rmcd function

Simulate from a Multivariate Cauchy Distribution

Produces one or more samples from the multivariate ($p$ variables) Cauchy distribution (MCD) with location parameter mu and scatter matrix Sigma.

rmcd(n, mu, Sigma, tol = 1e-6)

Arguments

• n: integer. Number of observations.
• mu: length $p$ numeric vector. The location parameter.
• Sigma: symmetric, positive-definite square matrix of order $p$. The scatter matrix.
• tol: tolerance for numerical lack of positive-definiteness in Sigma (for mvrnorm, see Details).

Returns

A matrix with $p$ columns and $n$ rows.

Details

A sample from a MCD with parameters $\boldsymbol{\mu}$ and $\Sigma$

can be generated using:

where $\mathbf{Y}$ is a random vector distributed among a centered Gaussian density with covariance matrix $\Sigma$ (generated using mvrnorm) and $u$ is distributed among a Chi-squared distribution with 1 degree of freedom.

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)
x <- rmcd(100, mu, sigma)
x
apply(x, 2, median)

References

S. Kotz and Saralees Nadarajah (2004), Multivariate $t$ Distributions and Their Applications, Cambridge University Press.

See Also

dmcd: probability density of a MCD.

estparmcd: estimation of the parameters of a MCD.

Author(s)

Pierre Santagostini, Nizar Bouhlel