= PROBLEM 1. Let {X1, ..., ,Xn} be a random sample from the Poisson (1) - distri- bution. a) Explain that S= X; is suffi

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Problem 1 Let X1 Xn Be A Random Sample From The Poisson 1 Distri Bution A Explain That S X Is Suffi 1 (134.35 KiB) Viewed 21 times

= PROBLEM 1. Let {X1, ..., ,Xn} be a random sample from the Poisson (1) - distri- bution. a) Explain that S= X; is sufficient for d. b) Show that ÎML = n-1S is the MLE for 1. c) Find the information matrix and argue that Îmi is a UMVUE for d. d) Describe shortly how to prove that S ~ Poisson (n)). We want to test Ho: 1 <lo versus Hj: 1 > lo with level a. A proposed test is : Reject H, iff S > ko. Suppose that to = 2, a = 0.05 and n = 4. . Table 1: The Poisson distribution with = 8 10 11 13 k P(S> k1 = 8) 12 14 0.1841 0.1192 0.0638 0.0341 0.0172 e) Use the table and choose ko so that the proposed test is admissable. f) Prove that S has the MLR property. g) Explain that the test stated above is UMP if the level is adjusted to some a'. What is a'? h) A modified randomised test is suggested, o=1(S > ko) + a 1(S = ko) where a € (0,1). Find the constant a so that this test is a UMP for the given а significance level. For the following three points ko and a are theoretical quantities. i) Let B(A) = P(S > kol 1) denote the power function for the fixed test. Prove that B is strictly increasing. j) Do as in the previous point for the randomised test. k) Explain that Bø(1) > B(A) unless a = 0. But this seems to be in conflict with the fact that the fixed test is UMP. Solve the apparantly contradicition here. 1) Use R and make a graphical illustration of the 2 different power functions.