Exercises 9

Therefore, for any value of θ, the variance of

¯n

X 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

X is eﬃcient.