Theory of Statistics - George Mason University

5.1 UMVUE 395
The risk of T ∗ is less than the maximum risk of T; therefore, T is not
minimax.
Now we might ask is T ∗ minimax?
We first note that the risk (5.12) is constant, so T ∗ is minimax if it is
admissible or if it is a Bayesian estimator (in either case with respect to the
squared-error loss). We can see that T ∗ is Bayesian estimator (with a beta
prior). (You are asked to prove this in Exercise 4.10 on page 379.) As we show
in Chapter 4, a Bayes estimator with a constant risk is a minimax estimator;
hence, δ ∗ is minimax. (This example is due to Lehmann.)
Although we may initially be led to consideration of UMVU estimators
by consideration of a squared-error loss, which leads to a mean squared-error
risk, the UMVUE may not minimize the MSE. It was the fact that we could
not minimize the MSE uniformly that led us to add on the requirement of unbiasedness.
There may, however, be estimators that have a uniformly smaller
MSE than the UMVUE. An example of this is in the estimation of the variance
in a normal distribution. In Example 5.6 we have seen that the UMVUE
of σ 2 in the normal distribution is S 2 , while in Example 3.13 we have seen
that the MLE of σ 2 is (n − 1)S 2 /n, and by equation (3.55) on page 239, we
see that the MSE of the MLE is uniformly less than the MSE of the UMVUE.
There are other ways in which UMVUEs may not be very good as estimators;
see, for example, Exercise 5.2. A further undesirable property of
UMVUEs is that they are not invariant to transformation.
5.1.5 Lower Bounds on the Variance of Unbiased Estimators
The three Fisher information regularity conditions (see page 168) play a major
role in UMVUE. In particular, these conditions allow us to develop a lower
bound on the variance of any unbiased estimator.
The Information Inequality (CRLB) for Unbiased Estimators
What is the smallest variance an unbiased estimator can have? For an unbiased
estimator T of g(θ) in a family of densities satisfying the regularity
conditions and such that T has a finite second moment, the answer results
from inequality (3.83) on page 252 for the scalar estimator T and estimand
g(θ). (Note that θ itself may be a vector.) That is the information inequality
or the Cramér-Rao lower bound (CRLB), and it results from the covariance
inequality.
If g(θ) is a vector, then ∂g(θ)/∂θ is the Jacobian, and we have
V(T(X))
∂
∂θ g(θ)
T (I(θ)) −1 ∂
g(θ), (5.13)
∂θ
where we assume the existence of all quantities in the expression.
Theory of Statistics c○2000–2013 James E. Gentle