Page 395 - Probability and Statistical Inference

372 7. Point Estimation

be more directly applicable. We state the result without proving it. Its proof
can be easily constructed from the preceding proof of Theorem 7.5.2.
Theorem 7.5.3 (Lehmann-Scheffé Theorem II) Suppose that U is a
complete sufficient statistic for ? where the unknown parameter θθ θθ θ ∈ Θ ⊆ ℜ .
k
Suppose that a statistic W = g(U) is an unbiased estimator of the real valued
parametric function T(θ). Then, W is the unique (w.p.1) UMVUE of T(θ).
In the Examples 7.5.6-7.5.8, neither the Rao-Blackwell Theorem
nor the Cramér-Rao inequality helps in identifying the UMVUE.
But, the Lehmann-Scheffé approach is right on the money.

Example 7.5.6 (Example 7.5.4 Continued) Suppose that X , ..., X are iid
1
n
Poisson(λ) where 0 < λ < ∞ is unknown with n ≥ 2. We wish to estimate T(µ) =
e unbiasedly. The Rao-Blackwellized unbiased estimator of T(λ) was
-λ
and its variance was strictly larger than the CRLB. But, W
depends only on the complete sufficient statistic U = Hence, in view of
the Lehmann-Scheffé Theorems, W is the unique (w.p.1) UMVUE for T(λ). !
Example 7.5.7 (Example 7.5.5 Continued) Suppose that X , ..., X are iid
1
n
N(µ, σ ) where µ is unknown but σ is known with −∞ < µ < ∞, 0 < σ < ∞
2
2
2
and χ = ℜ. We wish to estimate T(µ) = µ unbiasedly. We found the Rao-
Blackwellized unbiased estimator and but its vari-
ance was strictly larger than the CRLB. Now, W depends only on the com-
plete sufficient statistic Hence, in view of the Lehmann-Scheffé
Theorems, W is the unique (w.p.1) UMVUE for T(µ). !
Example 7.5.88 88 8 (Example 7.4.3 Continued) Suppose that X , ..., X are iid
1
n
Bernoulli(p) where 0 < p < 1 is unknown with n ≥ 2. We wish to estimate T(p)
= p(1 - p) unbiasedly. Recall that the Rao-Blackwellized version of the unbi-
ased estimator turned out to be But, W depends only on
the complete sufficient statistic Hence, in view of the Lehmann-
Scheffé Theorems, W is the unique (w.p.1) UMVUE for T(p). !
Let us add that the Rao-Blackwellized estimator in the Example 7.4.7 is
also the UMVUE of the associated parametric function. The verification is left
as Exercise 7.5.2.

Next we give examples where the Cramér-Rao inequality is not
applicable but we can conclude the UMVUE property of the
natural unbiased estimator via the Lehmann-Scheffé Theorems.

Example 7.5.9 Let X , ..., X be iid Uniform(0, θ) where θ(> 0) is the
n
1
unknown parameter. Now, U = X , the largest order statistic, is complete
n:n