Reference and Probability-Matching Priors for the Parameters of a Univariate Student $t$-Distribution

TL;DR

This paper derives reference and probability-matching priors for the univariate Student t-distribution, aiming to improve Bayesian inference with properties relatable to frequentist methods, validated through simulations and real data analysis.

Contribution

It introduces new priors for the Student t-distribution parameters that balance Bayesian validity with frequentist interpretability, supported by extensive simulation and real data testing.

Findings

Priors improve mean squared error of the posterior median.

Credibility intervals achieve nominal coverage levels.

Performance validated on stock return data.

Abstract

In this paper reference and probability-matching priors are derived for the univariate Student $t$-distribution. These priors generally lead to procedures with properties frequentists can relate to while still retaining Bayes validity. The priors are tested by performing simulation studies. The focus is on the relative mean squared error from the posterior median ($MSE(\nu)/\nu$) and on the frequentist coverage of the 95\% credibility intervals for a sample size of $n=30$. Average interval lengths of the credibility intervals as well as the modes of the interval lengths based on 2000 simulations are also considered. The performance of the priors are also tested on real data, namely daily logarithmic returns of IBM stocks.