Page 394 - Probability and Statistical Inference

7. Point Estimation 371

and in view of (7.5.15) we obtain

2
-2
2
-1
2
From (7.5.11), we have the CRLB = {T′(µ)} /(nσ ) = 4n µ σ and compar-
ing this with (7.5.16) it is clear that V [W] > CRLB for all µ. That is, the
µ
CRLB is not attained.!
7.5.2 The Lehmann-Scheffé Theorems and UMVUE
In situations similar to those encountered in the Examples 7.5.4-7.5.5, it is
clear that neither the Rao-Blackwell Theorem nor the Cramér-Rao inequality
may help in deciding whether an unbiased estimator W is the UMVUE of T(?).
An alternative approach is provided in this section.
If the Rao-Blackwellized version in the end always comes up with the same
refined estimator regardless of which unbiased estimator of T(θ) one initially
starts with, then of course one has found the UMVUE for T(θ). Lehmann and
Scheffé’s (1950) notion of a complete statistic, introduced in Section 6.6,
plays a major role in this area. We first prove the following result from Lehmann
and Scheffé (1950).
Theorem 7.5.2 (Lehmann-Scheffé Theorem I) Suppose that T is an
unbiased estimator of the real valued parametric function T(θ) where the un-
known parameter θθ θθ θ ∈ Θ ⊆ ℜ . Suppose that U is a complete (jointly) suffi-
k
cient statistic for θθ θθ θ. Define g(u) = E [T | U = u], for u belonging to U, the
θ
domain space of U. Then, the statistic W = g(U) is the unique (w.p.1) UMVUE
of T(θ).
Proof The Rao-Blackwell Theorem assures us that in order to search for
the best unbiased estimator of T(θ), we need only to focus on unbiased esti-
mators which are functions of U alone. We already know that W is a function
of U and it is an unbiased estimator of T(θ). Suppose that there is another
unbiased estimator W* of T(θ) where W* is also a function of U. Define h(U)
= W − W* and then we have

Now, we use the Definition 6.6.2 of the completeness of a statistic. Since U is
a complete statistic, from (7.5.17) it follows that h(U) = 0 w.p.1, that is we
must have W = W* w.p.1. The result then follows. !
In our quest for finding the UMVUE of T(θ), we need not always have to
go through the conditioning with respect to the complete sufficient statistic
U. In some problems, the following alternate and yet equivalent result may