stats (version 3.4.1)

rWishart: 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

n

integer sample size.

df

numeric parameter, “degrees of freedom”.

Sigma

positive definite (\(p\times p\)) “scale” matrix, the matrix parameter of the distribution.

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 \(W_p(\Sigma, m),\ \ m=\code{df},\ \Sigma=\code{Sigma}\).

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

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))