Theory of Statistics - George Mason University

2.4 The Exponential Class of Families 173
2.4.1 The Natural Parameter Space of Exponential Families
In the expression for the density, it might be more natural to think of the
parameter as η rather than θ; that way we would have an expression of form
η T T(x) rather than (η(θ)) T T(x). We call the form
pθ(x) = exp (η T T(x) − ζ(η) h(x) (2.11)
the canonical exponential form, and we call
H = {η : e ηTT(x) h(x)dx < ∞} (2.12)
the natural parameter space. (Notice that H is the upper-case form of η.) The
conditions in equation (2.12) are necessary to ensure that a ζ(η) exists such
that pθ(x) is a PDF. Another characterization of H is
H = {η : η = η(θ), θ ∈ Θ}
(under the assumption that Θ is properly defined, of course).
2.4.2 The Natural Exponential Families
An interesting subclass of exponential families is the class of exponential families
in which T(x) in the defining expression (2.7) is linear. This subclass
is variously called the “natural exponential families”, the “linear exponential
families”, or the “canonical exponential families”.
Given a random variable X whose distribution is in any exponential family,
the random variable Y = T(X) has a distribution in the natural exponential
family.
The cumulant-generating and probability-generating functions of natural
exponential families have several simple properties (see Brown (1986) or
Morris and Lock (2009)).
2.4.3 One-Parameter Exponential Families
An important subfamily of exponential families are those in which η(θ) ∈ IR,
that is, those whose parameter is a scalar (or effectively a scalar). This family
is called a one-parameter exponential.
Theorem 2.3
Suppose a PDF p(x|θ) can be written as exp(g(x; θ))h(x). where
g(x; θ) = η(θ)T(x) − ξ(θ),
with η(θ) ∈ IR, and Let x1, x2, x3, x4 be any values of x for which p(x|θ) > 0.
Then a necessary and sufficient condition that the distribution with the given
PDF is in a one-parameter exponential family is that
Theory of Statistics c○2000–2013 James E. Gentle