Objective prior for the number of degrees of freedom of a t distribution

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 dom- inate the data likelihood. We nd an objective criterion, based on loss functions, instead of trying to dene 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 dened 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.

[1]  S. Walker,et al.  An Objective Approach to Prior Mass Functions for Discrete Parameter Spaces , 2015 .

[2]  James O. Berger,et al.  Objective Priors for Discrete Parameter Spaces , 2012 .

[3]  P. Brown,et al.  Bayesian Priors from Loss Matching , 2012 .

[4]  李勇,et al.  Bayesian Analysis of Student t Linear Regression with Unknown Change-Point and Application to Stock Data Analysis , 2012 .

[5]  Svetlozar T. Rachev,et al.  Probability and Statistics for Finance , 2010 .

[6]  Miguel A. Juárez,et al.  Model-Based Clustering of Non-Gaussian Panel Data Based on Skew-t Distributions , 2010 .

[7]  Helio S. Migon,et al.  Objective Bayesian analysis for the Student-t regression model , 2008 .

[8]  Peter E. Rossi,et al.  Bayesian analysis of stochastic volatility models with fat-tails and correlated errors , 2004 .

[9]  M. Steel,et al.  Multivariate Student -t Regression Models : Pitfalls and Inference , 1999 .

[10]  Neri Merhav,et al.  Universal Prediction , 1998, IEEE Trans. Inf. Theory.

[11]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .

[12]  Stephen Blyth,et al.  Local divergence and association , 1994 .

[13]  J. Geweke,et al.  Bayesian Treatment of the Independent Student- t Linear Model , 1993 .

[14]  Jeremy MG Taylor,et al.  Robust Statistical Modeling Using the t Distribution , 1989 .

[15]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[16]  M. West.  Outlier Models and Prior Distributions in Bayesian Linear Regression , 1984 .

[17]  R. Maronna.  Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[18]  F. J. Anscombe,et al.  Topics in the Investigation of Linear Relations Fitted by the Method of Least Squares , 1967 .

[19]  R. Berk,et al.  Limiting Behavior of Posterior Distributions when the Model is Incorrect , 1966 .

[20]  J. T. Chu.  Errors in Normal Approximations to the $t,\tau,$ and Similar Types of Distribution , 1956 .

[21]  L. M. M.-T..  Theory of Probability , 1929, Nature.