (2004). “Bayesian analysis of stochastic volatility models with fat-tails and correlated errors.” Journal of Econometrics, 122: 185–212.
• Jeffreys, H. (1961). Theory of Probability. New York: Oxford University Press, 3rd edition.
• Juárez, M. A. and Steel, M. F. J. (2010). “Model-based clustering of non-Gaussian panel data based on skew-$t$ distributions.” Journal of Business and Economic Statistics, 28(1): 52–66.
• Kullback, S. and Leibler, R. A. (1951). “On information and sufficiency.” Annals of Mathematical Statistics, 22: 79–86.
• Lange, K. L., Little, R. J. A., and Taylor, J. M. G. (1989). “Robust statistical modelling using the $t$ distribution.” Journal of the American Statistical Association, 84(408): 881–896.
• Lin, J. G., Chen, J., and Lin, Y. (2012). “Bayesian analysis of Student $t$ linear regression with unknown change-point and application to stock data analysis.” Computational Economics, 40: 203–217.
• Maronna, R. A. (1976). “Robust $m$-estimators of multivariate location and scatter.” Annals of Statistics, 4: 51–67.
• Merhav, N. and Feder, M. (1998). “Universal prediction.” IEEE Transactions on Information Theory, 44: 2124–2147.
• Villa, C. and Walker, S. G. (2013). “An objective approach to prior mass functions for discrete parameter spaces.” Under Revision by Journal of the American Statistical Association.
• West, M. (1984). “Outlier models and prior distributions in Bayesian linear regression.” Journal of the Royal Statistical Society, Series B, 46: 431–439.