Question:
Let $X_1, · · · , X_n$ be a random sample from $Poisson(θ)$. The prior for θ is $G(α, β)$
Find the Bayesian estimator (rule) of θ under the SEL(squared error loss).
Find the generalized Bayesian estimator (rule) of θ under the loss $L(θ, a) = (a−θ)^2/θ$.
Solution:
My understanding:
Assume the prior distribution $θ ∼ G(α, β)$ and suppose that we observe a sample of n Poisson data. Then, we can derive the posterior distribution via Bayes theorem as:
$θ|x ∼ G(α + n \bar{x}, β + n)$
We have squared loss, $L(\theta-a)=(\theta-a)^2$. Then the Bayes rule will be $E[\theta|x]=?$
Then I couldn't go further from here. I wanted to make sense of the theory from Wikipedia. But it didn't help. I appreciate your suggestions!