Page 413 - Probability and Statistical Inference

390 7. Point Estimation

7.5.7 (Example 7.5.11 Continued) Let X , ..., X be iid N(µ, σ ) where µ, σ
2
n
1
are both unknown with µ ∈ ℜ, σ ∈ ℜ , n ≥ 2. Let θθ θθ θ = (µ σ) and T(θθ θθ θ) = µσ k
+
where k is a known and fixed real number. Derive the UMVUE for T(θθ θθ θ). Pay
particular attention to any required minimum sample size n which may be
needed. {Hint: Use (2.3.26) and the independence between and S to first
k
2
derive the expectation of S where S is the sample variance. Then make
some final adjustments.}
7.5.8 (Example 7.5.12 Continued) Let X , ..., X be iid having the common
1
n
pdf σ exp{−(x - µ)/σ}I(x > µ) where µ, σ are both unknown with −∞ < µ <
-1
∞, 0 < σ < ∞, n ≥ 2. Let θθ θθ θ = (µ, σ) and T(θθ θθ θ) = µσ where k is a known and
k
fixed real number. Derive the UMVUE for T(θθ θθ θ). Pay particular attention to any
required minimum sample size n which may be needed. {Hint: Use (2.3.26)
and the independence between X and to first derive
n:1
the expectation of Y . Then make some final adjustments.}
k
7.5.9 Suppose that X , ..., X are iid Uniform (−θ, θ) where θ is the un-
1 n
k
+
known parameter, θ ∈ ℜ . Let T(θ) = θ where k is a known and fixed positive
real number. Derive the UMVUE for T(θ). {Hint: Verify that U = |X | is
n:n
complete sufficient for θ. Find the pdf of U/θ to first derive the expectation of
U . Then make some final adjustments.}
k
7.5.10 In each Example 7.4.1-7.4.8, argue that the Rao-Blackwellized es-
timator W is indeed the UMVUE for the respective parametric function. In the
single parameter problems, verify whether the variance of the UMVUE attains
the corresponding CRLB.
7.5.11 Suppose that X , ..., X , X are iid N(µ, θ ) where µ is unknown
2
n
n+1
1
but σ is assumed known with µ ∈ ℜ, σ ∈ ℜ , n ≥ 2. Consider the parametric
+
function The problem is to find the UMVUE
for T(µ). Start with which is an unbiased estimator for
T(θ). Now, proceed along the following steps.
(i) Note that is complete sufficient for µ and that T
is an unbiased estimator of T(θ);
(ii) Observe that W = E {T |U = u} = P {T = 1| U = u}, and find
µ
µ
the expression for W. {Hint: Write down explicitly the bivari
ate normal distribution of and U. Then,
find the conditional probability, P {T > 0| U = u} utilizing the
1
µ
Theorem 3.6.1. Next, argue that T > 0 holds if and only if T
1
= 1.}.
7.5.12 Suppose that X , ..., X are iid Bernoulli(p) where 0 < p < 1 is an
n
1
2
unknown parameter. Consider the parametric function T(p) = p + qe with q
= 1 − p.