An identity concerning a Wishart matrix
Let S be a \(p\times p\) Wishart matrix with parameters n and \(\Sigma\). For a rational number \(\alpha =r/s\) with r and s integers and s positive, let \(S^{\alpha}\) denote a positive definite matrix such that \((S^{\alpha})^ s=S^ r\). Using a decision theoretic argument, we prove that \[ E[(tr S)^ 2tr S^{\alpha}]=(np+2+2\alpha)E[tr S tr S^{\alpha}] \] when \(\Sigma =I\) and \(np+2+2\alpha\) is positive.
scientific article; zbMATH DE number 3945095