Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.1. Location Models 571

Next, supposeY=aX.Ifa>0, thenFY(y)=FX(y/a) and, hence,

FY(aT(FX)) =FX(aT(FX)/a)=FX(T(FX)) = 1/ 2.

ThusT(FY)=aT(FX)whena>0. On the other hand, ifa<0, thenFY(y)=
1 −FX(y/a). Hence

FY(aT(FX)) = 1−FX(aT(FX)/a)=1−FX(T(FX)) = 1−

1

2

=

1

2

.

Therefore, (10.1.3) holds for alla = 0. Thus the median is a location functional.
Recall that the median is a percentile, namely, the 50th percentile of a distribu-
tion. As Exercise 10.1.1 shows, the median is the only percentile that is a location
functional.

We often continue to use parameter notation to denote functionals. For example,
θX=T(FX).
In Chapters 4 and 6, we wrote the location model for specified pdfs. In this
chapter, we write it for a general pdf in terms of a specified location functional.
LetXbe a random variable with cdfFX(x)andpdffX(x). LetθX=T(FX)bea
location functional. Define the random variableεto beε=X−T(FX). Then by
(10.1.2),T(Fε) = 0; i.e.,εhas location 0, according toT. Further, the pdf ofX
canbewrittenasfX(x)=f(x−T(FX)), wheref(x)isthepdfofε.

Definition 10.1.2(Location Model).LetθX=T(FX)be a location functional. We
say that the observationsX 1 ,X 2 ,...,Xnfollow alocation modelwith functional
θX=T(FX)if
Xi=θX+εi, (10.1.4)

whereε 1 ,ε 2 ,...,εnare iid random variables with pdff(x)andT(Fε)=0.Hence,
from the above discussion,X 1 ,X 2 ,...,Xnare iid with pdffX(x)=f(x−T(FX)).

Example 10.1.2.Letεbe a random variable with cdfF(x), such thatF(0) = 1/2.

Assume thatε 1 ,ε 2 ,...,εnare iid with cdfF(x). Letθ∈Rand define

Xi=θ+εi,i=1, 2 ,...,n.

ThenX 1 ,X 2 ,...,Xnfollow the location model with the locational functionalθ,

which is the median ofXi.

Note that the location model very much depends on the functional. It forces one
to state clearly which location functional is being used in order to write the model
statement. For the class of symmetric densities, though, all location functionals are
the same.

Theorem 10.1.1.LetXbe a random variable with cdfFX(x)and pdffX(x)such
that the distribution ofXis symmetric abouta.LetT(FX)be any location func-
tional. ThenT(FX)=a.