rmtd function

Simulate from a Multivariate $t$ Distribution

Produces one or more samples from the multivariate ($p$ variables) $t$ distribution (MTD) with degrees of freedom nu, mean vector mu and correlation matrix Sigma.

rmtd(n, nu, mu, Sigma, tol = 1e-6)

Arguments

• n: integer. Number of observations.
• 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 for numerical lack of positive-definiteness in Sigma (for mvrnorm, see Details).

Returns

A matrix with $p$ columns and $n$ rows.

Details

A sample from a MTD with parameters $\nu$, $\boldsymbol{\mu}$ and $\Sigma$

can be generated using:

where $Y$ is a random vector distributed among a centered Gaussian density with covariance matrix $\Sigma$ (generated using mvrnorm) and $u$ is distributed among a Chi-squared distribution with $\nu$ degrees of freedom.

Examples

nu <- 3
mu <- c(0, 1, 4)
Sigma <- matrix(c(1, 0.6, 0.2, 0.6, 1, 0.3, 0.2, 0.3, 1), nrow = 3)
x <- rmtd(10000, nu, mu, Sigma)
head(x)
dim(x)
mu; colMeans(x)
nu/(nu-2)*Sigma; var(x)

References

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

See Also

dmtd: probability density of a MTD.

estparmtd: estimation of the parameters of a MTD.

Author(s)

Pierre Santagostini, Nizar Bouhlel