Exercises 9 Therefore, for any value of θ, the variance of ¯ n achieves the Cram´er-Rao lower bound 1/In(θ) = σ2/n. The concept of the Fisher eﬃciency seems to be nice and powerful. In- deed, besides being unbiased, an eﬃcient estimator has the minimum pos- sible variance uniformly in θ ∈ Θ. Another feature is that it applies to any sample size n. Unfortunately, this concept is extremely restrictive. It works only in a limited number of models. The main pitfalls of the Fisher eﬃciency are discussed in the next chapter. Exercises Exercise 1.1. Show that the Fisher information can be computed by the formula In(θ) = − n Eθ ∂2 ln p (X, θ) ∂ θ2 . Hint: Make use of the representation (show!) ∂ ln p (x, θ) ∂ θ 2 p (x, θ) = ∂2 p (x, θ) ∂θ2 − ∂2 ln p (x, θ) ∂θ2 p (x, θ). Exercise 1.2. Let X1,...,Xn be independent observations with the N (μ, θ) distribution, where μ has a known value (refer to Example 1.1(b)). Prove that θn ∗ = 1 n n i = 1 (Xi − μ)2 is an eﬃcient estimator of θ. Hint: Use Exercise 1.1 to show that In(θ) = n/(2 θ2). When computing the variance of θn, ∗ first notice that the variable n i = 1 (Xi − μ)2/θ has a chi-squared distribution with n degrees of freedom, and, thus, its variance equals 2n. Exercise 1.3. Suppose that independent observations X1,...,Xn have a Bernoulli distribution with the probability mass function p (x, θ) = θ x (1 − θ)1−x, x ∈ { 0, 1 } , 0 θ 1. Show that the Fisher information is of the form In(θ) = n θ (1 − θ) , and verify that the estimator θn ∗ = ¯ n is eﬃcient.

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