rWishart
Random Wishart Distributed Matrices
Generate n random matrices, distributed according to the
Wishart distribution with parameters Sigma and df,
\(W_p(\Sigma, m),\ m=\code{df},\ \Sigma=\code{Sigma}\).
- Keywords
- multivariate
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.
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
\(W_p(\Sigma, m),\ \ m=\code{df},\ \Sigma=\code{Sigma}\).
References
Mardia, K. V., J. T. Kent, and J. M. Bibby (1979) Multivariate Analysis, London: Academic Press.
See Also
Examples
library(stats)
# 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))
# }