library(MCMCpack)
library(magrittr)
set.seed(123)

The question is whether an inverse Wishart distribution’s inverse \(\mathbf{X}^{-1}\) follows:

• \(\mathcal{W}(\mathbf \Psi^{-1}, \nu)\), or
• \(\mathcal{W}(\mathbf \Psi, \nu)\) (cf. Schaffer, 1997):

To determine which is correct, we start with drawing a \(\Sigma\)

#Make Sigma
Sigma <- c(1, rep(c(.1, .2, .3), 10 )) %>% 
  toeplitz 

to draw from the inverse Wishart distribution, with \(\nu = 31\)

# Draw inverse wishart
inv.wish <- riwish(31, Sigma)

Then, we create an estimated wishart from the generated inverse Wishart, simply by taking the inverse of the generated inverse Wishart:

# Create `estimated` wishart from inverse wishart
est.wish <- solve(inv.wish)

To test whether \(\mathbf{X}\) follows an inverse Wishart distribution if its inverse \(\mathbf{X}^{-1} \sim \mathcal{W}(\mathbf \Psi^{-1}, \nu)\) or if its inverse \(\mathbf{X}^{-1} \sim \mathcal{W}(\mathbf \Psi, \nu)\) (cf. Schaffer, 1997):

# Draw 2 versions of wishart - one with and one without inverse Sigma
drawn.wish <- rwish(31, solve(Sigma))
drawn.wish2 <- rwish(31, Sigma)

We can verify which distribution it follows by making a qqplot:

# Test whether estimated wishart from inverse wishart conforms to a drawn wishart 
qqplot(est.wish, drawn.wish); abline(coef = c(0,1)) #correct

qqplot(est.wish, drawn.wish2);  abline(coef = c(0,1))