Page 417 - Probability and Statistical Inference

394 7. Point Estimation

where θ(> 0) is the unknown parameter. Let with c > 0 and
consider estimating θ with T (c). First find the MSE of T (c) with c(> 0)
n
n
fixed. Then, minimize the MSE with respect to c. Denote the optimal choice
for c by Show that the minimum MSE estimator T (c*) is consistent
n
for θ.
7.7.12 (Example 7.4.6 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 T(µ) = P {X < a} where
µ
1
a is some known real number. First find the UMVUE T ≡ T for T(µ). Show
n
that T is consistent for T(µ).
n
7.7.13 (Example 7.4.6 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 T(µ) = P {|X | < a} where
+
µ
1
a is some known positive number. First find the UMVUE T ≡ T for T(µ).
n
Show that T is consistent for T(µ).
n
7.7.14 Suppose that X , ..., X are iid Uniform(0, θ) where θ(> 0) is the
n
1
unknown parameter. Let T (c) = cX with c > 0 and consider estimating θ
n
n:n
with T (c). First find the MSE of T (c) with c(> 0) fixed. Then, minimize the
n
n
MSE with respect to c. Denote the optimal choice for c by Show
that the minimum MSE estimator T (c*) is consistent for θ.
n
7.7.15 (Example 7.7.13 Continued) Let X , ..., X be iid N(µ, σ ) where µ,
2
1
n
σ are both unknown with µ ∈ ℜ, σ ∈ ℜ , n ≥ 2. Let T(µ) = P {|X | < a} where
+
µ
1
a is some known positive number. Obtain the expression for T(µ) and thereby
propose a consistent estimator U for T(µ). But, U must be different from
n
n
the UMVUE T proposed earlier in the Exercise 7.7.13. {Note: A consistent
n
estimator does not have to be unbiased.}