Page 360 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 337

that X and are independent. {Hint: Can the Example 4.4.12
n:1
be used here to solve the second part?}
6.6.6 Let X , ..., X be iid having a negative exponential distribution with
n
1
–2
the common pdf θ exp{–(x – θ)/θ }I(x > θ) where 0 < θ < ∞ is the unknown
2
parameter. Is the minimal sufficient statistic complete? Show that X and
n:1
are independent. {Hint: Can the Example 4.4.12 be used
here to solve the second part?}
6.6.7 Let X , ..., X be iid having the common Uniform distribution on the
1
n
interval (θ – ½, θ + ½) where –∞ < θ < ∞ is the unknown parameter. Is the
minimal sufficient statistic T = (X , X ) complete?
n:1 n:n
6.6.8 Let X , ..., X be iid having the common Uniform distribution on the
n
1
interval (–θ, θ) where 0 < θ < ∞ is the unknown parameter. Find the minimal
sufficient statistic T for θ. Is the statistic T complete? {Hint: Use indicator
functions appropriately so that the problem reduces to the common pdf’s of
the random variables | X |, i = 1, ..., n.}
i
6.6.9 Let X , ..., X be iid N(µ , σ ), Y , ..., Y be iid N(µ , σ ), the X’s are
2
2
1
m
2
n
1
1
independent of the Y’s where –∞ < µ , µ < ∞, 0 < σ < ∞ are all unknown
1
2
parameters. Is the minimal sufficient statistic for (µ , µ , σ ) complete? {Hint:
2
2
1
Is it possible to use the Theorem 6.6.2 here?}
6.6.10 (Exercise 6.3.13 Continued) Let X , ..., X be iid N(µ , σ ), Y , ...,
2
m
1
1
1
2
Y be iid N(µ , kσ ), the X’s be independent of the Y’s where –∞ < µ , µ < ∞,
2
2
n
1
0 < σ < ∞ are all unknown parameters, but k (> 0) is known. Is the minimal
2
sufficient statistic for (µ , µ , σ ) complete? {Hint: Is it possible to use the
2
1
Theorem 6.6.2 here?}
6.6.11 (Exercise 6.3.15 Continued) Let X , ..., X be iid Gamma(α, β), Y ,
m
1
1
..., Y be iid Gamma(α, kβ), the X’s be independent of the Y’s, with 0 < α, β
n
< ∞ where β is the only unknown parameter. Assume that the number k (> 0)
is known. Is the minimal sufficient statistic for β complete? {Hint: Is it pos-
sible to use the Theorem 6.6.2 here?}
6.6.12 (Exercise 6.3.16 Continued) Let X , ..., X be iid having a Beta
n
1
distribution with its parameters α = β = θ where θ (> 0) is unknown. Is the
minimal sufficient statistic for θ complete? {Hint: Is it possible to use the
Theorem 6.6.2 here?}
6.6.13 (Exercise 6.6.9 Continued) Let the X’s be independent of the
Y’s, X , ..., X be iid N(µ , σ ), Y , ..., Y be iid N(µ , σ ) where –∞ < µ ,
2
2
1
1
m
1
2
n
1
µ < ∞, 0 < σ < ∞ are all unknown parameters. Use Basu’s Theorem
2
along the lines of the Example 6.6.15 to show that is distributed