Page 411 - Probability and Statistical Inference

388 7. Point Estimation

This elegant proof is due to Kolmogorov. Observe that one has the sufficient
statistic U = ( , S*) for θ = (µ, σ) where
Let g(u) = E {T | , S*} = P {X > a| , S*} = P {Y > yo | , S*} with
θ
θ
θ
1
and
Next, verify that Y is distributed independently of U, and the pdf of Y is given
by k(1 − y ) for –1 < y < 1 and some known positive constant k. Hence,
2 (n-4)/2
}
7.4.7 (Exercise 7.4.6 Continued) Suppose that X , ..., X are iid N(µ, σ )
2
n
1
+
where µ, σ are both unknown with µ ∈ ℜ, σ ∈ ℜ , n ≥ 2. Let T(µ) = P {X >
1
µ
a} where a is some known real number. Consider the Rao-Blackwellized ver-
sion W from the Exercise 7.4.6 which estimates T(µ) unbiasedly. Derive the
form of W in its simplest form when n = 4, 6 and 8.
7.4.8 Suppose that X , ..., X , X are iid Bernoulli(p) where 0 < p < 1 is an
n+1
n
1
unknown parameter. Denote the parametric function T(p) =
Now, consider the initial unbiased estimator T = for T(p).
The problem is to find the Rao-Blackwellized version W of T. Here, it will be hard
to guess the form of the final estimator. Now, proceed along the following steps.
(i) Note that is sufficient for p and that T is an
unbiased estimator of T(p);
(ii) Observe that W = E {T |U = u} = P {T = 1| U = u}, and find
p
p
the expression for W; {Hint:
(iii) Directly evaluate the ratio of the two probabilities in part (ii),
and show that g(u) = 0, n/(n + 1), (n - 1)/(n + 1), respectively

when u = 0, 1, 2, 3, ..., n, n + 1;
(iv) Using the explicit distribution of W given in part (iii), show
n
that E [W] = 1 − np q − q where q = 1 − p;
2
p
(v) It is possible that so far the reader did not feel any urge to
think about the explicit form of the parametric function T(p).
In part (iv), one first gets a glimpse of the expression of T(p).
Is the expression of T(p) correct? In order to check, the reader
is now asked to find the expression of T(p) directly from its
definition. {Hint: