Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.9. Sufficiency, Completeness, and Independence 463

complete sufficient statistic forθ. HenceY 1 must be independent of each location-

invariant statisticu(X 1 ,X 2 ,...,Xn), enjoying the property that

u(x 1 +d, x 2 +d,...,xn+d)=u(x 1 ,x 2 ,...,xn)

for all reald. Illustrations of such statistics areS^2 , the sample range, and

1

n

∑n

i=1

[Xi−min(Xi)].

Example 7.9.3.LetX 1 ,X 2 denote a random sample of sizen= 2 from a distri-
bution with pdf

f(x;θ)=

1

θ

e−x/θ, 0 <x<∞, 0 <θ<∞,

= 0 elsewhere.

Thepdfisoftheform(1/θ)f(x/θ), wheref(w)=e−w, 0 <w<∞, zero else-
where. We know thatY 1 =X 1 +X 2 is a complete sufficient statistic forθ. Hence,
Y 1 is independent of every scale-invariant statisticu(X 1 ,X 2 ) with the property
u(cx 1 ,cx 2 )=u(x 1 ,x 2 ). Illustrations of these areX 1 /X 2 andX 1 /(X 1 +X 2 ), statis-
tics that haveF- and beta distributions, respectively.

Example 7.9.4.LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution
that isN(θ 1 ,θ 2 ),−∞<θ 1 <∞, 0 <θ 2 <∞. In Example 7.7.2 it was proved
that the meanXand the varianceS^2 of the sample are joint complete sufficient
statistics forθ 1 andθ 2. Consider the statistic

Z=

n∑− 1

1

(Xi+1−Xi)^2

∑n

1

(Xi−X)^2

=u(X 1 ,X 2 ,...,Xn),

which satisfies the property thatu(cx 1 +d,...,cxn+d)=u(x 1 ,...,xn). That is,
the ancillary statisticZis independent of bothXandS^2.

In this section we have given several examples in which the complete sufficient
statistics are independent of ancillary statistics. Thus, in those cases, the ancillary
statistics provide no information about the parameters. However, if the sufficient
statistics are not complete, the ancillary statistics could provide some information
as the following example demonstrates.

Example 7.9.5.We refer back to Examples 7.8.1 and 7.8.2. There the first and
nth order statistics,Y 1 andYn, were minimal sufficient statistics forθ,wherethe
sample arose from an underlying distribution having pdf (^12 )I(θ− 1 ,θ+1)(x). Often
T 1 =(Y 1 +Yn)/2 is used as an estimator ofθ, as it is a function of those sufficient
statistics that is unbiased. Let us find a relationship betweenT 1 and the ancillary
statisticT 2 =Yn−Y 1.