dmtd function

Density of a Multivariate $t$ Distribution

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

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.

Returns

The value of the density.

Details

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

When $p=1$ (univariate case) it is the location-scale $t$ distribution, with density function:

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)

References

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

See Also

rmtd: random generation from a MTD.

estparmtd: estimation of the parameters of a MTD.

plotmvd, contourmvd: plot of the probability density of a bivariate distribution.

Author(s)

Pierre Santagostini, Nizar Bouhlel