Page 158 - Elements of Distribution Theory

P1: JZP
052184472Xc05 CUNY148/Severini May 24, 2005 17:53

144 Parametric Families of Distributions

so that

E[Zh(T ); η] = E[g(Y)h(T ); η]
for all bounded h. Hence, by Theorem 2.6,

Z ≡ E[g(Y)|T ; η 0 ] = E[g(Y)|T ; η],
proving the result.

Theorem 5.4. Let Y denote a random variable with model function of the form
T
exp{c(θ) T (y) − A(η)}h(y), y ∈ Y,
m
where θ ∈ and c : → R . Let
H 0 ={η ∈ H: η = c(θ),θ ∈ },
where H denotes the natural parameter space of the exponential family, and let Z denote
areal-valued function on Y.
(i) If Z and T (Y) are independent, then the distribution of Z does not depend on θ ∈ .
m
(ii) If H 0 contains an open subset of R and the distribution of Z does not depend on
θ ∈ , then Z and T (Y) are independent.

Proof. We begin by reparameterizing the model in terms of the natural parameter η = c(θ)
so that the model function can be written
T
exp{η T (y) − k(η)}h(y), y ∈ Y,
with parameter space H 0 .
Suppose that Z and T (Y) are independent. Define
ϕ(t; η) = E[exp(it Z); η], t ∈ R,η ∈ H.

Then, by Lemma 5.2, for any η 0 ∈ H
T
ϕ(t; η) = exp{k(η 0 ) − k(η)}E[exp(it Z)exp{(η − η 0 ) T (Y)}; η 0 ], t ∈ R,η ∈ H.
Since Z and T (Y) are independent,
T
ϕ(t; η) = exp{k(η 0 ) − k(η)}E[exp(it Z); η 0 ]E[exp{(η − η 0 ) T (Y)}; η 0 ], t ∈ R,η ∈ H.
Since
T
E[exp{(η − η 0 ) T (Y)}; η 0 ] = exp{k(η) − k(η 0 )}
and E[exp(it Z); η 0 ] = ϕ(t; η 0 ), it follows that, for all η, η 0 ∈ H,

ϕ(t; η) = ϕ(t; η 0 ), t ∈ R
so that the distribution of Z does not depend on η ∈ H and, hence, it does not depend on
η ∈ H 0 . This proves (i).
Now suppose that the distribution of Z does not depend on η ∈ H 0 and that there exists
m
a subset of H 0 , H 1 , such that H 1 is an open subset of R . Fix η 0 ∈ H 1 and let g denote