16.3 Exercise 4

Let $\left\{X_{i} : 1 \leq i \leq n\right\}$ be a random sample, given H. Set $W = \left(X_{1} , X_{2} , \hdots , X_{n}\right)$. Suppose X conditionally Poisson $\left(u\right)$, given $H = u;$ i.e., suppose $P \left(\right. X = k \left|\right. H = u \left.\right) = e^{- u} u^{k} / k !$. If $H \sim$ gamma $\left(m , \lambda\right)$, determine the best mean square estimator for H, given W.