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2.3.3.Depthfunctionsinducequantilefunctions
LetD(x,F)beadepthfunctionmaximizedat“median”MF,andhavingne-
stedcontoursenclosingMFanddefinedastheboundariesof“levelsets”or“cen-
tralregions”ofform{x:D(x,F)≥α},α>0.
ThedepthcontoursinduceaquantilefunctionQ(u,F),u∈B
d−1(0),endo-
wingeachpointx∈R
dwithaquantilerepresentation:
1.Forx=MF,itisQ(0,F)=MF.
2.Forx≠MF,itisQ(u,F)=xwithu=pv,wherepistheprobabilityweightof
thecentralregionwithxonitsboundaryandvistheunitvectortowardx
fromMF.
Inthiscase,uindicatesthedirectiontowardQ(u,F)fromthemedianMF.
Thequantity||u||istheprobabilityweightofthecentralregionwithQ(u,F)on
itsboundaryandthusrepresentsanoutlyingnessparameter.
2.4.Centeredrankfunctions
A“centeredrankfunction”inR
dtakesvaluesintheunitball,withtheori-
ginassignedtoaselectedmultivariatemedianMF,i.e.,R(MF,F)=0.
Definition4.AcenteredrankfunctionR(x,F)takesvaluesinB
d−1(0)withori-
gin0assignedtomultivariatemedianx=MF,andforotherxdenotesa“direc-
tionalrank”inB
d−1(0).
Forx≠MF,thevectorR(x,F)representsa“directionalrank”associatedwi-
ththepointx.Forx≠MF,R(x,F)providesa“direction”insomesense,and
||R(x,F)||providesa“rank”,thusmeasuringoutlyingnessofx.Univariatecase:
R(x,F)=2F(x)−1,withthesignofR(x,F)givingthe“direction”(fromthe
medianMF=F
−1(1/2))andwith|R(x,F)|=|2F(x)−1|providingthe“rank”.For
testingH0:MF=
θ
0,thesampleversionofR(
θ
0,F)providesanaturalteststati-
stic,amultivariateversionoftheunivariatesigntest[Barnett,1976].
2.4.1.Quantilesandranksareequivalent
GivenaquantilefunctionQ(i,F),itsinversefunctionQ
−1(x,F),x∈Rd,i.e.,
thesolutionuofx=Q(u,F),isinterpretableasacenteredrankfunctionwhose
magnitude||u||=||Q
−1(x,F)||measurestheoutlyingnessofx.
Conversely,acenteredrankfunctionR(i,F)generatesacorrespondingqu-
antilefunctionasitsinverse:foruintheunitballinR
d,Q(u,F)isthesolutionx
oftheequationR(x,F)=u.
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