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)
| n | integer sample size. |
|---|---|
| df | numeric parameter, "degrees of freedom". |
| Sigma | positive definite \(p \times p\) "scale" matrix, the matrix parameter of the distribution. |
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.
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.
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 3.816392e-17 1.110223e-16 2.116363e-16 5.551115e-17 #> [2,] -9.020562e-17 1.000000e+00 -5.551115e-17 1.249001e-16 1.387779e-17 #> [3,] -5.551115e-17 0.000000e+00 1.000000e+00 -1.665335e-16 0.000000e+00 #> [4,] -5.551115e-17 2.220446e-16 0.000000e+00 1.000000e+00 0.000000e+00 #> [5,] 2.220446e-16 2.220446e-16 0.000000e+00 2.220446e-16 1.000000e+00