dist.Multivariate.t.Precision {LaplacesDemon}  R Documentation 
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 
n 
This is the number of random draws. 
mu 
This is a numeric vector representing the location parameter,

Omega 
This is a 
nu 
This is the degrees of freedom 
log 
Logical. If 
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: positivedefinite
k \times k
precision matrix\Omega
Parameter 3: degrees of freedom
\nu > 0
Mean:
E(\theta) = \mu
, for\nu > 1
, otherwise undefinedVariance:
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 onedimensional or univariate Student t distribution. A random vector is considered to be multivariate tdistributed if every linear combination of its components has a univariate Student tdistribution.
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 positivedefinite. 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. software@bayesianinference.com
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)