wishart {Boom}

Wishart Distribution

Description

Density and random generation for the Wishart distribution.

Usage

dWishart(W, Sigma, nu, logscale = FALSE)
rWishart(nu, scale.matrix, inverse = FALSE)

Arguments

W

Argument (random variable) for the Wishart density. A symmetric positive definite matrix.

Sigma

Scale or "variance" parameter of the Wishart distribution. See the "details" section below.

nu

The "degrees of freedom" parameter of the Wishart distribution. The distribution is only defined for nu >= nrow(W) - 1.

logscale

Logical. If TRUE then the density is returned on the log scale. Otherwise the density is returned on the density scale.

scale.matrix

For the Wishart distribution the scale.matrix parameter means the same thing as the Sigma parameter in dWishart. It is the variance parameter of the generating multivariate normal distribution.

If simulating from the inverse Wishart, scale.matrix is the INVERSE of the "sum of squares" matrix portion of the multivariate normal sufficient statistics.

inverse

Logical. If TRUE then simulate from the inverse Wishart distribution. If FALSE then simulate from the Wishart distribution.

Details

If nu is an integer then a W(\Sigma, \nu) random variable can be thought of as the sum of nu outer products: y_iy_i^T, where y_i is a zero-mean multivariate normal with variance matrix Sigma.

The Wishart distribution is

\frac{|W|^{\frac{\nu - p - 1}{2}} \exp(-tr(\Sigma^{-1}W) / 2)}{ 2^{\frac{\nu p}{2}}|\Sigma|^{\frac{\nu}{2}}\Gamma_p(\nu / 2)}%

where p == nrow(W) and \Gamma_p(\nu) is the multivariate gamma function (see lmgamma).

Value

dWishart returns the density of the Wishart distribution. It is not vectorized, so only one random variable (matrix) can be evaluated at a time.

rWishart returns one or more draws from the Wishart or inverse Wishart distributions. If n > 0 the result is a 3-way array. Unlike the rWishart function from the stats package, the first index corresponds to draws. This is in keeping with the convention of other models from the Boom package.

Author(s)

Steven L. Scott [e-mail]