rWishart

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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

cov, rnorm, rchisq.

Aliases
  • rWishart
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)) # }
Documentation reproduced from package stats, version 3.5.0, License: Part of R 3.5.0

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