8 1. The Fisher Efficiency
In the regular case, the differentiation and integration are interchangeable,
hence differentiating in θ , we get the equation,
1 + bn (θ) =
Rn
ˆ
θ n(x1,...,xn) ∂p (x1,...,xn,θ)/∂θ dx1 . . . dxn
=
Rn
ˆn(x1,...,xn)
θ
∂p (x1,...,xn,θ)/∂θ
p (x1,...,xn,θ)
p (x1,...,xn,θ) dx1 . . . dxn
= Eθ
ˆ
θ
n
Ln(θ) = Covθ
ˆ
θ
n
, Ln(θ)
where we use the fact that Eθ Ln(θ) = 0. The correlation coefficient ρn of
ˆn
θ and Ln(θ) does not exceed 1 in its absolute value, so that
1 ≥ ρn
2
=
(
Covθ
ˆn
θ , Ln(θ)
)2
Varθ[ˆn]
θ Varθ[Ln(θ)]
=
(1 +
bn(θ))2
Varθ[ˆn]
θ In(θ)
.
1.4. Efficiency of Estimators
An immediate consequence of Theorem 1.8 is the formula for unbiased esti-
mators.
Corollary 1.9. For an unbiased estimator
ˆ
θ
n
, the Cram´ er-Rao inequality
(1.1) takes the form
(1.2) Varθ
ˆn
θ ≥
1
In(θ)
, θ ∈ Θ.
An unbiased estimator θn
∗
= θn
∗(X1,
. . . , Xn) in a regular statistical
experiment is called Fisher efficient (or, simply, efficient) if, for any θ ∈ Θ,
the variance of θn
∗
reaches the Cram´ er-Rao lower bound, that is, the equality
in (1.2) holds:
Varθ θn
∗
=
1
In(θ)
, θ ∈ Θ.
Example 1.10. Suppose, as in Example 1.1(a), the observations X1,...,Xn
are independent N (θ,
σ2)
where
σ2
is assumed known. We show that the
sample mean
¯
X
n
= (X1 + · · · + Xn)/n is an efficient estimator of θ. Indeed,
¯n
X is unbiased and Varθ
¯n
X =
σ2/n.
On the other hand,
ln p (X, θ) = −
1
2
ln(2 π
σ2)
−
(X −
θ)2
2σ2
and
l (X , θ) =
∂ ln p (X, θ)
∂θ
=
X − θ
σ2
.
Thus, the Fisher information for the statistical experiment is
In(θ) = n Eθ
(
l (X , θ)
)2
=
n
σ4
Eθ (X −
θ)2
=
nσ2
σ4
=
n
σ2
.
