A weakly informative prior for the degrees of freedom of the t distribution based on a kurtosis measure

[1] proposed a general elicitation strategy for models that control kurtosis, such as the degrees of freedom parameter of the Student-t distribution, \(\nu>0\). The basic idea consists on assigning a prior distribution to a bounded kurtosis measure, as a function of the parameter of interest \(\nu\), say \(\kappa=\kappa(\nu)\). The boundedness assumption on \(\kappa\) allows us to assign a proper uniform prior on this quantity, while the injectivity is required for obtaining the induced prior on the parameter \(\nu\) by inverting this function. In general, most kurtosis measures are not available in close form as functions of the parameter \(\nu\), however, we can sample from the induced distribution on the parameter \(\nu\) and construct (parametric and nonparametric) approximations of this prior.

[1] propose to adopt the scalar kurtosis measure \(\kappa = 2\dfrac{f(\pi_R)}{f(\text{mode})}-1\) from [2], where \(\pi_R\) represents the positive mode of \(-f^{\prime}\) (the inflection point). This measure \(\kappa\) takes values in \(K\subset(-1,1)\), assigning the value \(\kappa=0.213\) to the normal distribution. Moreover, \(\kappa\) is an injective function of \(\delta\) for the Student-\(t\) distribution which takes values in \((0.213,0.633)\). This prior for \(\nu\) in the Student-\(t\) model is an alternative to others priors proposed for the Student-t distribution (see [3,4]). The algorithm to sample from the prior on \(\nu\), \(\pi(\nu)\), is as follows:

Algorithm

  1. For \(j=1,\dots,N\)

  1. Simulate \(u_j\sim U(0.213,0.633)\).

  2. Calculate \(\nu_j\) as the value that satisfies \(\kappa(\nu_j)=u_j\). This is a sample from \(\pi(\nu)\).

The following R code shows a handy approximation to the prior distribution obtained with this strategy. The idea is to fit a log-TPSAS distribution [6] to the samples obtained (not shown here) from \(\pi(\nu)\).

  1. Bayesian modelling of skewness and kurtosis with Two-Piece Scale and shape distributions

  2. Asymmetry and Gradient Asymmetry Functions: Density-Based Skewness and Kurtosis

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

  4. Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula

  5. TPSAS R package

  6. Survival and lifetime data analysis with a flexible class of distributions

rm(list=ls())
library(TPSAS)

# Parameters obtained by fitting the sample from \pi(\nu)
MLE <- c(0.84886337, 1.44326981, 0.05384718, 0.89544807)

# Prior on nu
prior <- Vectorize(function(x) dtpsas(log(x),MLE[1],MLE[2],MLE[3],MLE[4],param="eps")/x)
curve(prior,0,30,lwd=2,col="blue",n=1000,xlab=~nu,ylab="Density",main="Prior for the degrees of freedom",cex.axis=1.5,cex.lab=1.5)