Page 410 - Probability and Statistical Inference

7. Point Estimation 387

T(p) = p . Start with the estimator T = X X which is unbiased for T(p) and then
2
2
1
derive the Rao-Blackwellized version of T. {Hint: Proceed along the lines of
the Examples 7.4.1 and 7.4.3.}
7.4.2 (Example 7.4.3 Continued) Suppose that we have iid Bernoulli(p)
random variables X , ..., X where 0 < p < 1 is an unknown parameter with n
n
1
2
≥ 3. Consider the parametric function T(p) = p (1 − p). Start with the estima-
tor T = X X (1 − X ) which is unbiased for T(p) and then derive the Rao-
1 2 3
Blackwellized version of T. {Hint: Proceed along the lines of the Example
7.4.3.}
7.4.3 (Example 7.4.5 Continued) Suppose that we have iid Poisson(λ)
random variables X , ..., X where 0 < λ < ∞ is an unknown parameter with n
n
1
≥ 4. Consider the parametric function T(λ) and the initial estimator T defined
in each part and then derive the corresponding Rao-Blackwellized version W
of T to estimate T(λ) unbiasedly. Consider
-λ
(i) T(λ) = λe and start with T = I(X = 1). Verify first that T is
1
unbiased for T(λ). Derive W;
-2λ
(ii) T(λ) = e and start with T = I(X = 0∩ X = 0). Verify first that
1
2
T is unbiased for T(λ). Derive W;
-3λ
(iii) T(λ) = e and start with T = I(X = 0 ∩ X = 0∩ X = 0). Verify
1 2 3
first that T is unbiased for T(λ). Derive W;
2 -λ
(iv) T(λ) = λ e and start with T = 2I(X = 2). Verify first that T is
1
unbiased for T(λ). Derive W.
2
7.4.4 (Example 7.4.7 Continued) Suppose that X , ..., X are iid N(µ, σ )
n
1
where µ is unknown but σ is assumed known with µ ∈ ℜ, σ ∈ ℜ . Let T(µ) =
+
P {|X | ≤ a} where a is some known positive real number. Find an initial
1
µ
unbiased estimator T for T(µ). Next, derive the Rao-Blackwellized version of
T to estimate T(µ) unbiasedly.
7.4.5 (Exercise 7.4.4 Continued) Suppose that X , ..., X are iid N(µ, σ )
2
1
n
where µ is unknown but σ is assumed known with µ ∈ ℜ, σ ∈ ℜ , n ≥ 3. Let
+
T(µ) = P {X + X ≤ a} where a is some known real number. Find an initial
µ
1
2
unbiased estimator T for T(µ). Next, derive the Rao-Blackwellized version of
T to estimate T(µ) unbiasedly.
7.4.6 (Example 7.4.7 Continued) Suppose that X , ..., X are iid N(µ,
1
n
σ ) where µ, σ are both unknown with µ ∈ ℜ, σ ∈ ℜ , n ≥ 2. Let
+
2
T(µ) = P {X > a} where a is some known real number. Start with the
µ
1
initial unbiased estimator T = I(X > a) for T(µ). Next, derive the Rao-
1
Blackwellized version W of T to estimate T(µ) unbiasedly. {Hint:
Kolmogorov (1950a) first found the form of the final unbiased estimator W.