7.7. The Case of Several Parameters 449
- the support does not depend on the vector of parametersθ,
- the spaceΩcontains a nonempty,m-dimensional open rectangle,
- thepj(θ),j=1,...,m, are nontrivial, functionally independent, continuous
functions ofθ, - and, depending on whetherXis continuous or discrete, one of the following
holds, respectively:
(a) ifXis a continuous random variable, then themderivativesKj′(x),for
j=1, 2 ,...,m, are continuous fora<x<band no one is a linear
homogeneous function of the others, andH(x)is a continuous function
ofx, a < x < b.
(b) ifX is discrete, theKj(x),j=1, 2 ,...,m, are nontrivial functions of
xon the supportSand no one is a linear homogeneous function of the
others.
LetX 1 ,...,Xnbe a random sample onXwhere the pdf or pmf ofXis a regular
case of the exponential class with the same notation as in Definition 7.7.2. It follows
from (7.7.2) that the joint pdf or pmf of the sample is given by
∏n
i=1
f(xi;θ)=exp
⎡
⎣
∑m
j=1
pj(θ)
∑n
i=1
Kj(xi)+nq(θ)
⎤
⎦exp
[n
∑
i=1
H(xi)
]
, (7.7.3)
for allxi∈S. In accordance with the factorization theorem, the statistics
Y 1 =
∑n
i=1
K 1 (xi),Y 2 =
∑n
i=1
K 2 (xi),...,Ym=
∑n
i=1
Km(xi)
are joint sufficient statistics for them-dimensional vector of parametersθ.Itisleft
as an exercise to prove that the joint pdf ofY=(Y 1 ,...,Ym)′is of the form
R(y)exp
⎡
⎣
∑m
j=1
pj(θ)yj+nq(θ)
⎤
⎦, (7.7.4)
at points of positive probability density. These points of positive probability density
and the functionR(y) do not depend upon the vector of parametersθ.Moreover,
in accordance with a theorem in analysis, it can be asserted that in a regular case
of the exponential class, the family of probability density functions of these joint
sufficient statisticsY 1 ,Y 2 ,...,Ymis complete whenn>m. In accordance with a
convention previously adopted, we shall refer toY 1 ,Y 2 ,...,Ymasjoint complete
sufficient statisticsfor the vector of parametersθ.
Example 7.7.2.LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution
that isN(θ 1 ,θ 2 ),−∞<θ 1 <∞, 0 <θ 2 <∞.Thusthepdff(x;θ 1 ,θ 2 )ofthe
distribution may be written as
f(x;θ 1 ,θ 2 )=exp
(
− 1
2 θ 2
x^2 +
θ 1
θ 2
x−
θ^21
2 θ 2
−ln
√
2 πθ 2
)
.