6.8 Let N be an integer-valued random variable with distribution P (N = n) = P (n), n = 0,..., for which N is complete.

6.8 Let N be an integer-valued random variable with distribution Pθ (N = n) = Pθ (n), n = 0,..., for which N is complete. Given N = n, let X have the binomial distribution b(p, n) for n > 0, with p unknown, and let X = 0 when n = 0. For the observations

(N,X):

(a) Show that (N,X) is complete.

(b) Determine the UMVU estimator of pEθ (N).

(c) Show that no unbiased estimator of any function g(p) exists if Pθ (0) > 0 for some

θ.

(d) Determine the UMVU estimator of p if Pθ (0) for all θ.