Page 380 - Probability and Statistical Inference

7. Point Estimation 357

For any T ∈ D we have:
for all θθ θθ θ which shows that any estimator T chosen from D is an unbiased
estimator of T(θθ θθ θ). Hence, it is obvious that D ⊆ C. Now we wish to address
the following question.
What is the best estimator of T(θθ θθ θ) within the smaller class D?
That is, which estimator from class D has the smallest variance? From the
following theorem one will see that the answer is indeed very simple.
Theorem 7.3.2 Within the class of estimators D, the one which has the
-1
smallest variance corresponds to α = k , i = 1, ..., k. That is, the best unbi-
i
ased estimator of T(θθ θθ θ) within the class D turns out to be which
is referred to as the best linear (in T , ..., T ) unbiased estimator (BLUE) of
1 k
T(θθ θθ θ).
Proof Since the T ’s are pairwise uncorrelated, for any typical estimator T
i
from the class D, we have

From (7.3.6) it is now clear that we need to
minimize subjected to the restriction that
But, observe that

Hence, for all choices of α , i = 1, ..., k such that
i
But, from (7.3.7) we see that the smallest possible value, if

and only if That is, V (T) would be minimized if and
θ
-1
only if α = k , i = 1, ..., k. !
i
Example 7.3.4 Suppose that X , ..., X are iid N(µ, σ ) where µ, σ are both
2
1 n
unknown, −∞ < µ < ∞, 0 < σ < ∞. Among all linear (in X , ..., X ) unbiased
1 n
estimators of µ, the BLUE turns out to be , the sample mean. This follows
immediately from the Theorem 7.3.2.!