| rWishart {stats} | R Documentation |
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 ( |
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}.
Author(s)
Douglas Bates
References
Mardia KV, Kent JT, Bibby JM (1979). Multivariate Analysis. Academic Press, London.
See Also
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))