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) _ e( TE {0, 1, 2, . .. }, 020. a [2 marks] The statistic T(X) = _ _ 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) = logo, 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. Poisson belongs to the one-parameter exponential family with a(0) = e , b(x) = 1 c(0) = loge and d(x) = r.O Neyman-Fisher Factorization Criterion since f can be written as f(x; 0) = Le exp(x log 0) O Poisson belongs to the one-parameter exponential family with a(0) = e , b(x) = c(0) = 0 and d(x) =r.b) [3 marks] Suppose we are interested in the parameter h(0) = e . The estimator W = I(X1 0,X2 0,Xs=0) (X) is an unbiased estimator of h(0). Derive the UMVUE of h(0) = e-30. Hint: Use the interpretation that P(X1 = 0) = e- and therefore P(X1 = 0, X2 = 0, X3 = 0) = P(X1 = 0)3 = e-30. Also note that T(X) => X; ~ Poisson(no). O humvue = 1 co O humvue= n - 1 t n O humvue = e-# O humvue = 1 -\f() [2 marks] Find the Fisher Information contained in the statistic T = _ _, X; for the parameter 0. O IT(0) = O n IT(0) = 0(1 - 0) O n IT(0) = 7 O IT(0) = O IT(0) = Akd} [2 marks] Calculate the Cramer-Rao lower bound for the minimal variance of an unbiased estimator of M3} : 833. Use the notation [3(3) for the information contained in the sample X = {X1,X2,. . . , X\") about the parameter 5'. 3Ix{3}_ 9369 5:09) [3193(1 a) le'gl 3335 lef") 396(1 .9) 5:09)