dmtd {mstudentd}

Density of a Multivariate t Distribution

Description

Density of the multivariate (p variables) t distribution (MTD) with degrees of freedom nu, mean vector mu and correlation matrix Sigma.

Usage

dmtd(x, nu, mu, Sigma, tol = 1e-6)

Arguments

x

length p numeric vector.

nu

numeric. The degrees of freedom.

mu

length p numeric vector. The mean vector.

Sigma

symmetric, positive-definite square matrix of order p. The correlation matrix.

tol

tolerance (relative to largest variance) for numerical lack of positive-definiteness in Sigma.

Details

The density function of a multivariate t distribution with p variables is given by:

\displaystyle{ f(\mathbf{x}|\nu, \boldsymbol{\mu}, \Sigma) = \frac{\Gamma\left( \frac{\nu+p}{2} \right) |\Sigma|^{-1/2}}{\Gamma\left( \frac{\nu}{2} \right) (\nu \pi)^{p/2}} \left( 1 + \frac{1}{\nu} (\mathbf{x}-\boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x}-\boldsymbol{\mu}) \right)^{-\frac{\nu+p}{2}} }

When p=1 (univariate case) it becomes:

\displaystyle{ f(x|\nu, \mu, \sigma^2) = \frac{\Gamma\left( \frac{\nu+1}{2} \right)}{\Gamma\left( \frac{\nu}{2} \right) \sqrt{\nu \pi} \sigma} \left(1 + \frac{(x-\mu)^2}{\nu \sigma^2}\right)^{-\frac{\nu+1}{2}} }

Value

The value of the density.

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

S. Kotz and Saralees Nadarajah (2004), Multivariate t Distributions and Their Applications, Cambridge University Press.

Examples

nu <- 1
mu <- c(0, 1, 4)
Sigma <- matrix(c(0.8, 0.3, 0.2, 0.3, 0.2, 0.1, 0.2, 0.1, 0.2), nrow = 3)
dmtd(c(0, 1, 4), nu, mu, Sigma)
dmtd(c(1, 2, 3), nu, mu, Sigma)

# Univariate
dmtd(1, 3, 0, 1)
dt(1, 3)