ISTORTION OF PROBABILITY MOELS VÁVRA Frntišk (ČR), NOVÝ Pvl (ČR), MAŠKOVÁ Hn (ČR), NETRVALOVÁ Arnoštk (ČR) Abstrct. Th proposd ppr dls with on o possibl mthods or modlling th rltion o two probbility modls t distribution unctions lvl. Th ssnc o th usd concpt is distortion o th distribution unction. istortion unction is non-dcrsing mpping o th intrvl [,] into itsl. Som trnsormtion cts o th distribution unction o distortion unctions r drivd in this contribution. Th xploittion o th distortion unction concpt o probbility modl qulity msurmnt is lso proposd. Prcticl usg is rprsntd by th trnsormtion o mpiricl distribution unction o wkly orcst rror o PX5 indx vlu to norml (Gussin) modl. Ky words. istribution, distortion, dnsity, divrgnc Motivtion Otn usd mns o probbility modlling nd sttisticl vluting is trnsormtion o rndom vribl. Th xploittion o trnsormtion procdurs o distribution unctions nd dnsitis is lss rqunt. Such procdurs r usd in risk msurmnt, insurnc thory nd lso in othr ilds [,,3,4]. Our contribution ttmpts to suggst th link btwn this concpt nd clssicl inormtion thory [5,6]. Concpt ion Lt s dnot: x) is distribution unction, (x) is dnsity o rndom vribl, F (x) is nothr distribution unction, (x) is its corrsponding dnsity. Lt us hv unction G( clld distortion unction on condition tht: M: G( is non-dcrsing unction on th intrvl [,], M: G() =, G() =, M3: xcpt init numbr o points, g ( = G( dy d xists on th intrvl [,]. And or which it holds: F ( x) = G( ()
Rmrk: Th unction G( is distribution unction on th intrvl [,] nd so is its dnsity. Thror ( x) = () xcpt init numbr o points. It is obvious tht or vry distribution unction x) nd or vry unction G( stisying M, M nd M3, G( is gin distribution unction. Th coupl F, cn b intrprtd s distribution unction nd its dnsity, nd F, s som o thir stimts. Thn it is possibl to msur th distnc, (.g.) by th divrgnc [5] (or simplicity w will suppos tht x) is incrsing nd continuous): ( ) = lg dx = lg dx = lg dx x g F x x ( ) ( ( )) ( ) Atr substitution y = x) dy = dx dx = dy w gt: Likwis: ( Atr substitution ) = = ( ) lg dy. (3) ( x) ( x) lg dx = = lg dx. lg dx = y = x) dy = dx dx = dy w gt: ( ) = lg dy = H ( g) (4) J (, ) = ( ) + ( ) = lg dy H ( g) = ( ) lg dy (symmtricl divrgnc [5,6]). So th dirnc btwn th two probbility dscriptions dpnd only on th dnsity o distortion. 3 Momnts o th distortion For continuous nd incrsing distribution unction x), w cn sily dtrmin momnts o rndom vribl X tht ollows th distribution unction G(. I { E G( F (*)) X } xists, thn: E G( F (*)) { X } = = x ( F ( x) dx = ( x dx = x) dx = E G(*) {( F ( X )) whr F - (x) is gnrlizd invrsion o th distribution unction x) nd (F - ( is its rtionl powr. In prticulr, or =, : },
E G( F (*)) X} = ( F ( x) dx nd EG( F (*)){ X } = ( F ( x) { dx, rspctivly, (5) i thy xist, nd in such cs w gt: σ ( (*)){ } ( ( )) ( ) ( ( )) ( ) G F X = F x g x dx F x g x dx. (6) 4 Som typs o distortion unctions A wid rng o prmtric milis o distortion unctions is mntiond in [3]. In this ppr, w will dl with two o thm: ) proportionl: G ( = y α = α y α ; α >, y y b) xponntil: ( ) = λ G y = ; λ. λ For th lttr on, it holds: y y y lim G( = lim = lim = y, λ λ λ λ which is n idnticl distortion unction. For th proportionl distortion it holds tht: α ( ) = lgα nd ( ) = lgα + α nd α ( α ) J (, ) = ( ) + ( ) =. α For th xponntil distortion it holds tht: λ λ λ ( ) = lg + ( + λ) nd ( ) = lg λ + nd λ J(, ) = ( ) + ( ) = + ( + λ ). 5 Som pplictions nd usg Figs.,, 3 prsnt th mily o xponntil distortions.,,9,8,7,6,4,3,, istribution unction N(,) tr xponntil distortion lmbd= - lmbd= -5 lmbd= - lmbd= -, lmbd= -5 lmbd= 5 lmbd=, lmbd= lmbd= 5 lmbd= -4, -3,5-3, -,5 -, -,5 -, -,,5,,5 3, 3,5 4, Rndom vribl Fig.. istribution N(,) tr xponntil distortion. 3
nsity N(,) tr xponntil distortion, lmbd= -,9 lmbd= -5,8 lmbd= -,7 lmbd= -,,6 lmbd= -5 lmbd= 5,4 lmbd=,,3 lmbd=, lmbd= 5, lmbd= -4, -3,5-3, -,5 -, -,5 -, -,,5,,5 3, 3,5 4, Rndom vribl Fig.. nsity N(,) tr xponntil distortion. ivrgnc o xponntil distortion in dpndnc on lmbd 4, 3,5 3,,5,,5, - -9-8 -7-6 -5-4 -3 - - 3 4 5 6 7 8 9 Lmbd (,) (,) J(,) Fig. 3. ivrgnc msurs N(,) tr xponntil distortion. Possibl xploittion o th proposd concpt is modlling th distortion o mpiricl distribution unction o PX5 indx stimtion rror distribution up to on wk with th us o prdictor with on-yr mmory to norml probbility distribution, nd convrsly, s Figs. 4, 5, 6.,,9,8,7,6,4,3,, -3, -,5 -, -,5 -, -,,5,,5 3, Empiricl r7 Norml pproximtion r7, on-yr prdictor PX5 / Fig. 4. Empiricl distribution nd its norml pproximtion (r7, on-yr prdictor). 4
,,9 istortion unction or F E (x)=g(f N (,8,7,6,4,3,, F E (x) is th mpiricl distribution unction nd F N (x) is its norml pproximtion,,,3,4,6,7,8,9, Fig. 5. istortion rom norml to mpiricl (r7, on-yr prdictor).,,9,8 istortion unction or F N (x)=g(f E (,7,6,4,3,, F E (x) is th mpiricl distribution unction nd F N (x) is its norml pproximtion,,,3,4,6,7,8,9, Fig. 6. istortion rom mpiricl to norml (r7, on-yr prdictor). 6 Conclusions nd urthr progrss Th mntiond rltions wr drivd with strict ssumptions. It is obvious tht such ssumptions wr usd or simplicity o proos. It is lso vidnt tht th prsntd mthod cn b usd with mor r ssumptions which r lso mor vriibl in rl li. It could b lso intrsting to invstigt possibl dulity btwn trnsormtion (distortion) o distribution unction nd trnsormtion o vribls. Som trnsormtions r possibl s wll. S th xmpl in Fig. 7, whr distortion o normd norml probbility distribution into normd xponntil distribution pprs. 5
,,9 G(N(,)) = -xp(-x) <=> x >, othrwis =,8,7,6,4,3,,,,,3,4,6,7,8,9 Fig. 7. istortion rom normd norml probbility distribution to normd xponntil on. Rrncs [] Rsor, R. M., McLish, on L.: Risk, Entropy, nd th Trnsormtion o istributions. Bnk o Cnd, Working Ppr -. [] Hürlimnn, W.: istortion risk msurs nd conomic cpitl. Working Ppr, Wrnr Hürlimnn, Schönholzwg 4, CH-849 Wintrthur, Switzrlnd. [3] rkiwicz, G., hn, J., Goovrts, M.: Cohrnt istortion Risk Msurs A Pitll. Fculty o Economics nd Applid Economics, K.U.Luvn. July 5, 3. [4] Wng, S. S.: Equilibrium Pricing Trnsorms: Nw Rsults using Buhlmnn s 98 Economic Modl. Astin Bulltin, Vol. 33, No, 3, pp57-73. [5] Covr, T. M., Thoms, J. A.: Elmnts o Inormtion Thory. Wily, 99. [6] Vávr, F.: Inormc klsiikc. Hbilitční prác, ZČU Plzň, 995. [7] Vávr, F., Nový, P.: Inormc dzinormc. Sminář z plikovné mtmtiky. Ktdr plikovné mtmtiky, Přírodovědcká kult MU, Brno, 3.4.4. [8] Kotlíková, M., Mšková, H., Ntrvlová, A., Nový, P., Spírlová,., Vávr, F., Zmrhl,.: Inormc dzinormc - sttistický pohld. Ltní škol JČMF ROBUST 4, Třšť, 7.-.6.4. Acknowldgmnt This work ws supportd by th grnt o th Ministry o Eduction o th Czch Rpublic No: MSM-355 Inormtion Systms nd Tchnologis. Contct ddrss oc. Ing. Frntišk Vávr, CSc., vvr@kiv.zcu.cz Ing. Pvl Nový, Ph.., novyp@kiv.zcu.cz Ing. Hn Mšková, mskov@kiv.zcu.cz Ing. Arnoštk Ntrvlová, ntrvlo@kiv.zcu.cz prtmnt o Computr Scinc nd Enginring, Fculty o Applid Scincs Univrsity o Wst Bohmi Univrzitní, 36 4 Plzň, Czch Rpublic 6