dist.Multivariate.Cauchy | R Documentation |
These functions provide the density and random number generation for the multivariate Cauchy distribution.
dmvc(x, mu, S, log=FALSE)
rmvc(n=1, mu, S)
x |
This is either a vector of length |
n |
This is the number of random draws. |
mu |
This is a numeric vector representing the location parameter,
|
S |
This is a |
log |
Logical. If |
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.
dmvc
gives the density and
rmvc
generates random deviates.
Statisticat, LLC. software@bayesian-inference.com
dcauchy
,
dinvwishart
,
dmvcp
,
dmvt
, and
dmvtp
.
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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