Improved FFT Approximations of Probability. Functions Based on Modified Quadrature Rules

Transcription

1 Intrntonl thmtcl Form Vol. 8 3 no HIKAI td Improvd FFT Appromtons o Prolt Fnctons Bsd on odd Qdrtr ls Wrnr Hürlmnn Woltrs Klwr Fnncl Srvcs Sldstrss 69 H-88 Zürch Swtrlnd whrlmnn@lwn.ch Astrct In th lst tn rs th trdtonl Smpson qdrtr rl or nmrcl ntgrton hs n mprovd to n optml 3-pont qdrtr orml nd modd Smpson rl tht ts ddtonll th rst drvtv o th ppromtd ntgrnd t th two nd-ponts o ntgrton nto ccont. Th mpct o th mprovd qdrtr rls on th st Forr trnsorm FFT ppromton o prolt dnst nctons wth nown chrctrstc nctons s dscssd. Th qlt o th ppromtons s msrd wth dscrt vrson o th totl vrton dstnc. mrcl mpls sggst tht dscrt totl vrton dstnc o ppromtl.5 s th st possl ttnl vl cross ll th consdrd FFT ppromtons. thmtcs Sct lsscton: 4A55 6E T5 9G6 Kwords: dscrt Forr trnsorm st Forr trnsorm optml 3-pont qdrtr rl modd Smpson rl prolt dnst ppromton totl vrton dstnc. Introdcton Th dscrt Forr trnsorm ppromton o prolt dnst nctons wth nown chrctrstc nctons s spcll sl whn nltcl prssons or th dnst nctons r not vll. Ths s th cs or tmprd stl nd rltd dstrtons.g. chv t l. [4] Schrr t l. [5] Jlon [7] tc.. Usll th ppromton s sd on smpl nmrcl ntgrton qdrtr rl l th md-pont rl P or th

2 83 Wrnr Hürlmnn Smpson rl S. Howvr th qlt o ths ppromton s sldom qstond. In th lst tn rs som mprovmnts o th Smpson qdrtr rl hv n dscovrd. For mpl n optml 3-pont qdrtr O3Q orml o closd tp smlr n smplct to th Smpson rl hs n rvld Uvc [8]. Anothr ttmpt to mprov Smpson s rl s d to Uvc nd orts [9]. Th hv proposd modd Smpson rl S whch ts ddtonll th rst drvtv o th ppromtng ntgrnd t th two nd-ponts nto ccont. A r ccont o ths dvlopmnts s prsntd n Scton. Th corrspondng ormls rqrd to compt th st Forr trnsorm FFT ppromtons o prolt dnst nctons r prsntd n Scton 3. Scton 4 nvstgts th qlt o th otnd FFT ppromtons or nmr o lmntr prolt dstrtons. As qlt msr w s dscrt vrson o th totl vrton DTV dstnc whch s nown to n ppr ond or th Kolmogorov dstnc. It s shown tht DTV o.5 s th st ttnl vl or th consdrd FFT ppromtons t lst or th stndrd norml nd stndrd plc dstrtons. Th vrnc gmm dstrton whch s poplr dstrton sd n nncl pplctons nd th on-sdd ponntl dstrton r lso mnd. Thogh wth lss prcson DTV o ppromtl.5 sms th st possl ttnl vl cross ll FFT ppromtons. Th O3Q ppromton mpls n gnrl th st mprovd FFT ppromton whch s ollowd th S ppromton. Or osrvtons sggst tht thr s possl n optml smmtrc ntrvl wth st ttnl DTV vl or th O3Q ppromton nd tht th rt o convrgnc to ths vl s str or th O3Q ppromton thn th ltrntv ons. It s opn or tr rsrch to nls whthr ths holds mor gnrll nd whthr ct qntctons o ths nmrcl phnomn cn gvn.. mrcl ntgrton qdrtr rls nd mprovmnt. Smpl nd poplr ntgrl ppromtons o ncton dnd ovr nt ntrvl [ ] r th rctnglr rls nd th Smpson rl: t-pont rl P d d-pont rl P d ght-pont rl P d Smpson rl S d 4 } 6. lrl th Smpson rl s lnr comnton o th rst thr qdrtr rls so-clld 3-pont qdrtr orml o closd tp. Uvc [8] hs

3 Improvd FFT ppromtons o prolt nctons 83 drvd n optml 3-pont qdrtr orml o closd tp whch hs ttr rror stmt thn th Smpson rl. It s gvn Optml 3-pont rl O3: d 4 }.. I ' n trms o th -norm s [8] Thorm 6 s drntl ncton wth [ ] thn n rror stmt ' 3 / d 4 } d Th rror stmt or th Smpson rl s wors. It s o th orm. wth th constnt / 6 nstd o / 4 3 [8] mr 3. Othr ttmpts to mod nd mprov Smpson s rl hv n ndrtn. For mpl Uvc nd orts [9] hv proposd gnrld ppromton n trms o th rst drvtv ' t th two nd-ponts o th orm odd Smpson rl S: d } 6 ' ' }.3 Appld to nmrcl ntgrton th rror o th S s o sth ordr n grd spcng s Uvc nd orts [9] orollr. Th prpos o th prsnt not s to nvstgt th mpct o th O3 nd S modd qdrtr ormls on th st Forr trnsorm ppromton o prolt nctons. 3. odd FFT ppromtons o prolt nctons. Th rltonshp twn th chrctrstc ncton o prolt dnst ncton pd ssoctd to rndom vrl s gvn th nvrs Forr trnsorm. For scntl lrg ntrvl [ cc] t s possl to ppromt pd mns o nmrcl Forr nvrson s ollows c d d. 3. c Bsd on th qdrtr rls o Scton th nt ntgrl n 3. cn tht s dvdd nto ppromtd n drnt ws. onsdr n ntrvl [ ] dsonts sntrvls o ql lngth h nd ssm tht th

4 83 Wrnr Hürlmnn rndom vrl wth pd hs nown chrctrstc ncton ch. For... st h. For scntl lrg th constnt h c s lso lrg nd 3. mpls th pd ppromton. / / c c d d 3. In rst stp w consdr Dscrt Forr Trnsorm ppromtons o th pd or th P P nd P rctnglr rls rom whch w gt ppromtons or th S nd O3 rls. In scond stp w drv ppromtons or th S rl. In th rst stp st... nd consdr th mdponts... m. Applng th drnt qdrtr rls to th rght-hnd sd ntgrl n 3. on otns th ollowng nt sm ppromtons o : P: h P: m m h P:. h Snc on hs h ε ε ε. Insrtd nto th ov on gts ppromtons o : P: : P: : P: : whch on ntrprts s th -th componnts o vctors ssoctd to rsp nd... sch tht

5 Improvd FFT ppromtons o prolt nctons 833 P: P: P:. In ths sttng th S nd O3 qdrtr rls mpl th ppromtons: S: } O3: } To drv S sd ppromtons clclton o th rst drvtv } ' ' d d t nd s ncssr n ordr to gt smlrl to th ov th lt- nd rght-pont drvtv contrtons to th ovrll S ppromton o :... } } ' : ' d d d d d d d.... } } ' : ' d d d d d d d On otns th ollowng S ppromton } } d d. 3.5 An cnt sotwr mplmntton o th s sd on th Fst Forr Trnsorm FFT lgorthm ool nd T [3] s lso Schwrt [6] [7] Hdmn t l. [6] Dhml nd Vttrl [4] Btnov [] mong othrs. For nmrcl ppromton o th dstrton ncton dt t F on

6 834 Wrnr Hürlmnn drvs smlr ppromton n trms o th chrctrstc ncton.g. Km t l. [9] Proposton or on ss th rcrsv orml F F h... F 3.6 nd smpl pcws lnr ntrpolton or ntrmdt vls: F F h F F } [ ] Dscrt totl vrton dstnc nd FFT ppromtons. It s nstrctv to nvstgt th qlt o th FFT ppromtons or nmr o lmntr prolt dstrtons. Gvn s n soltl contnos rndom vrl wth pd nd dstrton ncton F. Th qlt o dscrt ppromton F to F... s msrd th dscrt totl vrton DTV dstnc dnd Δ DTV F F F F. 4. On nows tht th dscrt totl vrton dstnc s n ppr ond to th dscrt Kolmogorov dstnc Δ DK F F m... F F Δ DTV F F. 4. Thror th smllr th DTV th ttr s th qlt o ppromton. In th nt mpls th DTV s comptd or th FFT ppromtons sd on th O3Q S S nd P ppromtons s prsntd n Scton 3. Th FFT s q vltd ovr nt ntrvl [ ] wth nmr q 3 4 o qll spcd sntrvls. For smmtrc dstrton w choos. Empl 4.: stndrd norml dstrton Th ch s p ' } nd ts drvtv s. Th DTV or th FFT ppromtons drs rom.5 onl mrgnll whn 6. Sttng 6 on hs DTV.5 or th S nd P ppromtons n cs q 3 nd or th O3Q S nd S ppromtons n cs q 4. On otns DTV.5 or th P ppromton n cs q 4. It sms tht DTV.5 s th st ttnl qlt o ppromton.

7 Improvd FFT ppromtons o prolt nctons 835 Empl 4.: stndrd plc dstrton ' Th ch s / } nd ts drvtv s / }. Tl 4. low smmrs clcltons. Thogh th DTV cn slghtl smllr thn.5 O3 ppromton wth 8 th constnt.5 mst lso vwd s th st ttnl qlt n FFT ppromton. Ths sttmnt holds or th O3 ppromton wth scntl lrg smmtrc ntrvl hr wth. Th DTV dstnc s scond st or th S ppromton t drs slghtl rom.5. For smllr vls o th S nd P ppromtons dtrort rpdl ncrsng vl o q. For mpl wth 8 nd q 4 on otns DTV.3 or th P ppromton. Empl 4.3: vrnc gmm logrthmc rtrn dstrton On m s whthr DTV o.5 cn lso ttnd or dstrtons n th rl-l world. onsdr th dl closng prcs o th Swss rt nd th Stndrd & Poors 5 stoc mrt ndcs ovr th 3 rs -. Th osrvd smpl logrthmc rtrns r ngtvl swd nd hv mch hghr css rtoss thn s llowd norml dstrton. In ths stton th or prmtr shtd vrnc gmm dstrton wth ch } υ αβ ρ 4.3 αβ α β ts qt wll th logrthmc rtrns o ths two dt sts. ot tht th vrnc gmm dstrton hs n ntrodcd n nnc dn nd Snt [] nd dn nd ln[3]. Th mportnt vrnc gmm procss hs n stdd t mn plcs.g. dn t l. [] dn [] Kot t l. [8] Scton 8.4 rr t l. [] Gmn [5] tc.. Th prmtrs υ.5 ρ.85 α 5 β r rprsnttv o tpcl vrnc gmm logrthmc rtrn dstrton. Tl 4. low smmrs clcltons. It trns ot tht ll th qdrtr rls ld to FFT ppromtons wth DTV rthr clos to.5. Thogh th ovrll prrnc gos to th O3 ppromton th othr ons do not dr mch nd cn rslt n mrgnll smllr DTV n som spcc css. Agn th S ppromton s scond st n th ovrll.

8 836 Wrnr Hürlmnn Tl 4.: DTV dstnc o FFT ppromtons to stndrd plc dstrton qdrtr rl q O3 S S P Tl 4.: DTV dstnc o FFT ppromtons to vrnc gmm rtrn qdrtr rl q O3 S S P

9 Improvd FFT ppromtons o prolt nctons 837 Empl 4.4: stndrd ponntl dstrton Th prcdng mpls r ll dstrtons dnd ovr th whol rl ln. Wht ot th FFT ppromtons or on-sdd dstrtons? A smpl mpl s th stndrd ponntl dstrton wth ch }. Th otnd DTV vls r rportd n th Tl 4.3. Frst o ll snc th ponntl dstrton s onl dnd or postv nmrs t s not clr pror whthr th FFT ppromtons shold clcltd ovr n ntrvl o th orm [ ] or not. Th choc 6 shows tht th DTV o ppromtl.54 n ths cs s r w rom st ttnl qlt n FFT ppromton. Th stton chngs smmtrc ntrvls wth vls o < r llowd nd th DTV dstnc s rplcd th dstnc Δ DTV F F F F 4.4 / whch msrs th qlt n FFT ppromton ovr th postv nmrs onl s shold. As or mpls show th O3Q ppromton s st n ll consdrd css nd ll DTV vls rmn stl nd rthr clos to.5. Th comprson twn th ntrvls [ 4] nd [ 88] o ql lngth sggsts tht smmtrc ntrvl rslts n smllr DTV vls n ths cs. Scond st s gn th S ppromton. Th S nd P ppromtons cn dtrort rpdl s q ncrss s clcltons wth th smmtrc ntrvl [ 88] show. orovr wth ncrsng vl o q th DTV o th O3Q ppromton ncrss nd coms closr to.5. For lrgr smmtrc ntrvls l [ 66] th drncs n DTV vls twn th drnt FFT ppromtons coms smllr nd lmost nglgl. Th DTV vl o th O3Q ppromton wth q 4 cnnot mprovd whn nlrgng th ntrvl [ ] to [ 66]. Ths osrvtons sggst tht thr s possl n optml ntrvl [ ] wth st ttnl DTV vl or th O3Q ppromton nd tht th rt o convrgnc to ths vl s str or th O3Q ppromton thn or th othr ons. A closr loo t th Tl 4. shows tht ths osrvtons rmn tr or th stndrd plc dstrton. It s opn or tr rsrch to nls whthr ths holds mor gnrll nd whthr ct nltcl qntctons o ths nmrcl phnomn cn gvn.

10 838 Wrnr Hürlmnn Tl 4.3: DTV dstnc o FFT ppromtons to stndrd ponntl qdrtr rl q O3 S S P rncs [] D. Btnov Fst Forr Trnsorm K Pprs n omptr Scnc Smnr 5 [] P. rr H. Gmn D.B. dn nd. Yor Th n strctr o sst rtrns: n mprcl nvstgton J. Bsnss [3] J.W. ool nd J.W. T An lgorthm or th mchn clclton o compl Forr srs th. o omptton

11 Improvd FFT ppromtons o prolt nctons 839 [4] P. Dhml nd. Vttrl Fst Forr Trnsorms: ttorl rvw nd stt o th rt Sgnl Procssng [5] H. Gmn Pr mp év procsss or sst prc modllng J. o Bnng nd Fnnc [6].T. Hdmn D.H. Johnson nd.s. Brrs Gss nd th hstor o th Fst Forr Trnsorm Archv or th Hstor o Ect Scncs [7] P. Jlon Gnrtng tmprd stl rndom vrts rom mtr rprsntton Worng Ppr no. /4 Dpt. o Economcs Unvrst o cstr. [8] S. Kot T.J. Koows nd K. Podgors Th plc Dstrton nd Gnrltons: vst wth Applctons Brhäsr. [9] Y.S. Km T.S. chv.. Bnch nd F.J. Fo omptng V nd AV n nntl dvsl dstrtons Prolt thmtcl Sttstcs [] D. dn Prl dscontnos sst prcng procsss In: E. Jon J. vtnc nd. sl Eds. Opton Prcng Intrst ts nd s ngmnt 5-53 mrdg Unvrst Prss mrdg. [] D. dn nd E. Snt Th vrnc gmm modl or shr mrt rtrns Jornl o Bsnss [] D. dn P. rr nd E. hng Th vrnc gmm procss nd opton Prcng Eropn Fnnc vw [3] D. dn nd F. ln Opton prcng wth VG mrtngl componnts thmtcl Fnnc [4] S.T. chv Y.S. Km.. Bnch nd F.J. Fo Fnncl odls wth év Procsss nd Voltlt lstrng J. Wl & Sons. [5]. Schrr S.T. chv Y.S. Km nd F.J. Fo Appromton o swd nd lptortc rtrn dstrtons Appld Fnncl Economcs [6] H..Schwrt Elmntr Drstllng dr schnlln Forrtrnsormton omptng

12 84 Wrnr Hürlmnn [7] H.. Schwrt Th st Forr trnsorm or gnrl ordr omptng [8]. Uvc An optml qdrtr orml o closd tp Yoohom thmtcl Jornl [9]. Uvc nd A.J. orts A corrctd qdrtr orml nd Applctons Th Astrln nd w Zlnd Indstrl nd Appld thmtcs Jornl 45E 4 E4-E56 U: cvd: rch 4 3