Simultaneous estimation of Poisson means under entropy loss (Q1108701)

Let \(x=(x_ 1,...,x_ p)\) be p independent Poisson random variables with means \(\theta =(\theta_ 1,...,\theta_ p)\). For the problem of estimating \(\theta\) when the loss is the entropy distance given by \[ L(\theta,a)=\sum^{p}_{1}\theta_ i(a_ i/\theta_ i-\log (a_ i/\theta_ i)-1) \] where \(a=(a_ 1,...,a_ p)\), consider a class of estimators of the form \(XC+b\), where \(C=diag(c_ 1,...,c_ K)\) and \(b=(b_ 1,...,b_ K)\) are constants. It is shown that \(XC+b\) is an admissible estimator of \(\theta\) if and only if (i) \(b_ i>0\) and \(0\leq c_ i\leq 1\) for \(i=1,...,K\) and (ii) \(\sum_{c_ i=1}b_ i\leq 1\). It follows that \(X+\underset \tilde{} 1\) is inadmissible for \(K\geq 2\), where \(\underset \tilde{} 1\) denotes the vector (1,...,1). Monte Carlo results are given showing the risk improvement of certain estimators which dominate \(X+\underset \tilde{} 1\). It is noted that estimators of the form \(X+b\) are generalized Bayes with respect to improper (limiting gamma) priors.