Page 354 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 331

belong to the exponential family, it should be a 2-parameter family, but it is not
so. Does the parameter space include a 2-dimensional rectangle?}

6.3.11 Let X , ..., X be iid having a negative exponential distribution with
1 n
–2
the common pdf θ exp{–(x – θ)/θ }I(x > θ) where 0 < θ < ∞ is the unknown
2
parameter. Derive the minimal sufficient statistics for θ. Does the common
pdf belong to the exponential family (6.3.5)? {Hint: If it does belong to the
exponential family, it should be a 2-parameter family, but it is not so. Does the
parameter space include a 2-dimensional rectangle?}

6.3.12 Let X , ..., X be iid having the common Uniform distribution on the
1
n
interval (–θ, θ) where 0 < θ < ∞ is the unknown parameter. Derive the mini-
mal sufficient statistic for θ.
2
6.3.13 Let X , ..., X be iid N(µ , σ ), Y , ..., Y be iid N(µ , σ ), and also let
2
1
n
2
1
m
1
the X’s be independent of the Y’s where –∞ < µ , µ < ∞, 0 < σ < ∞ are the
1 2
unknown parameters. Derive the minimal sufficient statistics for (µ , µ , σ ).
2
1 2
6.3.14 (Exercise 6.3.13 Continued) Let X , ..., X be iid N(µ , σ ), Y , ...,
2
m
1
1
1
Y be iid N(µ , kσ ), and also let the X’s be independent of the Y’s where –∞
2
n
2
< µ , µ < ∞, 0 < σ < ∞ are the unknown parameters. Assume that the number
2
1
k (> 0) is known. Derive the minimal sufficient statistic for (µ , µ , σ ).
2
1 2
6.3.15 Let X , ..., X be iid Gamma(α, β), Y , ..., Y be iid Gamma(α, kβ),
1 m 1 n
and also let the X’s be independent of the Y’s with 0 < α, β < ∞ where β is the
only unknown parameter. Assume that the number k (> 0) is known. Derive
the minimal sufficient statistic for β.
6.3.16 (Exercise 6.2.12 Continued) Let X , ..., X be iid having a Beta
n
1
distribution with its parameters α = β = θ where θ (> 0) is unknown. Derive
the minimal sufficient statistic for θ.
6.3.17 Let X , ..., X be iid N(µ, σ ), Y , ..., Y be iid N(0, σ ), and also let
2
2
1
n
m
1
the X’s be independent of the Y’s where –∞ < µ < ∞, 0 < σ < ∞ are the
unknown parameters. Derive the minimal sufficient statistics for (µ, σ ).
2
2
2
6.3.18 Let X , ..., X be iid N(µ, σ ), Y , ..., Y be iid N(0, kσ ), and also let
n
1
m
1
the X’s be independent of the Y’s where –∞ < µ < ∞, 0 < σ < ∞ are the
unknown parameters. Assume that the number k (> 0) is known. Derive the
minimal sufficient statistics for (µ, σ ).
2
6.3.19 Suppose that X , ..., X are iid with the common pdf given by one
n
1
of the following: