Random Inverse Wishart Distributed Matrices

Generate n random matrices, distributed according to the inverse Wishart distribution with parameters Sigma and df, \(W_p(Sigma, df)\).

Note there are different ways of parameterizing the Inverse Wishart distribution, so check which one you need. Here, if \(X \sim IW_p(\Sigma, \nu)\) then \(X^{-1} \sim W_p(\Sigma^{-1}, \nu)\). Dawid (1981) has a different definition: if \(X \sim W_p(\Sigma^{-1}, \nu)\) and \(\nu > p - 1\), then \(X^{-1} = Y \sim IW(\Sigma, \delta)\), where \(\delta = \nu - p + 1\).

rInvWishart(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 realization of the inverse Wishart distribution \(IW_p(Sigma, df)\). Based on a modification of the existing code for the rWishart function.

References

Dawid, A. (1981). Some Matrix-Variate Distribution Theory: Notational Considerations and a Bayesian Application. Biometrika, 68(1), 265-274. doi:10.2307/2335827

Gupta, A. K. and D. K. Nagar (1999). Matrix variate distributions. Chapman and Hall.

Mardia, K. V., J. T. Kent, and J. M. Bibby (1979) Multivariate Analysis, London: Academic Press.

See also

rWishart, rCholWishart, and rInvCholWishart

Examples

set.seed(20180221)
A <- rInvWishart(1L, 10, 5 * diag(5L))[, , 1]
set.seed(20180221)
B <- stats::rWishart(1L, 10, .2 * diag(5L))[, , 1]

A %*% B
#>               [,1]          [,2]          [,3]          [,4]         [,5]
#> [1,]  1.000000e+00  4.780710e-17 -5.775969e-17  1.270195e-16 4.441151e-17
#> [2,]  2.076917e-17  1.000000e+00 -4.028034e-17  2.642691e-16 7.929804e-17
#> [3,] -8.992634e-17 -5.322340e-17  1.000000e+00 -1.298779e-16 2.514746e-17
#> [4,]  7.787488e-17  1.338887e-16  2.715793e-17  1.000000e+00 9.171614e-19
#> [5,]  1.938556e-16  2.752030e-16  7.210773e-17  3.054895e-16 1.000000e+00