Nonparametric Empirical Bayes Estimation of the Matrix Parameter of the Wishart Distribution

We consider independent pairs (X1, Σ1), (X2, Σ2), …, (Xn, Σn), where eachΣiis distributed according to some unknown density functiong(Σ) and, givenΣi=Σ,Xihas conditional density functionq(x∣Σ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observationXn+1is obtained, the objective is to estimateΣn+1corresponding toXn+1. This estimator is called the empirical Bayes (EB) estimator ofΣ. An EB estimator ofΣis constructed without any parametric assumptions ong(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal.