Description Usage Arguments Details Value Author(s) See Also Examples
These functions provide the density and random number generation for the multivariate Cauchy distribution.
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 scale matrix S. 
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. 
S 
This is a k x k positivedefinite scale matrix S. 
log 
Logical. If 
Application: Continuous Multivariate
Density:
p(theta) = Gamma[(1+k)/2] / {Gamma(1/2)1^(k/2)pi^(k/2)Sigma^(1/2)[1+(thetamu)^T*Sigma^(1)(thetamu)]^[(1+k)/2]}
Inventor: Unknown (to me, anyway)
Notation 1: theta ~ MC[k](mu, Sigma)
Notation 2: p(theta) = MC[k](theta  mu, Sigma)
Parameter 1: location vector mu
Parameter 2: positivedefinite k x 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.
dmvc
gives the density and
rmvc
generates random deviates.
Statisticat, LLC. [email protected]
dcauchy
,
dinvwishart
,
dmvcp
,
dmvt
, and
dmvtp
.
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)
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)

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