Page 400 - Probability and Statistical Inference

7. Point Estimation 377

we can write If one
first obtains the conditional pdf of T given X and evaluates E [T | X ]
n:n θ n:n
directly, then one will realize what a clever approach the present one has
been. !
Example 7.5.18 (Example 7.5.17 Continued) Let X , ..., X be iid random
1
n
variables distributed uniformly on (0, θ) where θ(> 0) is the unknown param-
eter with n ≥ 2. Again U = X is complete sufficient for θ. Consider T =
n:n
Obviously, E [T] = ¼θ (1 + 1/3n) = T(θ). One should verify that
2
θ
We leave it out as Exercise 7.5.21.
Again, if one first obtains the conditional pdf of T given X and evaluates
n:n
E [T | X ] directly, then one will realize what a clever approach the present
θ
n:n
one has been. !
7.6 Unbiased Estimation Under Incompleteness

Suppose that it we start with two statistics T and T’, both estimating the same
real valued parametric function T(θ) unbiasedly. Then, we individually condi-
tion T and T’ given the sufficient statistic U and come up with the refined
Rao-Blackwellized estimators W = E [T | U] and W’ = E [T’ | U] respectively.
θ
θ
It is clear that (i) both W, W’ would be unbiased estimators of T(θ), (ii) W will
have a smaller variance than that of T, and (iii) W’ will have a smaller variance
than that of T’.
An important question to ask here is this: how will the two
variances V (W) and V (W’) stack up against each other?
θ
θ
If the statistic U happens to be complete, then the Lehmann-Scheffé Theo-
rems will settle this question because W and W’ will be identical estimators
w.p.1. But if U is not complete, then it may be a different story. We highlight
some possibilities by means of examples.

7.6.1 Does the Rao-Blackwell Theorem Lead to UMVUE?
Suppose that X , ..., X are iid N(θ, θ ) having the unknown parameter θ ∈ Θ
2
n
1
χ
= (0, ∞), = (−∞, ∞), n ≥ 2. Of course, is sufficient for θ but
U is not complete since one can check that for
all θ ∈ Θ and yet is not identically zero w.p.1. Now, let us
work with where
From (2.3.26) it follows that E [S] = a θ and hence T’ is unbiased for
θ
n