Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

572 Nonparametric and Robust Statistics

Proof:By (10.1.2), we have

T(FX−a)=T(FX)−a. (10.1.5)

Since the distribution ofXis symmetric abouta, it is easy to show thatX−aand
−(X−a) have the same distribution; see Exercise 10.1.2. Hence, using (10.1.2) and
(10.1.3), we have

T(FX−a)=T(F−(X−a))=−(T(FX)−a)=−T(FX)+a. (10.1.6)

Putting (10.1.5) and (10.1.6) together gives the result.
The assumption of symmetry is very appealing, because the concept of “center”
is unique when it is true.

EXERCISES

10.1.1.LetXbe a continuous random variable with cdfF(x). For 0<p<1, let
ξpbe thepth quantile; i.e.,F(ξp)=p.Ifp =1/2, show that while property (10.1.2)
holds, property (10.1.3) does not. Thusξpis not a location parameter.

10.1.2.LetXbe a continuous random variable with pdff(x). Supposef(x)is
symmetric abouta; i.e.,f(x−a)=f(−(x−a)). Show that the random variables
X−aand−(X−a)havethesamepdf.

10.1.3. LetF̂n(x) denote the empirical cdf of the sampleX 1 ,X 2 ,...,Xn.The
distribution ofF̂n(x) puts mass 1/nat each sample itemXi. Show that its mean is
X.IfT(F)=F−^1 (1/2) is the median, show thatT(F̂n)=Q 2 , the sample median.

10.1.4.LetXbe a random variable with cdfF(x)andletT(F) be a functional.
We say thatT(F)isascale functionalif it satisfies the three properties

(i)

(ii)

(iii)

T(FaX)=aT(FX), fora> 0

T(FX+b)=T(FX), for allb

T(F−X)=T(FX).

Show that the following functionals are scale functionals.

(a)The standard deviation,T(FX)=(Var(X))^1 /^2.

(b)The interquartile range,T(FX)=FX−^1 (3/4)−FX−^1 (1/4).

10.2SampleMedianandtheSignTest....................

In this section, we consider inference for the median of a distribution using the
sample median. Fundamental to this discussion is the sign test statistic, which we
present first.
LetX 1 ,X 2 ,...,Xnbe a random sample that follows the location model

Xi=θ+εi, (10.2.1)