# rWishart

0th

Percentile

##### Random Wishart Distributed Matrices

Generate n random matrices, distributed according to the Wishart distribution with parameters Sigma and df, $W_p(Sigma, df)$.

Keywords
multivariate
##### Usage
rWishart(n, df, Sigma)
##### Arguments
n
integer sample size.
df
numeric parameter, “degrees of freedom”.
Sigma
positive definite ($p * p$) “scale” matrix, the matrix parameter of the distribution.
##### Details

If $X1,...,Xm, Xi in R^p$ 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, $W_1(sigma^2, m) = sigma^2 chi[m]^2$.

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 $p * p * n$, where each R[,,i] is a positive definite matrix, a realization of the Wishart distribution $W_p(Sigma, df)$.

##### References

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

cov, rnorm, rchisq.
library(stats) ## 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))