mixAK (version 5.8)

MVT: Multivariate Student t distribution

Description

Density and random generation for the multivariate Student t distribution with location equal to mu, precision matrix equal to Q (or scale matrix equal to Sigma).

Mentioned functions implement the multivariate Student t distribution with a density given by $$% p(\boldsymbol{z}) = \frac{\Gamma\bigl(\frac{\nu+p}{2}\bigr)}{\Gamma\bigl(\frac{\nu}{2}\bigr)\,\nu^{\frac{p}{2}}\,\pi^{\frac{p}{2}}}\, \bigl|\Sigma\bigr|^{-\frac{1}{2}}\, \Bigl\{1 + \frac{(\boldsymbol{z} - \boldsymbol{\mu})'\Sigma^{-1}(\boldsymbol{z} - \boldsymbol{\mu})}{\nu}\Bigr\}^{-\frac{\nu+p}{2}}, $$ where \(p\) is the dimension, \(\nu > 0\) degrees of freedom, \(\boldsymbol{\mu}\) the location parameter and \(\Sigma\) the scale matrix.

For \(\nu > 1\), the mean in equal to \(\boldsymbol{\mu}\), for \(\nu > 2\), the covariance matrix is equal to \(\frac{\nu}{\nu - 2}\Sigma\).

Usage

dMVT(x, df, mu=0, Q=1, Sigma, log=FALSE)
rMVT(n, df, mu=0, Q=1, Sigma)

Value

Some objects.

Arguments

df

degrees of freedom of the multivariate Student t distribution.

mu

vector of the location parameter.

Q

precision (inverted scale) matrix of the multivariate Student t distribution. Ignored if Sigma is given.

Sigma

scale matrix of the multivariate Student t distribution. If Sigma is supplied, precision is computed from \(\Sigma\) as \(Q = \Sigma^{-1}\).

n

number of observations to be sampled.

x

vector or matrix of the points where the density should be evaluated.

log

logical; if TRUE, log-density is computed

Value for dMVT

A vector with evaluated values of the (log-)density

Value for rMVT

A list with the components:

x

vector or matrix with sampled values

log.dens

vector with the values of the log-density evaluated in the sampled values

Author

Arnošt Komárek

See Also

dt, Mvt.

Examples

set.seed(1977)

### Univariate central t distribution
z <- rMVT(10, df=1, mu=0, Q=1)
ldz <- dMVT(z$x, df=1, log=TRUE)
boxplot(as.numeric(z$x))
cbind(z$log.dens, ldz, dt(as.numeric(z$x), df=1, log=TRUE))

### Multivariate t distribution
mu <- c(1, 2, 3)
Sigma <- matrix(c(1, 1, -1.5,  1, 4, 1.8,  -1.5, 1.8, 9), nrow=3)
Q <- chol2inv(chol(Sigma))

nu <- 3
z <- rMVT(1000, df=nu, mu=mu, Sigma=Sigma)
apply(z$x, 2, mean)              ## should be close to mu
((nu - 2) / nu) * var(z$x)       ## should be close to Sigma            

dz <- dMVT(z$x, df=nu, mu=mu, Sigma=Sigma)
ldz <- dMVT(z$x, df=nu, mu=mu, Sigma=Sigma, log=TRUE)

### Compare with mvtnorm package
#require(mvtnorm)
#ldz2 <- dmvt(z$x, sigma=Sigma, df=nu, delta=mu, log=TRUE)
#plot(z$log.dens, ldz2, pch=21, col="red3", bg="orange", xlab="mixAK", ylab="mvtnorm")
#plot(ldz, ldz2, pch=21, col="red3", bg="orange", xlab="mixAK", ylab="mvtnorm")