# dist.Multivariate.Cauchy: Multivariate Cauchy Distribution In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

These functions provide the density and random number generation for the multivariate Cauchy distribution.

## Usage

 ```1 2``` ```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.

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. [email protected]

`dcauchy`, `dinvwishart`, `dmvcp`, `dmvt`, and `dmvtp`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```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) ```