Construction of improved estimators in multiparameter estimation for continuous exponential families (Q792046)

For a multivariate normal distribution \(N(\mu,\sigma^ 2I_ p)\), \textit{W. James} and \textit{C. Stein} [Estimation with quadratic loss. Proc. 4th Berkeley Sympos. math. statist. Probab. 1, 361-379 (1961)] proved the existence of an estimator \(\delta\) (x) for \(\mu\) which dominates the usual maximum likelihood (UMVUE, best invariant) estimator \(\delta^ 0(x)=x\). \(\delta\) is obtained by shrinking \(\delta^ 0\) towards 0. Similar questions have been investigated for continuous exponential families with a p-variate parameter \(\mu\) by \textit{J. Berger} [Ann. Stat. 8, 545-571 (1980; Zbl 0447.62008)] and by the first author and \textit{A. Parsian} [J. Multivariate Anal. 10, 551-564 (1980; Zbl 0453.62007)]. The paper at hand generalizes these results by constructing, for a given initial estimator \(\delta^ 0\), an improved estimator \(\delta\) that shrinks \(\delta^ 0\) towards a prechosen point \(m\in {\mathbb{R}}^ p\) or towards the (geometric or arithmetic) mean of the p components. Typically, \(\delta\) has the form \(\delta_ i(x)=\delta^ 0_ i(x)- \eta(x)h_ i(x_ i)\Phi_ i(x)\) with, e.g., \(\Phi_ i(x):=- (\tau(S)/S)(g_ i(x_ i)-m_ i)\), \(S:=\Sigma_ jd_ j| g_ j(x_ j)-m_ j|^{\beta}\) and suitable functions \(\tau\) (S), \(g_ i(x_ i)\), \(\eta\) (x), \(h_ i(x_ i)\) and weights \(d_ i>0 (i=1,...,p).\) All results reside on the representation \(R(\vartheta,\delta)=R(\vartheta,\delta^ 0)+2E_{\vartheta}[D(X)]\) for the risk function with a differential operator D(X); each solution of \(D(X)<0\) gives a dominating estimator \(\delta\). Special examples are given for the normal, gamma, and Dirichlet distributions.