7.8. Minimal Sufficiency and Ancillary Statistics 459Example 7.8.6(Location- and Scale-Invariant Statistics).Finally, consider a ran-
dom sampleX 1 ,X 2 ,...,Xnthat follows a location and scale model as in Example
7.7.5. That is,
Xi=θ 1 +θ 2 Wi,i=1,...,n, (7.8.5)
whereWiare iid with the common pdff(t)whichisfreeofθ 1 andθ 2 .Inthiscase,
the pdf ofXiisθ− 21 f((x−θ 1 )/θ 2 ). Consider the statisticZ=u(X 1 ,X 2 ,...,Xn),
where
u(cx 1 +d,...,cxn+d)=u(x 1 ,...,xn).
Then
Z=u(X 1 ,...,Xn)=u(θ 1 +θ 2 W 1 ,...,θ 1 +θ 2 Wn)=u(W 1 ,...,Wn).Since neither the joint pdf ofW 1 ,...,WnnorZcontainsθ 1 andθ 2 , the distribution
ofZmust not depend uponθ 1 norθ 2. Statistics such asZ=u(X 1 ,X 2 ,...,Xn)are
calledlocation- and scale-invariant statistics. The following are four examples
of such statistics:
(a) T 1 =[max(Xi)−min(Xi)]/S;
(b) T 2 =
∑n− 1
i=1(Xi+1−Xi)(^2) /S (^2) ;
(c)T 3 =(Xi−X)/S;
(d) T 4 =|Xi−Xj|/S,,;i =j.
LetX−θ 1 =n−^1
∑n
i=1(Xi−θ^1 ). Then the location and scale invariance of the
statistic in (d) follows from the two identities
S^2 = θ^22
∑n
i=1
[
Xi−θ 1
θ 2
−
X−θ 1
θ 2
] 2
=θ^22
∑n
i=1
(Wi−W)^2
Xi−Xj = θ 2
[
Xi−θ 1
θ 2
−
Xj−θ 1
θ 2
]
=θ 2 (Wi−Wj).
See Exercise 7.8.6 for the other statistics.
Thus, these location-invariant, scale-invariant, and location- and scale-invariant
statistics provide good illustrations, with the appropriate model for the pdf, of an-
cillary statistics. Since an ancillary statistic and a complete (minimal) sufficient
statistic are such opposites, we might believe that there is, in some sense, no rela-
tionship between the two. This is true, and in the next section we show that they
are independent statistics.
EXERCISES
7.8.1.LetX 1 ,X 2 ,...,Xnbe a random sample from each of the following distribu-
tions involving the parameterθ. In each case find the mle ofθand show that it is
a sufficient statistic forθand hence a minimal sufficient statistic.