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 .