bayesestdft: An R package for Bayesian estimation of the number of degrees of the freedom of the Student's t-distribution

Contents

• Overview
• Installation
• Goal
• Jeffreys prior
• Exponential prior
• Gamma prior
• Log-normal prior

Overview

An R package bayesestdft includes tools to implement Bayesian estimation of the number of degrees of the freedom of the Student's t-distribution, developed by Dr. Se Yoon Lee ([email protected]). The package was developed to analyze simulated and real data from the published article "The Use of a Log-Normal Prior for the Student t-Distribution" Axioms 2022, 11, 462". Readers can see the paper for technical details about the package. At current version, the main functions are BayesLNP, BayesJeffreys, and BayesGA that implement Markov Chain Monte Carlo algorithms to sample from the posterior distribution of the degrees of freedom. To operatre the function BayesJeffreys, user needs to install R library(numDeriv). See the Slides that summarized the technical parts of the R package.

Required R version

R version 4.0.4 (or higher)

Installation

library(devtools)
devtools::install_github("yain22/bayesestdft")
library(bayesestdft)

Goal

The goal of R Package bayesestdft is the fully Bayesian estimation of the number of degrees of the freedom of the Student's t-distribution. More precisely, provided the $N$ number of independently and identically distributed samples $x = (x_1,\cdots,x_N)$ drawn from the Student t-distribution

$$ t_{\nu}(x) = \frac{\Gamma\left( \frac{\nu+1}{2} \right)}{\sqrt{\nu \pi} \Gamma\left( \frac{\nu}{2} \right)} \left(1 + \frac{x^2}{\nu} \right)^{-\frac{\nu +1}{2}}, \quad x \in \mathbb{R}, $$

and a prior distribution $\pi(\nu)$, the aim is to draw posterior samples from the posterior distribution

$$ \pi(\nu|\textbf{x}) = \frac{\prod t_{\nu}(x_i) \cdot \pi(\nu)}{\int \prod t_{\nu}(x_i) \cdot \pi(\nu) d\nu}, \quad \nu \in \mathbb{R}^+. $$

The current version of the package provides four options of the priors $\pi(\nu)$. They are the Jeffreys prior $\pi_J(\nu)$, an exponential prior $\pi_E(\nu)$, a gamma prior $\pi_G(\nu)$, and a log-normal prior $\pi_L(\nu)$.

Jeffreys prior

$$ \pi_{J}(\nu) \propto \left(\frac{\nu}{\nu+3} \right)^{1/2} \left( \psi'\left(\frac{\nu}{2}\right) -\psi'\left(\frac{\nu+1}{2}\right) -\frac{2(\nu + 3)}{\nu(\nu+1)^2}\right)^{1/2},\quad \nu \in \mathbb{R}^+$$

Estimation of the degrees of freedom from simulated data

x = rt(n = 100, df = 0.1)
nu1 = BayesJeffreys(x, sampling.alg = "MH")
nu2 = BayesJeffreys(x, sampling.alg = "MALA")
mean(nu1)
mean(nu2)

Estimation of the degrees of freedom of daily log-return rate of S&P500 index time series data

library(dplyr)
data(index_return)
index_return_US <- filter(index_return, Country == "United States")
x = index_return_US$log_return_rate
nu1 = BayesJeffreys(x, sampling.alg = "MH")
nu2 = BayesJeffreys(x, sampling.alg = "MALA")
mean(nu1)
mean(nu2)

Exponential prior

$$ \pi_{E}(\nu) =Ga(\nu|1,0.1) = Exp(\nu|0.1) = \frac{1}{10} e^{-\nu/10},\quad \nu \in \mathbb{R}^+$$

Estimation of the degrees of freedom from simulated data

x = rt(n = 100, df = 0.1)
nu = BayesGA(x, a = 1, b = 0.1)
mean(nu)

Estimation of the degrees of freedom of daily log-return rate of S&P500 index time series data

library(dplyr)
data(index_return)
index_return_US <- filter(index_return, Country == "United States")
x = index_return_US$log_return_rate
nu = BayesGA(x, a = 1, b = 0.1)
mean(nu)

Gamma prior

$$ \pi_{G}(\nu) =Ga(\nu|2,0.1) =\frac{\nu}{100} e^{-\nu/10},\quad \nu \in \mathbb{R}^+$$

Estimation of the degrees of freedom from simulated data

x = rt(n = 100, df = 0.1)
nu = BayesGA(x, a = 2, b = 0.1)
mean(nu)

Estimation of the degrees of freedom of daily log-return rate of S&P500 index time series data

library(dplyr)
data(index_return)
index_return_US <- filter(index_return, Country == "United States")
x = index_return_US$log_return_rate
nu = BayesGA(x, a = 2, b = 0.1)
mean(nu)

Log-normal prior

$$ \pi_{L}(\nu) =logN(\nu|1,1) =\frac{1}{\nu \sqrt{2\pi}} \exp\left[- \frac{(\log \nu - 1)^2}{2} \right],\quad \nu \in \mathbb{R}^+$$

Estimation of the degrees of freedom from simulated data

x = rt(n = 100, df = 0.1)
nu = BayesLNP(x)
mean(nu)

Estimation of the degrees of freedom of daily log-return rate of S&P500 index time series data

library(dplyr)
data(index_return)
index_return_US <- filter(index_return, Country == "United States")
x = index_return_US$log_return_rate
nu = BayesLNP(x)
mean(nu)

References

[1] Se Yoon Lee. (2022) “The Use of a Log-Normal Prior for the Student t-Distribution,” Axioms