Page 356 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 333

Find the pmf of X X and then using the fact that X X is independent of X ,
1
1
2
3
2
derive the pmf of U. Hence, find the expressions of I (p), and
X
I (p) where X = (X , X , X ). Show that I (p) < < I (p). This
X
2
1
3
U
U
shows that U or (X X , X ) can not be sufficient for p.
1 2 3
6.4.9 (Exercise 6.2.16 Continued) Suppose that X , ..., X are iid with the
n
1
Rayleigh distribution, that is the common pdf is
where θ(> 0) is the unknown parameter. Denote the statistic .
Evaluate I (θ) and I (θ). Are these two information contents the same? What
X
T
conclusions can one draw from this comparison?
6.4.10 (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.
n
2
Denote the statistic T=Σ X . Evaluate I (θ) and I (θ). Are these two infor-
i=1
i
T
X
mation contents the same? What conclusions can one draw from this com-
parison?
6.4.11 Prove the Theorem 6.4.3. {Hint: One may proceed along the lines
of the proof given in the case of the Theorem 6.4.1.}
6.4.12 Prove the Theorem 6.4.4 when
(i) X, θ are both vector valued, but X has a continuous distribution;
(ii) X, θ are both real valued, but X has a discrete distribution.
6.4.13 (Exercise 6.3.8 Continued) Let X , ..., X be iid N(θ, θ ) where 0 <
2
1
n
θ < ∞ is the unknown parameter. Evaluate I (θ) and , and compare
X
these two quantities. Is it possible to claim that is sufficient for θ?
6.4.14 (Exercise 6.3.9 Continued) Let X , ..., X be iid N(θ, θ) where 0 <
1
n
θ < ∞ is the unknown parameter. Evaluate I (θ) and , and compare
X
these two quantities. Is it possible to claim that is sufficient for θ?
6.4.15 (Proof of the Theorem 6.4.2) Let θ be a real valued parameter.
Suppose that X is the whole data and T = T(X) is some statistic. Then, show
that I (θ) ≥ I (θ) for all θ ∈ Θ. Also verify that the two information mea-
X
T
sures match with each other for all θ if and only if T is a sufficient statis-
tic for θ. {Hint: This interesting idea was included in a recent personal