Theory of Statistics - George Mason University

270 3 Basic Statistical Theory
and
hence, the risk is
Eθ(X) = − β′ (θ)
β(θ)
= 2θ
,
1 − θ2 Vθ(X) = d
dθ Eθ(X)
1 + θ2
= 2
(1 − θ2 ;
) 2
1 + θ2
R(g(θ), X) = 2
(1 − θ2 .
) 2
Now, consider the estimator Ta = aX. Its risk under squared-error is
R(θ, Ta) = Eθ(L(θ, Ta))
= Eθ((g(θ) − Ta) 2 )
= 2a 2 1 + θ 2
(1 − θ 2 ) 2 + 4(1 − a2 )
θ 2
(1 − θ2 .
) 2
If a = 0, that is, the estimator is the constant 0, the risk is 4θ 2 /(1 − θ 2 ) 2 ,
which is smaller than the risk for X for all θ ∈] − 1, 1[.
The natural sufficient statistic in this one-parameter exponential family is
inadmissible for its expectation!
Other Forms of Admissibility
We have defined admissibility in terms of a specific optimality criterion,
namely minimum risk. Of course, the risk depends on the loss function, so
admissibility depends on the particular loss function.
Although this meaning of admissibility, which requires a decision-theory
framework, is by far the most common meaning, we can define admissibility
in a similar fashion with respect to any optimality criterion; for example,
the estimator T(X) is Pitman-admissible for g(θ) if there does not exist an
estimator that is Pitman-closer to g(θ). In Example 3.3 on page 216 we saw
that the sample mean even in a univariate normal distribution is not Pitman
admissible. The type of estimator used in that example to show that the
univariate mean is not Pitman admissible is a shrinkage estimator, just as a
shrinkage estimator was used in Example 3.19.
3.3.4 Minimaxity
Instead of uniform optimality properties for decisions restricted to be unbiased
or equivariant or optimal average properties, we may just seek to find one with
the smallest maximum risk. This is minimax estimation.
Theory of Statistics c○2000–2013 James E. Gentle