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

integer sample size.

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.

Examples

Run this code

# NOT RUN {
## 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))
# }

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