Chapter 7: Problem 6
Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a Poisson distribution with parameter \(\theta, 0<\theta<\infty .\) Let \(Y=\sum_{1}^{n} X_{i}\) and let \(\mathcal{L}[\theta, \delta(y)]=[\theta-\delta(y)]^{2}\). If we restrict our considerations to decision functions of the form \(\delta(y)=b+y / n\), where \(b\) does not depend on \(y\), show that \(R(\theta, \delta)=b^{2}+\theta / n .\) What decision function of this form yields a uniformly smaller risk than every other decision function of this form? With this solution, say \(\delta\), and \(0<\theta<\infty\), determine \(\max _{\theta} R(\theta, \delta)\) if it exists.
Short Answer
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Key Concepts
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