LaplacesDemon (version 16.1.6)

dist.Multivariate.Cauchy: 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)

Value

dmvc gives the density and

rmvc generates random deviates.

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 scale matrix \(\textbf{S}\).

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 \(k \times k\) positive-definite scale matrix \(\textbf{S}\).

log

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

Author

Statisticat, LLC.

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: positive-definite \(k \times k\) 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 one-dimensional 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 Cauchy-distributed 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.

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)