Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

458 Sufficiency

for all reald. Hence

Z=u(W 1 +θ, W 2 +θ,...,Wn+θ)=u(W 1 ,W 2 ,...,Wn)

is a function ofW 1 ,W 2 ,...,Wnalone (not ofθ). HenceZmust have a distribution
that does not depend uponθ.WecallZ=u(X 1 ,X 2 ,...,Xn)alocation-invariant
statistic.
Assuming a location model, the following are some examples of location-invariant
statistics: the sample variance =S^2 , the sample range = max{Xi}−min{Xi},the
mean deviation from the sample median = (1/n)

∑
|Xi−median(Xi)|,X 1 +X 2 −
X 3 −X 4 ,X 1 +X 3 − 2 X 2 ,(1/n)

∑

[Xi−min(Xi)], and so on. To see that the range

is location-invariant, note that

max{Xi}−θ =max{Xi−θ}=max{Wi}

min{Xi}−θ =min{Xi−θ}=min{Wi}.

So,

range = max{Xi}−min{Xi}=max{Xi}−θ−(min{Xi}−θ)=max{Wi}−min{Wi}.

Hence the distribution of the range only depends on the distribution of theWis
and, thus, it is location-invariant. For the location invariance of other statistics, see
Exercise 7.8.4.

Example 7.8.5(Scale-Invariant Statistics).Consider a random sampleX 1 ,...,Xn

that follows ascale model, i.e., a model of the form

Xi=θWi,i=1,...,n, (7.8.4)

whereθ>0andW 1 ,W 2 ,...,Wnare iid random variables with pdff(w), which
does not depend onθ. Then the common pdf ofXiisθ−^1 f(x/θ). We callθascale
parameter. Suppose thatZ=u(X 1 ,X 2 ,...,Xn) is a statistic such that

u(cx 1 ,cx 2 ,...,cxn)=u(x 1 ,x 2 ,...,xn)

for allc>0. Then

Z=u(X 1 ,X 2 ,...,Xn)=u(θW 1 ,θW 2 ,...,θWn)=u(W 1 ,W 2 ,...,Wn).

Since neither the joint pdf ofW 1 ,W 2 ,...,WnnorZcontainsθ, the distribution of
Zmust not depend uponθ.WesaythatZis ascale-invariant statistic.
The following are some examples of scale-invariant statistics:X 1 /(X 1 +X 2 ),
X 12 /

∑n

1 X

2
i,min(Xi)/max(Xi), and so on. The scale invariance of the first statistic
follows from
X 1
X 1 +X 2

=

(θX 1 )/θ

[(θX 1 )+(θX 2 )]/θ

=

W 1

W 1 +W 2

.

The scale invariance of the other statistics is asked for in Exercise 7.8.5.