Page 412 - Probability and Statistical Inference

7. Point Estimation 389

> 1 ∩ X = 1} = (1 − q )q + (1 − q − npq )p = 1 − np q −
n-1
2
n
n
n+1
q . Even if one had first found the expression of T (p), could
n
one intuitively guess the form of W found in part (ii)?}
7.4.9 (Example 7.4.8 Continued) Suppose that X , ..., X are iid N(µ, σ )
2
n
1
+
where µ is unknown but σ is assumed known with µ ∈ ℜ, σ ∈ ℜ , n ≥ 2. Let
3
T (µ) = µ . Find an initial unbiased estimator T for T (µ). Next, derive the Rao-
Blackwellized version of T to estimate T (µ) unbiasedly. {Hint: Start with the
3
fact that E [(X − µ) ] = 0 and hence show that is an unbi-
1
µ
ased estimator for T(µ). Next, work with the third moment of the conditional
distribution of X given that }
1
7.5.1 Prove Theorem 7.5.3 exploiting the arguments used in the proof of
Theorem 7.5.2.
7.5.2 (Example 7.4.7 Continued) Show that the Rao-Blackwellized estima-
tor W is the UMVUE of T(µ).
7.5.3 (Exercise 6.2.16 Continued) Suppose that X , ..., X are iid with the
1
n
Rayleigh distribution, that is the common pdf is
where θ(> 0) is the unknown parameter. Find the UMVUE for (i) θ, (ii) θ and
2
-1
(iii) θ . {Hint: Start with which is sufficient for θ. Show that
the distribution of U/θ is a multiple of Hence, derive unbiased estimators
for θ, θ and θ which depend only on U. Can the completeness of U be
-1
2
justified with the help of the Theorem 6.6.2?}
7.5.4 (Exercise 7.5.3 Continued) Find the CRLB for the variance of unbi-
ased estimators of (i) θ, (ii) θ and (iii) θ . Is the CRLB attained by the vari-
-1
2
ance of the respective UMVUE obtained in the Exercise 7.5.3?
7.5.5 (Exercise 6.2.17 Continued) Suppose that X , ..., X are iid with the
n
1
Weibull distribution, that is the common pdf is
where α(> 0) is the unknown parameter, but β(> 0) is assumed known. Find the
2
-1
UMVUE for (i) α, (ii) α and (iii) α . {Hint: Start with U = which is
sufficient for θ. Show that the distribution of U/α is a multiple of Hence,
-1
2
derive unbiased estimators for α, α and α which depend only on U. Can the
completeness of U be justified with the help of the Theorem 6.6.2?}.
7.5.6 (Exercise 7.5.5 Continued) Find the CRLB for the variance of
unbiased estimators of (i) α, (ii) α and (iii) α . Is the CRLB attained
2
-1
by the variance of the respective UMVUE obtained in the Exercise 7.5.5?