Page 152 - Elements of Distribution Theory

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

138 Parametric Families of Distributions

Example 5.10 (Poisson distributions). As in Example 5.5, let P denote the set of all
Poisson distributions with mean λ> 0; the model function is therefore given by
x
p(x; λ) = λ exp(−λ)/x!, x = 0, 1,....
This can be written
1
exp{x log(λ) − λ} , x ∈{0, 1, 2,...};
x!
hence, this is a one-parameter exponential family with c(λ) = log(λ), T (x) = x, A(λ) = λ,
h(x) = 1/x!, and X ={0, 1, 2,...}.

One important consequence of the exponential family structure is that if Y 1 , Y 2 ,..., Y n
are independent random variables such that the marginal distribution of each Y j has model
function of the form (5.1), then the model function for Y = (Y 1 ,..., Y n )is also of the form
(5.1). The situation is particularly simple if Y 1 ,..., Y n are identically distributed.

Example 5.11 (Exponential distributions). Let Y denote a random variable with density
function
1
exp(−y/θ), y > 0
θ
where θ> 0; this is an exponential distribution with mean θ. This density function may be
written in the form (5.1) with T (y) = y, c(θ) =−1/θ, A(θ) = log θ, and h(y) = 1. Hence,
this is a one-parameter exponential family distribution.
Now suppose that Y = (Y 1 ,..., Y n ) where Y 1 ,..., Y n are independent random variables,
each with an exponential distribution with mean θ. Then the model function for Y is given
by
n
1 1
exp − y j .
θ n θ
j=1
n
This is of the form (5.1) with T (y) = y j , c(θ) =−1/θ, A(θ) = n log θ, and
j=1
h(y) = 1; it follows that the distribution of Y is also a one-parameter exponential family
distribution.
Natural parameters
It is often convenient to reparameterize the models in order to simplify the structure of the
exponential family representation. For instance, consider the reparameterization η = c(θ)
so that the model function (5.1) becomes
T
exp{η T (y) − A[θ(η)]}h(y), y ∈ Y.
Writing k(η) for A[θ(η)], the model function has the form
T
exp{η T (y) − k(η)}h(y), y ∈ Y; (5.2)
the parameter space of the model is given by

m
H 0 ={η ∈ R : η = c(θ),θ ∈ }.