dmtd {mstudentd}R Documentation

Density of a Multivariate tt Distribution

Description

Density of the multivariate (pp variables) tt 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 pp numeric vector.

nu

numeric. The degrees of freedom.

mu

length pp numeric vector. The mean vector.

Sigma

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

tol

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

Details

The density function of a multivariate tt distribution with pp variables is given by:

f(xν,μ,Σ)=Γ(ν+p2)Σ1/2Γ(ν2)(νπ)p/2(1+1ν(xμ)TΣ1(xμ))ν+p2 \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=1p=1 (univariate case) it becomes:

f(xν,μ,σ2)=Γ(ν+12)Γ(ν2)νπσ(1+(xμ)2νσ2)ν+12 \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 tt 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)


[Package mstudentd version 1.1.1 Index]