Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

448 Sufficiency

and equals zero elsewhere. Accordingly, the joint pdf ofX 1 ,X 2 ,...,Xncan be
written, for all points in its support (allxisuch thatθ 1 −θ 2 <xi<θ 1 +θ 2 ),

(

1

2 θ 2

)n

=

n(n−1)[max(xi)−min(xi)]n−^2

(2θ 2 )n

(

1

n(n−1)[max(xi)−min(xi)]n−^2

)

.

Since min(xi)≤xj ≤max(xi),j=1, 2 ,...,n, the last factor does not depend
upon the parameters. Either the definition or the factorization theorem assures us
thatY 1 andYnare joint sufficient statistics forθ 1 andθ 2.

The concept of a complete family of probability density functions is generalized
as follows: Let
{f(v 1 ,v 2 ,...,vk;θ):θ∈Ω}

denote a family of pdfs ofkrandom variablesV 1 ,V 2 ,...,Vkthat depends upon the
p-dimensional vector of parametersθ∈Ω. Letu(v 1 ,v 2 ,...,vk) be a function of
v 1 ,v 2 ,...,vk(but not a function of any or all of the parameters). If

E[u(V 1 ,V 2 ,...,Vk)] = 0

for allθ∈Ω implies thatu(v 1 ,v 2 ,...,vk) = 0 at all points (v 1 ,v 2 ,...,vk), except on
a set of points that has probability zero for all members of the family of probability
density functions, we shall say that the family of probability density functions is a
complete family.
In the case whereθis a vector, we generally consider best estimators of functions
ofθ, that is, parametersδ,whereδ=g(θ) for a specified functiong. For example,
suppose we are sampling from aN(θ 1 ,θ 2 ) distribution, whereθ 2 is the variance. Let
θ√=(θ 1 ,θ 2 )′and consider the two parametersδ 1 =g 1 (θ)=θ 1 andδ 2 =g 2 (θ)=
θ 2. Hence we are interested in best estimates ofδ 1 andδ 2.
The Rao–Blackwell, Lehmann–Scheff ́e theory outlined in Sections 7.3 and 7.4
extends naturally to this vector case. Briefly, supposeδ=g(θ) is the parameter
of interest andYis a vector of sufficient and complete statistics forθ.LetTbe
a statistic that is a function ofY,suchasT=T(Y). IfE(T)=δ,thenTis the
unique MVUE ofδ.
The remainder of our treatment of the case of several parameters is restricted to
probability density functions that represent what we shall call regular cases of the
exponential class. Herem=p.

Definition 7.7.2. Let X be a random variable with pdf or pmff(x;θ),where
the vector of parametersθ∈Ω⊂Rm.LetSdenote the support ofX.IfXis
continuous, assume thatS=(a, b),whereaorbmay be−∞or∞, respectively. If
Xis discrete, assume thatS={a 1 ,a 2 ,...}.Supposef(x;θ)is of the form

f(x;θ)=

{

exp

[∑

m

j=1pj(θ)Kj(x)+H(x)+q(θ^1 ,θ^2 ,...,θm)

]
for allx∈S
0 elsewhere.
(7.7.2)
Then we say this pdf or pmf is a member of theexponential class. We say it is
aregular caseof the exponential family if, in addition,