Quantile-locatingQuantile-locatingQuantile-locating functionsfunctionsfunctions andandand thethethe distancedistancedistance betweenbetweenbetween thethethe meanmeanmean andandand quantilesquantilesquantiles

Gilat’D.D.D. Gilat*Gilat* SchoolSchoolSchool ofofof MathematicalMathematicalMathematical Sciences,Sciences,Sciences, TelTelTel AvivAvivAviv UniversityUniversityUniversity

RamatRamatRamat AvivAvivAviv 69978,69978,69978, IsraelIsraelIsrael

T.T. P.P.P. Hill**Hill**Hill**T. SchoolSchoolSchool ofofof Mathematics,Mathematics, GeorgiaGeorgiaGeorgia InstituteInstituteInstitute ofofof TechnologyTechnologyTechnology

Atlanta,Atlanta,Atlanta, GAGAGA 30332,30332,30332, USAUSAUSA Mathematics.

GivenGivenGiven aaa randomrandomrandom variablevariablevariable XXX withwithwith finitefinitefinite mean,mean,mean, forforfor eacheacheach 000

ThroughoutThroughoutThroughout thisthis note,note, ififif aaa andandand bbb areareare realrealreal numbers,numbers,numbers, aaa vvv bbb (a(a(a 1\1\ b)b)b) standstandstand forforfor theirtheirtheir maximaximummummum (minimum)(minimum)(minimum) and,and,and, asasas isisis customary,customary,customary, aaa' ++ === aaa vvv 000 andandand a-a-a- === (-(-(- a )a)a) +.+.'. RecallRecallRecall thatthatthat thethethe realrealreal

this note, A maxi-

numbernumbernumber mmm === mpmmpp isisis aaap-quantile ofX(O

FirstFirstFirst applyapplyapply integrationintegrationintegration bybyby partspartsparts tototo rewriterewriterewrite Up,Up,Up, defineddefineddefined ininin (2),(2),(2), ininin thethethe formsformsforms

Up(x)Up(x) =p(EX=p(EX -x)-x) +++ E(XE(X -x)U,(x)=p ( E X -x) E (X-x)-x)xxr;

= p ( E X - x ) +x)x) ++ ff P ( X < t } d tpjxpjx ~~ (5i)(5i)== p(EXp(EX -- tltl dtdt (59 --m

(l-p)(EX(l-p)(EX -x)-x) +++ E ( XE(XE(X - x ) +-x)+-x)+=== --- (1 - p ) ( E X -x> oooo

== --=- (1 - -X)x)x) +++ JJP ( Xpjxpjx >>> t)tltl dt.dt.d t m

(1-(1- p)(EXp)(EXp)(EX -- (5ii)(5ii)(5ii) Xxx

Next,Next,Next, distinguishdistinguishdistinguish betweenbetweenbetween twotwotwo cases.cases.cases. IfIfIf xxx 2~~ mpmmpp useuseuse (5i)(5i)(5i) tototo obtainobtainobtain mm pp xxmP r;

U,(X)= p ( E X -X ) j P(X5 t ) f P ( X 5 t)Up(x)Up(x) == p(EXp(EX -- x)x) +++ JJ pixpix ~~ tltl dtdtdt +++ JJ pixpix ~~ tltl dtdtdt --m mP mmpp

~p(EX~p(EX -x)-x) +++ rr pixpix ~~-> p ( E X -x) me P ( X tltlt )dtdtdt +++ p ( xp(xp(x ---mmm,)pp)) ( 6 )(6)(6) --m

== p(EXp(EX -- mmpp)) +++ E ( XE(XE(X -- mmpp)-)- == Up(mUp(mpp),),EX -m,) -m,)- = U,,(m,), wherewherewhere thethethe inequaltyinequaltyinequalty isisis validvalidvalid becausebecausebecause mmmp ofX,pp isisis aaa p-quantilep-quantilep-quantile ofofX,X, andandand thethethe lastlastlast equalityequalityequality followsfollowsfollows

mppp isisis similarsimilarsimilar usingusingusing (5ii).(5ii).(5ii). ThisThisThis completescompletescompletes thethethe proofproofprooffromfromfrom (5i).(5i).(5i). TheTheThe proofproofproof forforfor thethethe casecasecase xxx

PROOF:PROOF:PROOF: (i)(i)(i) ByByBy definition,definition,definition,

~p(x)~p(x)Ap(x)== Up(p)Up(p) --- Vp(x)Vp(x) == x)+x)+ -- ( E X(EX(EX --- x ) + ]x)+}x)+}= Up(P) V,(X)= p { E ( Xp!E(Xp!E(X --x )+ -EXEXEX -xtl-xtl+++ (1-(1-(1-p )p)!E(Xp)!E(X( E(x-x)--x)--x)- - -x)- ]

=pl[E(x=pl[E(x -x)+-x)+ -E(X-E(X -xtJ-xtJ -- [(EX[(EX -x)+-x)+ -- (EX(EX -x)-]I-x)-]I= p ( [ E ( X - x ) + - E ( X - x ) - ] - [ ( E X - x ) + - ( E X - x ) - ] } (EX(EX -x)-}-x)-}+++ lE(XlE(X{ E ( X -x)--x)--x)- --- ( E X - x ) - )

== p!(EXp!(EX -- x)x) -- (EX(EX -- x)lx)l ++ !E(X!E(X -- x)-x)- -- (EX(EX -- x)-Ix)-I === !E(X!E(X( E ( X ---x)-x)-x)- --- ( E X(EX(EX ---x)-} = p { ( E X - x ) - ( E X - x ) ] + { E ( X - x ) - - ( E X - x ) - )

x)-Ix)-I

xsEXxsEX {{

E(XE(XE ( X - x ) - ,-x)-,-x)-, X S E X

== {E ( XE(XE(X -- x)+,x)+, xx "2EX."2EX.= -XI+,x 2 E X . (ii)(ii)(ii) TheTheThe inequalityinequalityinequality followsfollowsfollows fromfrom (i)(i)(i) (or(or(or fromfromfrom Jensen).Jensen).Jensen). T h eTheThe asymptoticasymptoticasymptotic statementstatementstatement

followsfollowsfollows fromfromfrom (i)(i)(i) usingusingusing monotonemonotonemonotone convergence.convergence.convergence. From

(iii)(iii)(iii) Assume,Assume,Assume, withoutwithoutwithout losslossloss ofofof generality,generality,generality, thatthatthat EXEXEX ===O.O.0. ByByBy (i)(i)(i) m 000 '"'"m'"'"

A,(x)dx= JJ E(XE(X -- eLy x)+x)+ dx.dx.x)-x)- ely ++ JJE(XE(X JJ ~p(x)~p(x) dxdx == E ( X - x ) - d u + j E ( X - x ) + d x .-m - m oo0

ApplyingApplyingApplying integrationintegrationintegration bybyby partspartsparts twicetwicetwice andandand usingusingusing FubiniFubiniFubini ininin betweenbetweenbetween tototo changechangechange thethethe orderorderorder ofofof integration,integration,integration, oneoneone obtainsobtainsobtains

m 0 m'"'" 00 '"'" A,(x) II (p!X(p!Xt P { X > tltlt ] dtdtdtII ~p(x)~p(x) dxdxdx ===

--OD - m oo0

= (1/2){E(X-)* + E ( X + ) 2 ]=(3E X 2 .-oo

'REMARK.‘REMARK.ApplyingApplyingApplying (iii)(iii)(iii) withwithwith ppp === f·REMARK. tt ititit followsfollowsfollows thatthatthat m'"'"

JJj IEIE( E I X - X ~IIXX -- xx II --- lEXlEX -- xx IIII dxdx == VarVar X.X.I E X - x I ] d x = V a r X .

--m

A pFinally,Finally,Finally, PropositionPropositionProposition 222 andandand thethethe aboveaboveabove propertiespropertiesproperties ofofof ~p~p willwillwill bebebe usedusedused tototo obtainobtainobtain boundsboundsbounds ononon thethethe distancedistancedistance betweenbetweenbetween anyanyany p-quantilep-quantilep-quantile ofofof aaa randomrandomrandom variablevariablevariable andandand itsitsits meanmeanmean ininin termstermsterms ofof itsitsits centralcentralcentral absoluteabsoluteabsolute firstfirstfirst moment.moment.moment. TheseTheseThese boundsboundsbounds areareare analogousanalogousanalogous tototo thethethe standardstandardstandard deviationdeviationdeviation

OF

DHARMADHIKARI boundsboundsbounds ofofof DHARMADHlKARIDHARMADHlKARI (1991)(1991)(1991)

E XEXEX --- a{qJPa{qJP SS mpsmps E XEXEX +++ afPTQafP{Q0msmp 4 .fi whichwhichwhich bothbothboth generalizegeneralizegeneralize (1)(1)(1) andandand strengthenstrengthenstrengthen thethethe symmetricsymmetricsymmetric versionversionversion ofofof O'CINNEIDE(1990)O'CINNEIDE(1990)O’CINNEIDE (1990)

IIIE XEXEX ---mp III sasa ymaxymax Ip/q,Ip/q, q/pl.q/pl.mmpp 5 0 i m a x b/(7.cI/Pl.

For 0 mp ofof thethe Exisfinite,isfinite,THEOREMTHEOREMTHEOREM 1.1.1. ForOForO