LaplacesDemon (version 16.1.1)

dist.Multivariate.Cauchy.Precision: 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 $k$ or a matrix with a number of columns, $k$, equal to the number of columns in precision matrix $\mathrm{\Omega }$.

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 $k\times k$ positive-definite precision matrix $\mathrm{\Omega }$.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Value

dmvcp gives the density and rmvcp generates random deviates.

Details

• Application: Continuous Multivariate

• Density: $$p(\theta )=\frac{\mathrm{\Gamma }((1+k)/2)}{\mathrm{\Gamma }(1/2)1^{k/2}{\pi }^{k/2}}|\mathrm{\Omega }{|}^{1/2}(1+(\theta -\mu {)}^T\mathrm{\Omega }(\theta -\mu ){)}^{-(1+k)/2}$$

• Inventor: Unknown (to me, anyway)

• Notation 1: $\theta \sim {\mathcal{MC}}_k(\mu ,{\mathrm{\Omega }}^{-1})$

• Notation 2: $p(\theta )={\mathcal{MC}}_k(\theta |\mu ,{\mathrm{\Omega }}^{-1})$

• Parameter 1: location vector $\mu$

• Parameter 2: positive-definite $k\times k$ precision matrix $\mathrm{\Omega }$

• Mean: $E(\theta )=\mu$

• Variance: $var(\theta )=undefined$

• Mode: $mode(\theta )=\mu$

The multivariate Cauchy distribution is a multidimensional extension of the one-dimensional or univariate Cauchy distribution. A random vector is considered to be multivariate Cauchy-distributed 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 $\mathrm{\Omega }$, which must be positive-definite.

See Also

dcauchy, dmvc, dmvt, dmvtp, and dwishart.

Examples

# NOT RUN {
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)
# }