Page 414 - Probability and Statistical Inference

7. Point Estimation 391

(i) Find a suitable unbiased estimator T for T(p);
(ii) Since the complete sufficient statistic is use the
Lehmann-Scheffé theorems and evaluate the conditional expec
tation, E [T | U = u];
p
(iii) Hence, derive the UMVUE for T(p).
{Hint: Try and use the mgf of the X’s appropriately.}
7.5.13 Suppose that X , ..., X are iid Bernoulli(p) where 0 < p < 1 is an
n
1
unknown parameter. Consider the parametric function T(p) = (p+qe ) with
3 2
q = 1 - p.
(i) Find a suitable unbiased estimator T for T for T(p);
(ii) Since the complete sufficient statistic is use the
Lehmann-Scheffé theorems and evaluate the conditional expec
tation, E [T | U = u];
p
(iii) Hence, derive the UMVUE for T(p).
{Hint: Try and use the mgf of the X’s appropriately.}
7.5.14 Suppose that X , ..., X are iid N(µ, σ ) where µ is unknown but σ
2
n
1
is assumed known with µ ∈ ℜ, σ ∈ ℜ . Consider the parametric function T(µ)
+
2µ
= e . Derive the UMVUE for T(µ). Is the CRLB attained in this problem?
2
7.5.15 Suppose that X , ..., X are iid N(µ, σ ) where µ and σ are both
n
1
assumed unknown with µ ∈ ℜ, σ ∈ ℜ , θθ θθ θ = (µ, σ), n ≥ 2. Consider the
+
parametric function T(θθ θθ θ) = e 2(µ+σ2) . Derive the UMVUE for T(θθ θθ θ).
7.5.16 Prove the Theorem 7.5.4 by appropriately modifying the lines of
proof given for the Theorem 7.5.1.
7.5.17 Let X , ..., X be iid N(0, σ ), Y , ..., Y be iid N(2, 3σ ) where σ is
2
2
m
n
1
1
unknown with σ ∈ ℜ . Assume also that the X’s are independent of the Y’s.
+
Derive the UMVUE for σ and check whether the CRLB given by the Theo-
2
rem 7.5.4 is attained in this case.
7.5.18 Suppose that X , ..., X are iid Gamma(2, β), Y , ..., Y are iid
n
m
1
1
+
Gamma(4, 3β) where β is unknown with β ∈ ℜ . Assume also that the X’s
are independent of the Y’s. Derive the UMVUE for β and check whether the
CRLB given by the Theorem 7.5.4 is attained in this case.
7.5.19 Suppose that X is distributed as N(0, σ ), Y has its pdf given by g(y;
2
2
2
+
2 -1
σ ) = (2σ ) exp{− |y| /σ }I(y ∈ ℜ) where σ is unknown with σ ∈ ℜ . Assume
also that the X is independent of the Y. Derive the UMVUE for σ and check
2
whether the CRLB given by the Theorem 7.5.4 is attained in this case.
7.5.20 Let X , ..., X be iid having the common pdf σ exp{ −(x
-1
n
1
− µ)/σ}I(x > µ) where µ is known but σ is unknown with −∞ < µ <