dist.Multivariate.Cauchy {LaplacesDemon}  R Documentation 
Multivariate Cauchy Distribution
Description
These functions provide the density and random number generation for the multivariate Cauchy distribution.
Usage
dmvc(x, mu, S, log=FALSE)
rmvc(n=1, mu, S)
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,

S 
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}\Sigma^{1/2}[1+(\theta\mu)^{\mathrm{T}}\Sigma^{1}(\theta\mu)]^{(1+k)/2}}
Inventor: Unknown (to me, anyway)
Notation 1:
\theta \sim \mathcal{MC}_k(\mu, \Sigma)
Notation 2:
p(\theta) = \mathcal{MC}_k(\theta  \mu, \Sigma)
Parameter 1: location vector
\mu
Parameter 2: positivedefinite
k \times k
scale matrix\Sigma
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. The multivariate Cauchy distribution is equivalent to a multivariate t distribution with 1 degree of freedom. A random vector is considered to be multivariate Cauchydistributed if every linear combination of its components has a univariate Cauchy distribution.
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.
Value
dmvc
gives the density and
rmvc
generates random deviates.
Author(s)
Statisticat, LLC. software@bayesianinference.com
See Also
dcauchy
,
dinvwishart
,
dmvcp
,
dmvt
, and
dmvtp
.
Examples
library(LaplacesDemon)
x < seq(2,4,length=21)
y < 2*x+10
z < x+cos(y)
mu < c(1,12,2)
Sigma < matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
f < dmvc(cbind(x,y,z), mu, Sigma)
X < rmvc(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)