n
random matrices, distributed according to the
Wishart distribution with parameters Sigma
and df
,
$W_p(Sigma, df)$.
rWishart(n, df, Sigma)
array
, say R
, of dimension
$p * p * n$, where each R[,,i]
is a
positive definite matrix, a realization of the Wishart distribution
$W_p(Sigma, df)$.
Consequently, the expectation of $M$ is
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
cov
, rnorm
, rchisq
.
## 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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