Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

598 Nonparametric and Robust Statistics

1.‖v‖≥0and‖v‖=0ifandonlyifv= 0.

2.‖av‖=|a|‖v‖, for allasuch that−∞<a<∞.

3.‖u+v‖≤‖u‖+‖v‖, for allu,v∈Rn.

For the triangle inequality, use the anti-rank version, that is,

‖v‖=

∑n

j=1

j|vij|. (10.3.39)

Then use the following fact: If we have two sets of nnumbers, for example,
{t 1 ,t 2 ,...,tn}and{s 1 ,s 2 ,...,sn}, then the largest sum of pairwise products, one
from each set, is given by

∑n
j=1tijskj,where{ij}and{kj}are the anti-ranks for
thetiandsi, respectively, i.e.,ti 1 ≤ti 2 ≤···≤tinandsk 1 ≤sk 2 ≤···≤skn.

10.3.8.Consider the norm given in Exercise 10.3.7. For a location model, define
the estimate ofθto be
θ̂=Argminθ‖Xi−θ‖. (10.3.40)

Show thatθ̂is the Hodges–Lehmann estimate, i.e., satisfies (10.4.27).

Hint: Use the anti-rank version (10.3.39) of the norm when differentiating with

respect toθ.

10.3.9.Prove that a pdf (or pmf)f(x) is symmetric about 0 if and only if its mgf

is symmetric about 0, provided the mgf exists.

10.3.10.In Exercise 10.1.4, we defined the term scale functional. Show that the

parameterτW, (10.3.26), is a scale functional.

10.4Mann–Whitney–WilcoxonProcedure..................

SupposeX 1 ,X 2 ,...,Xn 1 is a random sample from a distribution with a continuous
cdfF(x)andpdff(x)andY 1 ,Y 2 ,...,Yn 2 is a random sample from a distribution
with a continuous cdfG(x)andpdfg(x). For this situation there is a natural null
hypothesis given byH 0 :F(x)=G(x) for allx; i.e., the samples are from the same
distribution. What about alternative hypotheses besides the general alternative
notH 0? An interesting alternative is thatXisstochastically largerthanY,
which is defined byG(x)≥F(x), for allx, with strict inequality for at least onex.
This alternative hypothesis is discussed in the exercises.
For the most part in this section, however, we consider the location model. In
this case,G(x)=F(x−Δ) for some value of Δ. Hence the null hypothesis becomes
H 0 : Δ = 0. The parameter Δ is often called theshiftbetween the distributions
and the distribution ofYis the same as the distribution ofX+Δ; that is,

P(Y≤y)=P(X+Δ≤y)=F(y−Δ). (10.4.1)

If Δ>0, thenYis stochastically larger thanX; see Exercise 10.4.8.