dist.Multivariate.t.Precision {LaplacesDemon}

Multivariate t Distribution: Precision Parameterization

Description

These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution. These functions use the precision parameterization.

Usage

dmvtp(x, mu, Omega, nu=Inf, log=FALSE)
rmvtp(n=1, mu, Omega, nu=Inf)

Arguments

x

This is either a vector of length k or a matrix with a number of columns, k, equal to the number of columns in precision matrix \Omega.

n

This is the number of random draws.

mu

This is a numeric vector representing the location parameter, \mu (the mean vector), of the multivariate distribution (equal to the expected value when df > 1, otherwise represented as \nu > 1). It must be of length k, as defined above.

Omega

This is a k \times k positive-definite precision matrix \Omega.

nu

This is the degrees of freedom \nu, which must be positive.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Details

• Application: Continuous Multivariate

• Density:

  p(\theta) = \frac{\Gamma((\nu+k)/2)}{\Gamma(\nu/2)\nu^{k/2}\pi^{k/2}} |\Omega|^{1/2} (1 + \frac{1}{\nu} (\theta-\mu)^T \Omega (\theta-\mu))^{-(\nu+k)/2}

• Inventor: Unknown (to me, anyway)

• Notation 1: \theta \sim \mathrm{t}_k(\mu, \Omega^{-1}, \nu)

• Notation 2: p(\theta) = \mathrm{t}_k(\theta | \mu, \Omega^{-1}, \nu)

• Parameter 1: location vector \mu

• Parameter 2: positive-definite k \times k precision matrix \Omega

• Parameter 3: degrees of freedom \nu > 0

• Mean: E(\theta) = \mu, for \nu > 1, otherwise undefined

• Variance: var(\theta) = \frac{\nu}{\nu - 2} \Omega^{-1}, for \nu > 2

• Mode: mode(\theta) = \mu

The multivariate t distribution, also called the multivariate Student or multivariate Student t distribution, is a multidimensional extension of the one-dimensional or univariate Student t distribution. A random vector is considered to be multivariate t-distributed if every linear combination of its components has a univariate Student t-distribution.

It is usually parameterized with mean and a covariance matrix, or in Bayesian inference, with mean and a precision matrix, where the precision matrix is the matrix inverse of the covariance matrix. These functions provide the precision parameterization for convenience and familiarity. It is easier to calculate a multivariate t density with the precision parameterization, because a matrix inversion can be avoided.

This distribution has a mean parameter vector \mu of length k, and a k \times k precision matrix \Omega, which must be positive-definite. When degrees of freedom \nu=1, this is the multivariate Cauchy distribution.

Value

dmvtp gives the density and rmvtp generates random deviates.

Author(s)

Statisticat, LLC. [e-mail]

See Also

dwishart, dmvc, dmvcp, dmvt, dst, dstp, and dt.

Examples

library(LaplacesDemon)
x <- seq(-2,4,length=21)
y <- 2*x+10
z <- x+cos(y) 
mu <- c(1,12,2)
Omega <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
nu <- 4
f <- dmvtp(cbind(x,y,z), mu, Omega, nu)
X <- rmvtp(1000, c(0,1,2), diag(3), 5)
joint.density.plot(X[,1], X[,2], color=TRUE)