Page 387 - Probability and Statistical Inference

364 7. Point Estimation

Example 7.4.7 (Example 7.4.6 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
and χ = ℜ. We wish to estimate the parametric function T(µ) = P {X > a}
µ
1
unbiasedly where a is a fixed and known real number. Consider T = I(X > a)
1
and obviously T is an unbiased estimator of T(µ). Consider
the sufficient statistic for µ. The domain space for U is u = ℜ. For u ∈ u,
conditionally given U = u, the distribution of the statistic T is N (u, σ (1 - n )).
-1
2
In other words, we have

That is, the Rao-Blackwellized version of the initial unbiased estimator T is
Was there a way to guess the form
of the improved estimator W before we used Rao-Blackwellization? The Exer-
cise 7.4.4 poses a similar problem for estimating the two-sided probability. A
much harder problem is to estimate the same parametric function T(µ) when
both µ and σ are unknown. Kolmogorov (1950a) gave a very elegant solution
for this latter problem. The Exercise 7.4.6 shows the important steps. !
Example 7.4.88 88 8 (Example 7.4.6 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
χ
2
and = ℜ. We wish to estimate T(µ) = µ unbiasedly. Consider
which is an unbiased estimator of T(µ). Consider again a
sufficient statistic for µ. The domain space for U is u = ℜ. As before, for u
∈ u, conditionally given U = u, the distribution of the statistic T is N (u, σ (1
2
-1
- n )). In other words, we can express E [T | U = u] as
µ
That is, the Rao-Blackwellized version of the initial unbiased estimator T of µ 2
is What initial estimator T would one start with if we wished
to estimate T(µ) = µ unbiasedly? Try to answer this question first before
3
looking at the Exercise 7.4.9. !
Remark 7.4.1 In the Example 7.4.8, we found as
the final unbiased estimator of µ . Even though W is unbiased for µ , one
2
2
may feel somewhat uneasy to use this estimator in practice. It is true that
P {W < 0} is positive whatever be n, µ and σ. The parametric function
µ
µ is non-negative, but the final unbiased estimator W can be negative
2
with positive probability! From the Figure 7.4.1 one can see the behavior