Question One Let X = (X1, X2, ..., Xn) be a sample of i.i.d. Poisson(@) random variables with probability mass function f(x; 0) = Po(X = x) = TE {0, 1, 2, ...}, 0>0.a) SUBMITTED [2 marks] The statistic T(X) = >i_ X; is complete and sufficient for 0. Which of the following provides justification for why this statement is true. Poisson belongs to the one-parameter exponential family with a(0) = loge, b(x) = x, c(0) = e # and d(x) =- Poisson belongs to the one-parameter exponential family with a(0) = loge, b(x) = - c (0 ) = e- and d(x) = x. ONeyman-Fisher Factorization Criterion since f can be written as f(x; 0) = _e exp(xlog #) Poisson belongs to the one-parameter exponential family with a(0) = et, b(x) = 1 c(0) = loge and d(x) =x. Poisson belongs to the one-parameter exponential family with a(0) = et, b(x) = c(0) = 0 and d(x) =z.b) [3 marks] Suppose we are interested in the parameter h(0) = de *. The estimator W = I(x,=1) (X) is an unbiased estimator of h(0). Derive the UMVUE of h(0) = de-e. Hint: Use the interpretation that P(X] = 1) = de . Also note that T(X) =>Xi =nx ~ Poisson(ne). i= 1 O humvue = X (1 - 1 )2x n O humvue = X (1 _ 1)2x-1 n O humvue = Xe-X O humvue = (1 - 2 ) O humvue = (1 - _ ) nx n() SUBMITTED [2 marks] Find the Fisher Information contained in the statistic T = _!_ X, for the parameter 0. O Ir(0) = O n IT(0) = 0(1 - 0) O n IT(0) = 1 -0 O IT(0) =; O IT(0) = -dj [2 marks] Calculate the Cramer-Rao lower bound for the minimal variance of an unbiased estimator of h(0) = de . Use the notation Ix (0) for the information contained in the sample X = (X1, X2, . .., Xn) about the parameter 0. Show items as icons, in a list, in columns or in a gallery e (1 - 0) Ix (0) O e-20 (1 - 0)2 Ix(0) O -20 (1 - 0) Ix (0) O e- 20 (1 - 0)' Ix (0) O e (1 -0)2 IX(0)