436 Sufficiency(b) ifXis a discrete random variable, thenK(x)is a nontrivial function of
x∈S.For example, each member of the family{f(x;θ):0<θ<∞},wheref(x;θ)
isN(0,θ), represents a regular case of the exponential class of the continuous type
because
f(x;θ)=1
√
2 πθe−x(^2) / 2 θ
=exp
(
−
1
2 θ
x^2 −log
√
2 πθ
)
, −∞<x<∞.
On the other hand, consider the uniform density function given by
f(x;θ)=
{
exp{−logθ} x∈(0,θ)
0elsewhere.
This can be written in the form (7.5.1), but the support is the interval (0,θ), which
depends onθ. Hence the uniform family is not a regular exponential family.
LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution that represents
a regular case of the exponential class. The joint pdf or pmf ofX 1 ,X 2 ,...,Xnis
exp
[
p(θ)
∑n
1
K(xi)+
∑n
1
H(xi)+nq(θ)
]
forxi∈S,i=1, 2 ,...,nand zero elsewhere. At points in theSofX,thisjoint
pdf or pmf may be written as the product of the two nonnegative functions
exp
[
p(θ)
∑n
1
K(xi)+nq(θ)
]
exp
[n
∑
1
H(xi)
]
.
In accordance with the factorization theorem, Theorem 7.2.1,Y 1 =
∑n
1 K(Xi)isa
sufficient statistic for the parameterθ.
Besides the fact thatY 1 is a sufficient statistic, we can obtain the general form
of the distribution ofY 1 and its mean and variance. We summarize these results in
a theorem. The details of the proof are given in Exercises 7.5.5 and 7.5.8. Exercise
7.5.6 obtains the mgf ofY 1 inthecasethatp(θ)=θ.
Theorem 7.5.1.LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution
that represents a regular case of the exponential class, with pdf or pmf given by
(7.5.1). Consider the statisticY 1 =
∑n
i=1K(Xi).Then
- The pdf or pmf ofY 1 has the form
fY 1 (y 1 ;θ)=R(y 1 )exp[p(θ)y 1 +nq(θ)], (7.5.2)fory 1 ∈SY 1 and some functionR(y 1 ). NeitherSY 1 norR(y 1 )depends onθ.2.E(Y 1 )=−nq′(θ)
p′(θ).