
Comparing Proportions: A Modern Solution to a Classical Problem 69
The corresponding reference posterior expected loss fr om using σ
0
as a
proxy for σ, given a random sample of size n, is
d(σ
0
|z) =
Z
∞
0
n δ
z
{σ
0
|σ, µ}π(σ |z) dσ
=
n
2
ψ
n − 1
2
− 1 +
ns
2
(n −3)σ
2
0
+ log
2σ
2
0
ns
2
.
By definition, the Bayes point estimator with respect to this loss function,
the intrinsic reference estimato r of σ is that value of σ
0
which minimizes
d(σ
0
|z); this is found to be σ
∗
(z) =
√
ns/
√
n − 3. Thus, the intrinsic reference
estimator of the va riance is
σ
2
∗
(z) =
n s
2
n − 3
,
an estimator already suggested by Stein [16], which is always larger than
both the MLE and the conventiona l unbiased ...