We derive the first and the second moments of the Moore-Penrose generalized inverse of a singular standard Wishart matrix without relying on a density. Instead, we use the moments of an inverse Wishart distribution and an invariance argument which is related to the literature on tensor functions. We also find the order of the spectral norm of the generalized inverse of a Wishart matrix as its dimension and degrees of freedom diverge.
"On the mean and variance of the generalized inverse of a singular Wishart matrix." Electron. J. Statist. 5 146 - 158, 2011. https://doi.org/10.1214/11-EJS602