10 1. The Fisher Eﬃciency Exercise 1.4. Assume that X1,...,Xn are independent observations from a Poisson distribution with the probability mass function p (x, θ) = θ x x! e−θ, x ∈ { 0, 1,... }, θ 0. Prove that the Fisher information in this case is In(θ) = n/θ, and show that ¯ n is an eﬃcient estimator of θ. Exercise 1.5. Let X1,...,Xn be a random sample from an exponential distribution with the density p (x, θ) = 1 θ e− x/θ , x 0, θ 0. Verify that In(θ) = n/θ2, and prove that ¯ n is eﬃcient. Exercise 1.6. Show that in the exponential model with the density p(x , θ) = θ exp{−θ x} , x , θ 0, the MLE θ ∗ n = 1/ ¯ n has the expected value Eθ[ θ ∗ n ] = n θ/(n − 1). What is the variance of this estimator? Exercise 1.7. Show that for the location parameter model with the density p(x , θ) = p 0 (x − θ), introduced in Example 1.2, the Fisher information is a constant if it exists. Exercise 1.8. In the Exercise 1.7, find the values of α for which the Fisher information exists if p 0 (x) = C cosα x , −π/2 x π/2 , and p 0 (x) = 0 otherwise, where C = C(α) is the normalizing constant. Note that p 0 is a probability density if α −1 .

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