Theory of Statistics - George Mason University

396 5 Unbiased Point Estimation
Note the meaning of this relationship in the multiparameter case: it says
that the matrix
∂
V(T(X)) −
∂θ g(θ)
T (I(θ)) −1 ∂
g(θ) (5.14)
∂θ
is nonnegative definite. (This includes the zero matrix; the zero matrix is
nonnegative definite.)
Example 5.11 Fisher efficiency in a normal distribution
Consider a random sample X1, X2, . . ., Xn from the N(µ, σ2 ) distribution. In
Example 3.9, we used the parametrization θ = (µ, σ). Now we will use the
parametrization θ = (µ, σ2 ). The joint log density is
logp(µ,σ)(x) = c − n
2 log(σ2 ) −
(xi − µ) 2 /(2σ 2 ). (5.15)
The information matrix is diagonal, so the inverse of the information matrix
is particularly simple:
I(θ) −1 2
σ
n =
0
0
σ 4
.
2(n−1)
(5.16)
For the simple case of g(θ) = (µ, σ2 ), we have the unbiased estimator,
n
T(X) = X, (Xi − X) 2
/(n − 1) ,
and
i=1
V(T(X)) =
σ 2
i
n
0
0
σ 4
2(n−1)
, (5.17)
which is the same as the inverse of the information matrix. The estimators
are Fisher efficient.
It is important to know in what situations an unbiased estimator can
achieve the CRLB. Notice this would depend on both p(X, θ) and g(θ). Let
us consider this question for the case of scalar θ and scalar function g. The
necessary and sufficient condition that an estimator T of g(θ) attain the CRLB
is that (T − g(θ)) be proportional to ∂ log(p(X, θ))/∂θ a.e.; that is, for some
a that does not depend on X,
∂ log(p(X, θ))
∂θ
= a(θ)(T − g(θ)) a.e. (5.18)
This means that the CRLB can be obtained by an unbiased estimator only in
the one-parameter exponential family.
For example, there are unbiased estimators of the mean in the normal,
Poisson, and binomial families that attain the CRLB. There is no unbiased
Theory of Statistics c○2000–2013 James E. Gentle