Page 352 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 329

where b(α, β) = Γ(α)Γ(β){Γ(α + β)} . Refer to (1.7.35) as needed. Use the
–1
Neyman Factorization Theorem to show that
(i) is sufficient for α if β is known;
(ii) is sufficient for β if α is known;
(iii) is jointly sufficient for (α, β) if both
parameters are unknown.
6.2.13 Suppose that X , ..., X are iid with the Uniform distribution on the
1
n
interval (θ - ½, θ + ½), that is the common pdf is given by f(x; θ) = I(θ – ½
< x < θ + ½) where θ(> 0) is the unknown parameter. Show that (X , X ) is
n:n
n:1
jointly sufficient for θ.
6.2.14 Solve the Examples 6.2.8-6.2.11 by applying the Theorem 6.2.2
based on the exponential family.
6.2.15 Prove the Theorem 6.2.2 using the Neyman Fctorization Theorem.
6.2.16 Suppose that X , ..., X are iid with the Rayleigh distribution, that is
1
n
the common pdf is

where θ(> 0) is the unknown parameter. Show that is sufficient for
θ.
6.2.17 Suppose that X , ..., X are iid with the Weibull distribution, that is
n
1
the common pdf is

where α(> 0) is the unknown parameter, but β(> 0) is assumed known that
is sufficient for α.
6.3.1 (Exercise 6.2.10 Continued) Let X , ..., X be iid N(µ, σ ) where –∞
2
1
n
< µ < ∞, 0 < σ < ∞. Show that
(i) is minimal sufficient for µ if σ is known;
(ii) is minimal sufficient for σ if µ is known.
6.3.2 (Exercise 6.2.11 Continued) Let X , ..., X be iid having the common
1
n
pdf σ exp{–(x – µ)/σ}I(x > µ) where –∞ < µ < ∞, 0 < σ < ∞. Show that
–1
(i) X , the smallest order statistic, is minimal sufficient for µ if
n:1
σ is known;
(ii) is minimal sufficient for s if µ is known;
(iii) is minimal sufficient for (µ, σ) if both
parameters are unknown.