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

Author： Pensky M.  

Publisher： Academic Press

ISSN： 0047-259X

Source： Journal of Multivariate Analysis, Vol.69, Iss.2, 1999-05, pp. : 242-260

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Abstract

We consider independent pairs (X1, Σ1), (X2, Σ2), …, (Xn, Σn), where each Σi is distributed according to some unknown density function g(Σ) and, given Σi=Σ, Xi has conditional density function q(x∣Σ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observation Xn+1 is obtained, the objective is to estimate Σn+1 corresponding to Xn+1. This estimator is called the empirical Bayes (EB) estimator of Σ. An EB estimator of Σ is constructed without any parametric assumptions on g(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal. Copyright 1999 Academic Press.

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