Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.7. The Case of Several Parameters 449

1. the support does not depend on the vector of parametersθ,


2. the spaceΩcontains a nonempty,m-dimensional open rectangle,


3. thepj(θ),j=1,...,m, are nontrivial, functionally independent, continuous
   functions ofθ,


4. 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

)

.