Two kurtosis measures in a simulation study

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Phillip Pearson
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1 Two kuto meue n multon tudy Ann M Fo Deptment o Quntttve Method o Economc nd Bune Scence Unvety o Mlno-Bcocc nn.o@unmb.t CNAM, P COMPSTAT 00 Augut 3, 00

2 Ovevew The IF (SIF) nd t ole n kuto tude Fom nequlty (Loenz) odeng to ght/let kuto meue Two kuto meue: SIF compon / numecl expement Ann M. Fo Two kuto meue COMPSTAT 00

3 Bckgound Fom Peon (905) onwd tttc textbook hve dened kuto opetonlly the chctetc o dtbuton meued by t tnddzed outh moment. Howeve Exce/deecto equencyne the men comped to noml cuve (Peon, ) Non-Guntype o ymmety (Ft Ed. ESS) Revee tendency towd bmodlty (Dlngton, 970) X β E σ 4 Tledne only (Al, 974) Dpeonound the two vlue +/ σ(moo, 986) Pekedne + tledne (Dyon, 943 Fnucn, 964) Ann M. Fo Two kuto meue COMPSTAT 00 3

4 Kuto nd the IF An ely ntuton o L. Flechn Flechn (Stttc, 948) Hmpel (968) Ruppet (987) Bckgound & method Mn eult Tke equency dtbuton: {, ; Σ P} To nvetgte the behvouo β when equency lteed, we compute the ptl devtveo β wt β ( ) z β β( β ) 4γz P whee: z ; σ γ σ 3 3 Conde pobblty dtbuton F nd the unctonl β β (F) How doe β chnge we thow n n ddtonl obevton t ome pont x? Contmnted dtbuton: F ε ( ε) F + ε δ x (0 < ε < ) IF F β, ( x ( z wth: β( F ) lm ε 0 β ) z β x σ ε ) β( F) ε ( β ) 4γ z Ann M. Fo Two kuto meue COMPSTAT 00 4

5 5 Kutond the IF Flechn devtve computtonl detl 4 β ( ) ( ) ( ) P ( ) ( ) ( ) m P m, 4 ( ) ( ) m 0 ( ) ( ) { } P ( ) m 0 Rwmoment oode n P m -thpoweo n P Ann M. Fo Two kuto meue COMPSTAT 00

6 Kuto nd the IF Kuto explned In the ymmetc ce: IF SIF ( ) z β β ( ) IF( x; F, β ) β Qutcuncton wth ou el oot: x ( ),,3,4 ± σ β ± β β T F C x x x 3 x 4 F T Unbounded Locl mxmum: β t x Mnm: β ( β ) t x ± σ β KURTOSIS pekedne + tledne but β domnted by tlweght Th IF ugget tht β lkely to be oveetmted by mple kuto o xdtnt om, but undeetmted t ntemedte vlue o x Ann M. Fo Two kuto meue COMPSTAT 00 6

7 Kutond the IF The noml ce nd mple kuto β P [( ) ] z 3 6 P σ 4 6 σ + 3 <- IF SIF cente Root o the qutc: lnk lnk () (),334σ 0,74σ (3) (4) + 0,74σ +,334σ tl tl Ann M. Fo Two kuto meue COMPSTAT 00 7

8 Kutond the IF The noml ce nd mple kuto β P [( ) ] z 3 6 P σ 4 6 σ + 3 <- IF SIF Thee e two ntevl (lnk) o ubtntl pobblty (% ech) n whch the IF h eltvely lge negtve vlue (mnmum 6) -6-6 Thee pobly coepond to mlle vlue o mple kuto -> Content wth undeetmton o β by mple kuto (on vege) nd undecovegeo condence lnk cente 0,54 tl 0, 0, ntevl o β lnk tl Ann M. Fo Two kuto meue COMPSTAT 00 8

9 Kutobynequlty Zeng (ESS, 006); Fo (Comm. Sttt., 008) Septe nly o D nd S: Kuto-nceng tnomton Medn Pgou-Dlton tne pncple Let devton S γ X X < γ wth E(S) δ S < Rght devton D X γ X γ wth E(D) δ D < Non-egltn tne on D nd S o xed γ, δ D, δ S Ann M. Fo Two kuto meue COMPSTAT 00 9

10 Kutobynequlty Zeng kuto odeng(ess, 006) L S (p) Mnmum kuto L D (p) Let kuto Loenz cuveo S : LS ( p) δ S 0 p F S ( t) dt Zeng kuto cuve Repeent kuto by mo plot o the Loenz cuveo Dnd S Rght kuto Loenz cuveo D : L D ( p) δ D 0 p F D ( t) dt Uned tetment o ymmetc nd ymmetc dtbuton Kuto odeng dened v neted Loenz cuve Lu, Pelu nd Sngh(Ann. Sttt., 999) n multvte ettng Ann M. Fo Two kuto meue COMPSTAT 00 0

11 Kutobynequlty Zeng kuto meue(ess, 006) K + [ C( D) C( )] S wth: ght let ght let δd C( D) E( D ) δs C( S) E( S ) [ R( D) R( S) ] K + R( D) R( S) 0 0 wth: [ p L ( p) ] D dp [ p L ( p) ]dp S (to o ght/let cle unctonl) (ght & letgnndexe) Nomlzedmeue, wthvluebetween0 (mnmum kuto) nd (mxmumkuto) Ann M. Fo Two kuto meue COMPSTAT 00

12 Kutomeue A look t the SIF Symmetcdtbuton(γ 0): Symmetc contmnnt(ruppet, 987): Ft meue o(ght) kuto Ann M. Fo Two kuto meue COMPSTAT 00

13 Kutomeue A look t the SIF Second meue o(ght) kuto Ann M. Fo Two kuto meue COMPSTAT 00 3

14 Kuto meue SIF compon t tndd noml Allthe meuee nceedbycontmntonn the tlnd t the cente nd e deceed by contmnton n the houlde/lnk. HvngunboundedSIF, theye entve tothe locton o tl outle welltheequency. Howeve, conventonl kutomuchmoe entve (qutc SIF). The mgntudeomn SIF condebly lge o conventonl kuto. Ann M. Fo Two kuto meue COMPSTAT 00 4

15 Kutomeue Monte Clo expement (N 0,000; n 0 to640) Reltve b Reltve RMSE Noml Lplce Tukey λ hghe pek heve tl Ann M. Fo Two kuto meue COMPSTAT 00 5

16 Kutomeue Smll/medum mple behvou Boottpcondencentevlt 95% level(pecentle method) B,000 boottp emple; N 0,000 eplcton Empcl covege Avege length K the meuewhchlkelytobeetmtedwththe hghetccucy The mple peomnce ok mpovethe pentdtbuton become moe peked(lplce) Ann M. Fo Two kuto meue COMPSTAT 00 6

17 Dectonoeech Rgoouymptotcneenceothe newmeuek, poblyn vew o pctcl(nncl?) pplcton o ght/let kuto Check comptblty between the nequlty-bed concept o kuto nd ome obut(e.g. quntle-bed) meue ecently popoedn ltetue(goeneveld, 998; Schmd& Tede, 003; By, Hubet& Stuy, 006; Kotz& See, 007; ) Ann M. Fo Two kuto meue COMPSTAT 00 7

18 Two kuto meue n multon tudy Ann M Fo Deptment o Quntttve Method o Economc nd Bune Scence Unvety o Mlno-Bcocc nn.o@unmb.t CNAM, P COMPSTAT 00 Augut 3, 00

E-Companion: Mathematical Proofs

E-Companion: Mathematical Proofs E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth