dist.Multivariate.Cauchy.Precision {LaplacesDemon}  R Documentation 
Multivariate Cauchy Distribution: Precision Parameterization
Description
These functions provide the density and random number generation for the multivariate Cauchy distribution. These functions use the precision parameterization.
Usage
dmvcp(x, mu, Omega, log=FALSE)
rmvcp(n=1, mu, Omega)
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 
log 
Logical. If 
Details
Application: Continuous Multivariate
Density:
p(\theta) = \frac{\Gamma((1+k)/2)}{\Gamma(1/2)1^{k/2}\pi^{k/2}} \Omega^{1/2} (1 + (\theta\mu)^T \Omega (\theta\mu))^{(1+k)/2}
Inventor: Unknown (to me, anyway)
Notation 1:
\theta \sim \mathcal{MC}_k(\mu, \Omega^{1})
Notation 2:
p(\theta) = \mathcal{MC}_k(\theta  \mu, \Omega^{1})
Parameter 1: location vector
\mu
Parameter 2: positivedefinite
k \times k
precision matrix\Omega
Mean:
E(\theta) = \mu
Variance:
var(\theta) = undefined
Mode:
mode(\theta) = \mu
The multivariate Cauchy distribution is a multidimensional extension of the onedimensional or univariate Cauchy distribution. A random vector is considered to be multivariate Cauchydistributed if every linear combination of its components has a univariate Cauchy distribution. The multivariate Cauchy distribution is equivalent to a multivariate t distribution with 1 degree of freedom.
The Cauchy distribution is known as a pathological distribution because its mean and variance are undefined, and it does not satisfy the central limit theorem.
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 Cauchy 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.
Value
dmvcp
gives the density and
rmvcp
generates random deviates.
Author(s)
Statisticat, LLC. software@bayesianinference.com
See Also
dcauchy
,
dmvc
,
dmvt
,
dmvtp
, and
dwishart
.
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)
f < dmvcp(cbind(x,y,z), mu, Omega)
X < rmvcp(1000, rep(0,2), diag(2))
X < X[rowSums((X >= quantile(X, probs=0.025)) &
(X <= quantile(X, probs=0.975)))==2,]
joint.density.plot(X[,1], X[,2], color=TRUE)