Department of Statistical Science
Duke University

presents:

Mark F.J. Steel & Carmen Fernández
Department of Statistics, Purdue University

"On Bayesian Modelling of Fat Tails and Skewness and the Dangers of Continuous Distributions"

Abstract:

We consider a Bayesian analysis of linear regression models that can account for skewed error distributions with fat tails. A general procedure for introducing skewness into symmetric distributions is first proposed. Even though this allows for a great deal of flexibility in distributional shape, tail behaviour is not affected. Applying this skewness procedure to a Student-t distribution, we generate a "skewed Student" distribution, which displays both flexible tails and possible skewness, each entirely controlled by a separate scalar parameter. The linear regression model with a skewed Student error term is the main focus of this talk: we first characterize existence of the posterior distribution and its moments, using standard improper priors and allowing for inference on skewness and tail parameters. For posterior inference with this model, a numerical procedure is suggested, using Gibbs sampling with data augmentation. The latter proves very easy to implement and renders the analysis of quite challenging problems a practical possibility. An example for stock returns illustrates the use of this model in empirical data analysis.

In a general context, we also point out that Bayesian inference on the basis of a given sample is not always possible with continuous sampling models, even under a proper prior. The reason for this paradoxical situation is explained, and its empirical relevance is liked to coarse gathering of data, such as rounding. A solution, inspired by the way observations are recorded, is proposed. Use of a Gibbs sampler makes the solution practically feasible. The case of independent sampling from (possibly skewed) scale mixtures of Normals is analysed in detail for a location-scale model with a commonly used noninformative prior. We analyse the same stock returns example in this light and show that Bayesian inference based on set observations is possible, whereas the "usual" inference is precluded.

November 15, 1996

4:00 pm - 5:00 pm

116 Old Chem Building

Any questions concerning the seminar may be addressed to Cheryl McGhee @ [919] 684-8029 or e-mail cheryl@stat.duke.edu