dmcd {mcauchyd}

Density of a Multivariate Cauchy Distribution

Description

Density of the multivariate (p variables) Cauchy distribution (MCD) with location parameter mu and scatter matrix Sigma.

Usage

dmcd(x, mu, Sigma, tol = 1e-6)

Arguments

x

length p numeric vector.

mu

length p numeric vector. The location parameter.

Sigma

symmetric, positive-definite square matrix of order p. The scatter matrix.

tol

tolerance (relative to largest eigenvalue) for numerical lack of positive-definiteness in Sigma.

Details

The density function of a multivariate Cauchy distribution is given by:

\displaystyle{ f(\mathbf{x}|\boldsymbol{\mu}, \Sigma) = \frac{\Gamma\left(\frac{1+p}{2}\right)}{\pi^{p/2} \Gamma\left(\frac{1}{2}\right) |\Sigma|^\frac{1}{2} \left[ 1 + (\mathbf{x}-\boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x}-\boldsymbol{\mu}) \right]^\frac{1+p}{2}} }

Value

The value of the density.

Author(s)

Pierre Santagostini, Nizar Bouhlel

See Also

rmcd: random generation from a MCD.

plotmcd, contourmcd: plot of a bivariate Cauchy density.

Examples

mu <- c(0, 1, 4)
sigma <- matrix(c(1, 0.6, 0.2, 0.6, 1, 0.3, 0.2, 0.3, 1), nrow = 3)
dmcd(c(0, 1, 4), mu, sigma)
dmcd(c(1, 2, 3), mu, sigma)