dist.Multivariate.Cauchy {LaplacesDemon}

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 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.

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.

Value

dmvc gives the density and rmvc generates random deviates.

Author(s)

Statisticat, LLC. [e-mail]

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)