stats (version 3.6.2)

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

n

integer sample size.

df

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.

cov, rnorm, rchisq.

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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