Page 386 - Probability and Statistical Inference

7. Point Estimation 363

the sufficient statistic for λ. The domain space for U is U =
{0, 1, 2, ..}. Now, for all u ∈ u, conditionally given U = u, the statistic T can
take one of the possible values 0 or 1. Thus, for u ∈ u, we can write

Note that is Poisson(nλ), is Poisson((n − 1)λ) whereas X 1
and are independently distributed. Thus, from (7.4.8) we rewrite E [T
λ
| U = u] as

and hence the Rao-Blackwellized version of the estimator T is W = (1 -
Now we know the expression for W and so we can directly
evaluate E [W]. We use the form of the mgf of a Poisson random variable,
λ
s
namely E [e ] = exp{nλ(e - 1)}, and then replace s with log(1 - n ) to write
-1
sU
λ
-λ
In other words, W is an unbiased estimator of e , but this should not be
surprising. The part (i) of the Rao-Blackwell Theorem leads to the same con-
clusion. Was there any way to guess the form of the estimator W before
-2λ
actually going through Rao-Blackwellization? How should one estimate e or
e ? One should attack these problems via Rao-Blackwellization or mgf as we
-3λ
just did. We leave these and other related problems as Exercise 7.4.3. !
Example 7.4.6 Suppose that X , ..., X are iid N(µ, σ ) where µ is un-
2
1
n
χ
known but σ is known with −∞ < µ < ∞, 0 < σ < ∞ and = ℜ. We wish to
2
estimate T(µ) = µ unbiasedly. Consider T = X which is an unbiased estimator
1
of µ. Consider the sufficient statistic for µ. The domain
space for U is u = ℜ. Now, for u ∈ u, conditionally given U = u, the distribu-
tion of the statistic T is N (u, σ (1 - n )). Refer to the Section 3.6 on the
2
-1
bivariate normal distribution as needed. Now, That
is, the Rao-Blackwellized version of the initial unbiased estimator T turns out
to be . !