Random Wishart Distributed Matrices

Description

Generate n random matrices, distributed according to the Wishart distribution with parameters Sigma and df, W_p(\Sigma, m),\ m=\code{df},\ \Sigma=\code{Sigma}.

Usage

rWishart(n, df, Sigma)

Arguments

Details

If X_1,\dots, X_m, \ X_i\in\mathbf{R}^p is a sample of m independent multivariate Gaussians with mean (vector) 0, and covariance matrix \Sigma, the distribution of M = X'X is W_p(\Sigma, m).

Consequently, the expectation of M is

E[M] = m\times\Sigma.

Further, if Sigma is scalar (p = 1), the Wishart distribution is a scaled chi-squared (\chi^2) distribution with df degrees of freedom, W_1(\sigma^2, m) = \sigma^2 \chi^2_m.

The component wise variance is

\mathrm{Var}(M_{ij}) = m(\Sigma_{ij}^2 + \Sigma_{ii} \Sigma_{jj}).

Value

a numeric array, say R, of dimension p \times p \times n, where each R[,,i] is a positive definite matrix, a realization of the Wishart distribution W_p(\Sigma, m),\ \ m=\code{df},\ \Sigma=\code{Sigma}.

Author(s)

Douglas Bates

References

Mardia, K. V., J. T. Kent, and J. M. Bibby (1979) Multivariate Analysis, London: Academic Press.

See Also

cov, rnorm, rchisq.

Examples

## Artificial
S <- toeplitz((10:1)/10)
set.seed(11)
R <- rWishart(1000, 20, S)
dim(R)  #  10 10  1000
mR <- apply(R, 1:2, mean)  # ~= E[ Wish(S, 20) ] = 20 * S
stopifnot(all.equal(mR, 20*S, tolerance = .009))

## See Details, the variance is
Va <- 20*(S^2 + tcrossprod(diag(S)))
vR <- apply(R, 1:2, var)
stopifnot(all.equal(vR, Va, tolerance = 1/16))