R: Multivariate Cauchy Distribution: Precision Parameterization

dist.Multivariate.Cauchy.Precision {LaplacesDemon} | R Documentation

## Multivariate Cauchy Distribution: Precision Parameterization

### Description

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

### Usage

    dmvcp(x, mu, Omega, log=FALSE)
    rmvcp(n=1, mu, Omega)

### 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 precision matrix Omega.
`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.
`Omega` | This is a k x k positive-definite precision matrix Omega.
`log` | Logical. If `log=TRUE`, then the logarithm of the density is returned.

### Details

- Application: Continuous Multivariate

- Density:

  p(theta) = (Gamma((nu+k)/2) / (Gamma(1/2)*1^(k/2)*pi^(k/2))) * |Omega|^(1/2) * (1 + (theta-mu)^T Omega (theta-mu))^(-(1+k)/2)

- Inventor: Unknown (to me, anyway)

- Notation 1: theta ~ MC[k](mu, Omega^(-1))

- Notation 2: p(theta) = MC[k](theta | mu, Omega^(-1))

- Parameter 1: location vector mu

- Parameter 2: positive-definite k x k precision matrix Omega

- 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. A random vector is considered to be multivariate Cauchy-distributed if every linear combination of its components has a univariate Cauchy distribution. The multivariate Cauchy distribution is equivalent to a multivariate t distribution with 1 degree of freedom.

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.

It is usually parameterized with mean and a covariance matrix, or in Bayesian inference, with mean and a precision matrix, where the precision matrix is the matrix inverse of the covariance matrix. These functions provide the precision parameterization for convenience and familiarity. It is easier to calculate a multivariate Cauchy density with the precision parameterization, because a matrix inversion can be avoided.

This distribution has a mean parameter vector mu of length k, and a k x k precision matrix Omega, which must be positive-definite.

### Value

`dmvcp` gives the density and `rmvcp` generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`dcauchy`, `dmvc`, `dmvt`, `dmvtp`, and `dwishart`.

### Examples

    library(LaplacesDemon)
    x <- seq(-2,4,length=21)
    y <- 2*x+10
    z <- x+cos(y)
    mu <- c(1,12,2)
    Omega <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
    f <- dmvcp(cbind(x,y,z), mu, Omega)

    X <- rmvcp(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)

[Package LaplacesDemon version 16.1.1 Index]