dist.Multivariate.Cauchy: Multivariate Cauchy Distribution

In benmarwick/LaplacesDemon: Complete Environment for Bayesian Inference

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 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 positive-definite scale matrix S.
log	Logical. If log=TRUE, then the logarithm of the density is returned.

Details

• Application: Continuous Multivariate

• Density:

  p(theta) = Gamma[(1+k)/2] / {Gamma(1/2)1^(k/2)pi^(k/2)|Sigma|^(1/2)[1+(theta-mu)^T*Sigma^(-1)(theta-mu)]^[(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: positive-definite 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 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.

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)

benmarwick/LaplacesDemon documentation built on May 12, 2019, 12:59 p.m.