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
X 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
X 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 θn

= 1/
¯n
X has the expected value Eθ[ θn

] =
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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