Description Usage Arguments Details Value Author(s) See Also Examples
These functions provide the density and random number generation for the multivariate Cauchy distribution. These functions use the precision parameterization.
1 2 
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. It must be of length k, as defined above. 
Omega 
This is a k x k positivedefinite precision matrix Omega. 
log 
Logical. If 
Application: Continuous Multivariate
Density:
p(theta) = (Gamma((nu+k)/2) / (Gamma(1/2)*1^(k/2)*pi^(k/2))) * Omega^(1/2) * (1 + (thetamu)^T Omega (thetamu))^((1+k)/2)
Inventor: Unknown (to me, anyway)
Notation 1: theta ~ MC[k](mu, Omega^(1))
Notation 2: p(theta) = MC[k](theta  mu, Omega^(1))
Parameter 1: location vector mu
Parameter 2: positivedefinite k x 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 x k precision matrix Omega, which must be positivedefinite.
dmvcp
gives the density and
rmvcp
generates random deviates.
Statisticat, LLC. software@bayesianinference.com
dcauchy
,
dmvc
,
dmvt
,
dmvtp
, and
dwishart
.
1 2 3 4 5 6 7 8 9 10 11 12  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)

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