Let X1, . . . , Xn be a random sample from the Poisson distribution with mean θ. Let Y =
a. Prove that there is no unbiased estimator of 1/θ. (Write the equation that is equivalent to Eθ (r(X)) = 1/θ. Simplify it, and then use what you know from calculus of infinite series to show that no function r can satisfy the equation.)
b. Suppose that we wish to estimate 1/θ. Consider r(Y) = n/(Y + 1) as an estimator of θ. Find the bias of r(Y), and show that the bias goes to 0 as n → ∞.
c. Use the delta method to find the asymptotic (as n→ ∞) distribution of n/(Y + 1).