quantile-locatingquantile-locatingquantile-locating … · 2020. 1. 7. ·...
TRANSCRIPT
-
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