Entropy loss and risk of improved estimators for the generalized variance and precision (Q1118291)

Let the distributions of X(p\(\times r)\) and S(p\(\times p)\) be \(N(\xi,\Sigma \otimes I_ r)\) and \(W_ p(n,\Sigma)\) respectively and let them be independent. The risk of the improved estimator for \(| \Sigma |\) or \(| \Sigma^{-1}|\) based on X and S under entropy loss \[ L(d,| \Sigma |)=d/| \Sigma | -\log (d/| \Sigma |)-1\quad or\quad =d| \Sigma | -\log (d| \Sigma |)-1 \] is evaluated in terms of incomplete beta functions of matrix arguments and its derivative. Numerical comparison for the reduction of risk over the best affine equivariant estimator is given.