Page 353 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 330

6.3.3 Solve the Exercises 6.2.5-6.2.8 along the lines of the Examples 6.3.2
and 6.3.5.
6.3.4 Show that the pmf or pdf corresponding to the distributions such as
Binomial(n, p), Poisson(λ), Gamma(α, β), N(µ, σ ), Beta(α, β) belong to the
2
exponential family defined in (6.3.5) when
(i) 0 < p < 1 is unknown;
(ii) λ ∈ ℜ is unknown;
+
(iii) µ ∈ ℜ is unknown but σ ∈ ℜ is known;
+
+
(iv) σ ∈ ℜ is unknown but µ ∈ ℜ is known;
+
(v) µ ∈ ℜ and σ ∈ ℜ are both unknown;
+
(vi) α ∈ ℜ is known but β ∈ ℜ is unknown;
+
(vii) α ∈ ℜ , β ∈ ℜ are both unknown.
+
+
In each case, obtain the minimal sufficient statistic(s) for the associated
unknown parameter(s).
6.3.5 (Exercise 6.2.1 Continued) Let X , ..., X be Geometric(p), that is the
1
n
common pmf is given by f(x;p) = p(1 - p) , x = 0, 1, 2, ... where 0 < p < 1 is the
x
unknown parameter. Show that this pmf belongs to the exponential family de-
fined in (6.3.5). Hence, show that is minimal sufficient for p.
6.3.6. Show that the common pdf given in the Exercises 6.2.16-6.2.17
respectively belongs to the exponential family.
6.3.7 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 minimal sufficient for θ.
6.3.8 Let X , ..., X be iid N(θ, θ ) where 0 < θ < ∞ is the unknown
2
n
1
parameter. Derive the minimal sufficient statistic for ∞. 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?}
6.3.9 Let X , ..., X be iid N(θ, θ) where 0 < θ < ∞ is the unknown param-
1
n
eter. Derive the minimal sufficient statistic for θ. Does the common pdf be-
long to the exponential family (6.3.5)? {Hint: If it does belong to the exponen-
tial family, it should be a 2-parameter family, but it is not so. Does the param-
eter space include a 2-dimensional rectangle?}
6.3.10 Let X , ..., X be iid having a negative exponential distribution
1
n
with the common pdf θ exp{–(x – θ)/θ}I(x > θ) where 0 < θ < ∞ is the
–1
unknown parameter. Derive the minimal sufficient statistics for θ. Does
the common pdf belong to the exponential family (6.3.5)? {Hint: If it does