Page 415 - Probability and Statistical Inference

392 7. Point Estimation

∞, 0 < σ < ∞, n ≥ 5. Let U = X , the smallest order statistic, which is
n:1
complete sufficient for µ.
(i) Find the conditional expectation, E [ | U];
µ
(ii) Find the conditional expectation, E [X | U];
µ 1
(iii) Find the conditional expectation,

{Hint: Try and use the Theorem 7.5.5 appropriately.}
7.5.21 (Example 7.5.18 Continued) Let X , ..., X be iid random vari-
n
1
ables distributed uniformly on (0, θ) where θ(> 0) is the unknown param-
eter with n ≥ 2. Show that {Hint:
Try and use the Theorem 7.5.5 appropriately.}
7.6.1 (Example 7.6.1 Continued) Suppose that X , ..., X are iid with
n
1
the common pdf f(x;θ) = θ exp{ −(x − θ)/θ}I(x > θ) with θ ∈ Θ = ℜ +
-1
where θ is the unknown parameter and n ≥ 2. Recall from Exercise 6.3.10
that is minimal sufficient for θ but U is not
complete. Here, we wish to estimate θ unbiasedly.
(i) Show that T = n(n + 1) X is an unbiased estimator for θ;
-1
n:1
(ii) Show that is also an unbiased
estimator for θ;
(iii) Along the lines of our discussions in the Section 7.6.1, derive
the Rao-Blackwellized versions W and W′ of the estimators T
and T′ respectively;
(iv) Compare the variances of W and W′. Any comments?
(v) Define a class of unbiased estimators of θ which are convex
combinations of W and W′ along the lines of (7.6.4). Within this
class find the estimator T* whose variance is the smallest;
(vi) Summarize your comments along the lines of the remarks made
in (7.6.6).
7.7.1 Show that the estimators derived in the Examples 7.2.1-7.2.5 are
consistent for the parametric function being estimated.
7.7.2 Show that the MLE’s derived in the Examples 7.2.6-7.2.7 are con-
sistent for the parametric function being estimated.
7.7.3 Show that the MLE’s derived in the Examples 7.2.9-7.2.12 are con-
sistent for the parametric function being estimated.
7.7.4 Show that the UMVUE’s derived in the Section 7.5 are consistent for
parametric function being estimated.
7.7.5 Show that the estimators W, W′ and T*(α*) defined in (7.6.4)-(7.6.5)
are consistent for the parameter θ being estimated.