Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.2. A Sufficient Statistic for a Parameter 425

Example 7.2.6.In Example 7.2.3 withf(x;θ)=e−(x−θ)I(θ,∞)(x), it was found
that the first order statisticY 1 is a sufficient statistic forθ. To illustrate our point
about not considering the domain of the function, taken= 3 and note that

e−(x^1 −θ)e−(x^2 −θ)e−(x^3 −θ)=[e−3maxxi+3θ][e−x^1 −x^2 −x^3 +3 maxxi]

or a similar expression. Certainly, in the latter formula, there is noθin the second
factor and it might be assumed thatY 3 =maxXiis a sufficient statistic forθ.Of
course, this is incorrect because we should have written the joint pdf ofX 1 ,X 2 ,X 3
as
∏^3

i=1

[e−(xi−θ)I(θ,∞)(xi)] = [e^3 θI(θ,∞)(minxi)]

[

exp

{

−

∑^3

i=1

xi

}]

becauseI(θ,∞)(minxi)=I(θ,∞)(x 1 )I(θ,∞)(x 2 )I(θ,∞)(x 3 ). A similar statement can-
not be made with maxxi.ThusY 1 =minXiis the sufficient statistic forθ,not
Y 3 =maxXi.

EXERCISES

7.2.1.LetX 1 ,X 2 ,...,Xnbe iidN(0,θ), 0 <θ<∞. Show that

∑n

1 X

2
i is a
sufficient statistic forθ.

7.2.2.Prove that the sum of the observations of a random sample of sizenfrom a

Poisson distribution having parameterθ, 0 <θ<∞, is a sufficient statistic forθ.

7.2.3.Show that thenth order statistic of a random sample of sizenfrom the
uniform distribution having pdff(x;θ)=1/θ, 0 <x<θ, 0 <θ<∞, zero
elsewhere, is a sufficient statistic forθ. Generalize this result by considering the pdf
f(x;θ)=Q(θ)M(x), 0 <x<θ, 0 <θ<∞, zero elsewhere. Here, of course,

∫θ

0

M(x)dx=

1

Q(θ)

.

7.2.4.LetX 1 ,X 2 ,...,Xnbe a random sample of sizenfrom a geometric distribu-
tion that has pmff(x;θ)=(1−θ)xθ, x=0, 1 , 2 ,..., 0 <θ<1, zero elsewhere.
Show that

∑n

1 Xiis a sufficient statistic forθ.

7.2.5. Show that the sum of the observations of a random sample of sizenfrom
a gamma distribution that has pdff(x;θ)=(1/θ)e−x/θ, 0 <x<∞, 0 <θ<∞,
zero elsewhere, is a sufficient statistic forθ.

7.2.6.LetX 1 ,X 2 ,...,Xnbe a random sample of sizenfrom a beta distribution
with parametersα=θandβ= 5. Show that the productX 1 X 2 ···Xnis a sufficient
statistic forθ.

7.2.7.Show that the product of the sample observations is a sufficient statistic for

θ>0 if the random sample is taken from a gamma distribution with parameters

α=θandβ=6.