Page 154 - Elements of Distribution Theory

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

140 Parametric Families of Distributions

with probability 1 under the distribution of Y with parameter η 0 .It follows that η is not
identifiable if and only if, with probability 1 under the distribution with parameter η 0 ,
T
(η 1 − η 2 ) T (Y) = k(η 2 ) − k(η 1 ).
That is, η is not identifiable if and only if T (Y) has a degenerate distribution under the
distribution with parameter η 0 . Since η 0 is arbitrary, it follows that η is not identifiable if and
only if T (Y) has a degenerate distribution under all η 0 ∈ H. Equivalently, η is identifiable
if and only if T (Y) has a nondegenerate distribution for some η 0 ∈ H.

An important property of the natural parameter space H is that it is convex; also, the
function k is a convex function.

Theorem 5.1. Consider an m-dimensional exponential family of probability distributions
T
exp{η T (y) − k(η)}h(y), y ∈ Y.
Then the natural parameter space H is a convex set and k is a convex function.

Proof. We give the proof for the case in which the distribution is absolutely continuous.
Let η 1 and η 2 denote elements of H and let 0 < t < 1. By the H¨older inequality,

T T
exp tη +(1 − t)η T (y) h(y) dy
1 2
Y

T T
= exp tη T (y) exp (1 − t)η T (y) h(y) dy
1
2
Y

t (1−t)
T T
= exp η T (y) exp η T (y) h(y) dy
2
1
Y
t (1−t)

T T
≤ exp η T (y) h(y) dy exp η T (y) h(y) dy < ∞.
1 2
Y Y
It follows that tη 1 + (1 − t)η 2 ∈ H and, hence, that H is convex. Furthermore,
exp{k(tη 1 + (1 − t)η 2 )}≤ exp{tk(η 1 ) + (1 − t)k(η 2 )},
proving that k is a convex function.
The function k is called the cumulant function of the family. This terminology is based
on the fact that if the natural parameter space is open set, in which case the exponential
family is said to be regular, then the cumulant-generating function of T (Y) may be written
in terms of k.
Theorem 5.2. Let Y denote a random variable with model function of the form
T
exp{η T (y) − k(η)}h(y), y ∈ Y,
where η ∈ H and H is an open set. Then the cumulant-generating function of T (Y) under
the distribution with parameter η ∈ H is given by
m
K T (t; η) = k(η + t) − k(η), t ∈ R , t <δ
for some δ> 0.