Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

450 Sufficiency

Therefore, we can takeK 1 (x)=x^2 andK 2 (x)=x. Consequently, the statistics

Y 1 =

∑n

1

X^2 i and Y 2 =

∑n

1

Xi

are joint complete sufficient statistics forθ 1 andθ 2. Since the relations

Z 1 =

Y 2

n

=X, Z 2 =

Y 1 −Y 22 /n

n− 1

=

∑

(Xi−X)^2

n− 1

define a one-to-one transformation,Z 1 andZ 2 are also joint complete sufficient
statistics forθ 1 andθ 2 .Moreover,

E(Z 1 )=θ 1 and E(Z 2 )=θ 2.

From completeness, we have thatZ 1 andZ 2 are the only functions ofY 1 andY 2
that are unbiased estimators ofθ 1 andθ 2 , respectively. HenceZ 1 andZ 2 are the
unique minimum variance estimators ofθ 1 andθ 2 , respectively. The MVUE of the
standard deviation

√

θ 2 is derived in Exercise 7.7.5.

In this section we have extended the concepts of sufficiency and completeness
to the case whereθis ap-dimensional vector. We now extend these concepts to
the case whereXis ak-dimensional random vector. We only consider the regular
exponential class.
SupposeXis ak-dimensional random vector with pdf or pmff(x;θ), where
θ∈Ω⊂Rp.LetS⊂Rkdenote the support ofX. Supposef(x;θ)isoftheform

f(x;θ)=

{

exp

[∑

m

j=1pj(θ)Kj(x)+H(x)+q(θ)

]

for allx∈S

0elsewhere.

(7.7.5)

Then we say this pdf or pmf is a member of theexponential class. If, in addition,
p=m, the support does not depend on the vector of parametersθ, and conditions
similar to those of Definition 7.7.2 hold, then we say this pdf is aregular caseof
the exponential class.
Suppose thatX 1 ,...,Xnconstitute a random sample onX. Then the statistics,

Yj=

∑n

i=1

Kj(Xi), forj=1,...,m, (7.7.6)

are sufficient and complete statistics forθ.LetY=(Y 1 ,...,Ym)′. Supposeδ=g(θ)
is a parameter of interest. IfT=h(Y) for some functionhandE(T)=δthenT
is the unique minimum variance unbiased estimator ofδ.

Example 7.7.3 (Multinomial). In Example 6.4.5, we consider the mles of the
multinomial distribution. In this example we determine the MVUEs of several of
the parameters. As in Example 6.4.5, consider a random trial that can result in one,
and only one, ofkoutcomes or categories. LetXjbe 1 or 0 depending on whether
thejth outcome does or does not occur, forj=1,...,k. Suppose the probability