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

We consider independent pairs (X1, [Sigma]1), (X2, [Sigma]2), ..., (Xn, [Sigma]n), where each[Sigma]iis distributed according to some unknown density functiong([Sigma]) and, given[Sigma]i=[Sigma],Xihas conditional density functionq(x|[Sigma]) 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 estimate[Sigma]n+1corresponding toXn+1. This estimator is called the empirical Bayes (EB) estimator of[Sigma]. An EB estimator of[Sigma]is constructed without any parametric assumptions ong([Sigma]). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal.