On a Conjecture of Krishnamoorthy and Gupta

We consider the problem of estimating the precision matrix (Σ−1) under a fully invariant convex loss. Suppose that there exists a minimax constant risk estimatorΦ(say) for this problem. K. Krishnamoorthy and A. K. Gupta have proposed an operation which transforms this estimator into an orthogonally invariant estimatorΦ* (say) and they have a conjecture saying thatΦ* is minimax as well. This paper contains two parts. In the first part, we present counterexamples. In the second part, we elaborate a technique which can be used to prove that certain estimators are minimax. This technique is then applied successfully to some of the estimators proposed in the Krishnamoorthy and Gupta paper.