ESTIMATION OF THE STRUCTURAL PARAMETERS IN THE GENERAL SEMI-LINEAR CREDIBILITY MODEL

Transcription

1 U.P.B. Sci Bull. Seies A Vol. 69 No. 4 7 ISSN 3-77 ESTIMATION OF THE STRUCTURAL PARAMETERS IN THE GENERAL SEMI-LINEAR CREDIBILITY MODEL Vigii ATANASIU O luce oigilă ce eziă şi lizeză esimoii meilo suculi di modelul de cedibilie semi-liiă imlicâd oieăţi memice comlice le vloilo medii codiţioe şi le coviţelo codiţioe. Deci eu ue olosi ezulele sueioe de cedibilie semiliiă obţiue i ces model vom oei esimoi uili i meilo de sucuă. Di uc de vedee cic ese evideţiă oiee civă de edelse ceso esimoi. A oigil e which eses d lyses he esimos o he sucul mees i he semi-lie cedibiliy model ivolvig comliced mhemicl oeies o codiiol execios d o codiiol covices. Thus o be ble o use he sueio semi-lie cedibiliy esuls obied i his model we will ovide useul esimos o he sucue mees. Fom he cicl oi o view he cive oey o he ubisedess o hese esimos is highlighed. Key wods: cocs ubised esimos sucue mees semi-lie cedibiliy heoy. AMS Subec Clssiicio: 6P5.. Ioducio I his icle we is give he geel semi-lie cedibiliy model see Secio which ivolves oly oe isoled coc. We deive he oiml lieized cedibiliy esime o he isk emium o his cse d we coside s licios o his esul: he secil semi-lie cedibiliy model obied om he geel semi-lie cedibiliy model o he oximio o μ θ -he e emium o coc wih isk mee θ -bsed o uiue oiml oximig ucio 3 he secil hiechicl semi-lie cedibiliy model. Lecue Mhemics Deme Acdemy o Ecoomic Sudies Buches ROMANIA

2 3 Vigii Asiu I us ou h his ocedue does o ovide us wih sisics comuble om he obsevios sice he esul ivolves ukow mees o he sucue ucio. To obi esimes o hese sucue mees o he geel semilie cedibiliy model we embed he coc i collecive o cocs ll ovidig ideede iomio o he sucue disibuio see Secio. We is deive he ubised esimos o he sucue mees o he secil semi-lie cedibiliy model. We close his secio givig s licio o his esimio he ubised esimos obied o he sucul mees o he geel semi-lie cedibiliy model. Secio The semi-lie cedibiliy model Coside iie seuece θ. + o dom vibles. Assume h o ixed θ he vibles. + e codiiolly ideede d ideiclly disibued codiiolly i.i.d. The vibles e obsevble d θ is he sucue vible. The vible + is cosideed s beig o ye obsevble. We ssume h ; + hve iie vice. Fo we ke he ucio o + we w o oecs. We use he oio: μ θ E. [ ] ; + This exessio does o deed o. Fo his model we deie he ollowig sucue mees: m E μ θ E E E. [ ] Cov [ μ θ μ θ ] [ ] θ { [ ]} [ ] { [ θ ]} E.3 b Cov.4 c Cov.5 [ ] d Cov μ.6 o. These exessios do o deed o +. The sucue mees e coeced by he ollowig elios: c + b.7 d b.8 o. This ollows om he covice elios obied i he obbiliy heoy whee hey e vey well-kow.

3 Esimio o he sucul mees i he geel semi-lie cedibiliy model 33 Jus s i he cse o cosideig lie combiios o he obsevble vibles hemselves we c lso obi o-homogeeous cedibiliy esimes kig s esimos he clss o lie combiios o give ucios o he obsevble vibles s show i he ollowig heoem: Theoem. Oiml o-homogeeous lieized esimos The lie combiio o d he dom vibles ; closes o μ θ E [ + ] d o + i he les sues sese euls: M z + m z m.9 whee z z. z is soluio o he lie sysem o euios: [ c + d ] z d o o he euivle lie sysem o euios: + b z b.. Alicios o Theoem.: Le us coside he cse o oe give ucio i ode o oxime μ θ. We omule he ollowig heoem. So o he secil whe Theoem. eds: Theoem. Oiml o-homogeeous lieized esimo The lie combiio o d he dom vibles closes o μ θ i he les sues sese euls: M z + m zm whee m E[ ] z d [ ] / c + d wih d Cov[ ' ] Cov[ ' ] Cov[ ] The esimo M o θ M +. + whee x z x + m z m...3 d ' ' ; c..4 μ o Theoem. c be dislyed s:.5 Le us oge ow bou his sucue o d look o y ucio θ. I e cosideed oly ucios such such h.5 is closes o μ h hs iie vice he he oiml oximig ucio esuls om he ollowig heoem:

4 34 Vigii Asiu Theoem.3 Oiml oximig ucio o uiue oiml ucio +. + is closes o μ θ d o + i he les sues sese i d oly i is soluio o he euio: g E [ ] E[ ].6 Mi E g + Poo: we hve o solve he ollowig miimizio oblem: + g. g.7 Suose h deoes he soluio o his oblem he we coside: + αh h biy like i viiol clculus. Le: {[ ] } wih ϕ α E [. αh. αh ] Clely o o be oiml ' {[ ][ h +. + h ]} { } +.8 ϕ so o evey choice o h : E +..9 mus hold. This c be ewie s: E [ h h h ]. o: E h E + E. [ { [ ] [ ]}] Becuse his euio hs o be sisied o evey choice o he ucio h oe obis he exessio i bckes i. mus be ideicl o zeo which oves.6. The ollowig exmle is licio o Theoem.3: I. + c oly ke he vlues. d P +. + [ ] μ θ d o o: he is closes o + i he les sues sese i d oly i o is soluio o he lie sysem: +. Ideed: : P E ; E P Iseig hese exessios o: ; [ ] P [ ] E [ ] d [ ].6 leds o 3 The secil hiechicl semi-lie cedibiliy heoy is ohe licio o he secil semi-lie cedibiliy heoy. E io

5 Esimio o he sucul mees i he geel semi-lie cedibiliy model 35 Like i Jewell s hiechicl model we coside oolio o cocs which c be boke u io P secos ech seco cosisig o k gous o cocs. Ised o esimig: + μ θ θ E θ θ he ue e isk emium o he coc [ + ] θ E[ θ ] ν + he ue e isk emium o he seco we ow esime: θ θ E[ θ θ ] θ E[ θ ] + μ he ue e isk emium o he coc + ν + he ue e isk emium o he seco whee P d k. I semi-lie cedibiliy heoy he ollowig clss P k o esimos is cosideed: + α i i α whee. i e ucios give i dvce. Le us coside he cse o oe give ucio i ode o oxime + o ν θ d μ θ θ. We omule he ollowig heoem: Theoem.4 Hiechicl semi-lie cedibiliy Usig he sme oios s ioduced o he hiechicl model o Jewell d deoig s s d s s oe obis he ollowig les sues esimes o he ue e isk emiums: ν θ m z m + z μ θ θ m z m + z zw w w w z 3. k whee: w zw w z w. d /[ c + w. d]. z. he cedibiliy co o coc level wih: d Cov ' d Cov ' ' c Cov V d: z z. D /[ C + z. D] he cedibiliy co seco level wih: D Cov w ' w D Cov w ' ' w C Cov V. w w w Remk. -he lie combiio o d he dom vibles P k closes o + d o ν θ i he les sues sese euls ν θ d he lie combiio o d he dom vibles P k sues sese euls μ θ θ. closes o μ θ θ i he les

6 36 Vigii Asiu Remk. -i should be oed h he soluio.9 o he lieized cedibiliy oblem oly yields sisics comuble om he obsevios i he sucue mees e kow. Geelly howeve he sucue ucio U is o kow. The he esimo s i sds is o sisic. Is iees is meely heoeicl bu i will be he bsis o uhe esuls o semi-lie cedibiliy. I he ollowig secio we coside diee cocs ech wih he sme sucue mees so we c esime hese uiies usig he sisics o he diee cocs. Secio 3 Pmee esimio The esimo obied i he evious secio coied sucue mees. I his secio we ssume he sucue mees e ukow so he exessios o hese seudo- esimos e o loge sisics. Bu sice he cocs e embedded i collecive o ideicl cocs we ow hve moe h oe obsevio vilble o he isk mee θ so we c elce he ukow sucue mees by esimes. So ow h we embedded he see coc i collecive o ideicl cocs i is ossible o give ubised esimos o hese uiies. I should be oed h he oximio o + o o μ θ bsed o uiue oiml oximig ucio is lwys bee h he oe uished i Secio bsed o escibed oximig ucios.. The useuless o he le oximio is h i is esy o ly sice i is suicie o kow esimes o he mees b eig i he cedibiliy cos z. I his secio we give some ubised esimos o he mees. Fo his uose we coside k cocs k k ideede d ideiclly disibued dom vecos d ' θ. θ o k. The coc idexed is dom veco cosisig o dom sucue mee θ d obsevios. whee k. Fo evey coc k d o θ ixed he vibles. e codiiolly ideede d ideiclly disibued. Hee we will oly deive esimos o he ollowig mees: m E[ ]. E{ Cov[ ]}. b Cov{ E[ ] E[ ]}.3 Oe c ove he ollowig heoem o hold. Theoem. Ubised esimos o sucue mees Le:

7 Esimio o he sucul mees i he geel semi-lie cedibiliy model 37 k k k m.4 k.. k.5 b k k. k. k.6 he: E m m E E b b.7 We close his secio givig s licio o his esimio he ubised esimos obied o he sucul mees o he geel semilie cedibiliy model. A licio o Theoem.: The esimos k m.8 k k k. k..9 k b... k k k e ubised esimos o he coesodig sucue mees i.e.: E m m E E b b. Ideed we hve: k k k E m E[ ] m m m. k k k see.3. So he veiicio o he is euliy. is edily eomed. Remk Noe h he usul deiiios o he sucue mees ly wih θ elcig θ d elcig so: m E μ θ E E E.3 b [ ] { [ ]} [ ] { Cov[ ]} [ μ θ μ θ ] Cov{ E[ ] E[ ]} [ ] E.4 Cov.5 c Cov.6 Nex: k E E.. k

8 38 Vigii Asiu [ Cov. k E E. Cov. E. E + + Cov.. + E. E. Bu: Cov Cov[ ] k Cov + E E ' ' b see.8 ' becuse: + b see.9 ' o ' we hve: Cov[ ' ] E[ Cov ' θ ] + Cov[ E θ E θ ] E{ E θ E θ E θ }+ + Cov '.7 [ ' ] [ ] [ ' ] E{ E[ θ ] E[ θ ] E[ θ ] E[ [ μ θ μ θ ] ' ' ]} + b b.8 o we hve: Cov E Cov θ + Cov E θ { [ ]} [ ] ] + b [ E.9 Cov ' δ ' + b. Nex: E E [ ] m. E [ ] E m. Also we hve: Cov. Cov ' δ ' + b + b ' ' Cov. + b.3 Nex: E. E m m m.4 Cov. Cov ' δ ' + b. + b ' ' see he clculios om.3.

9 Esimio o he sucul mees i he geel semi-lie cedibiliy model 39 Cov. + b.5 Nex: E. E m m m.6 d: Cov.. Cov ' δ ' + b + b ' ' Cov.. + b.7 Iseig d.7 i.7 oe obis: k E [ b m m b m m k + b m m + + b + m m ] s ws o be ove see Filly we hve: E b k Cov.. + E. E. Cov. k k E. E Cov. E E. + k k k + Cov + E E k k k k.8 Bu: Cov.. + b.9 see.7 Nex: k k Cov. ' ' ' ' Cov δ δ + b k k ' ' k ' ' + b becuse: k k ' ' ' ' Cov ' ' Cov ' ' ' δ + ' b see. ' δ ' δ ' + b.3 see.3 '

10 4 Vigii Asiu whee: Cov ' ' E[ Cov ' ' ] + Cov[ E E ' ' ] E [ E ' ' E E ' ' ] + Cov[ E E ' ' ] E[ E E ' ' - E E ' ' ] + E. θ i ' Cov ' ' '.3 d: Cov. + b.3 k k k Nex: k k Cov. Cov ' ' δ ' δ ' + b k k ' ' k ' ' + b k k see.3. Cov. + b.33 k k k Nex: k k k k Cov Cov ' ' δ ' δ ' k k k ' ' k ' ' + b + b k k Cov + b.34 k k k k Also we hve: E. m.35 see.6. E. m.36 see.4. k E E E km m k k.37 k k see

11 Esimio o he sucul mees i he geel semi-lie cedibiliy model 4 k E E E km m k k.38 k k see Iseig he vlues o he covices d o he execios see d.38 i.8 ovides us wih he desied esuls. Ideed: k E b + b + m m + b k k k + b m m + + b + m m k k k s ws o be ove see 4. Coclusios b m m k This e comlees he soluio o he semi-lie cedibiliy model i cse o o homogeeous lie esimo o + o wh mous o he sme o μ θ. I view o he ssumio bou he ideedece o cocs i migh come s suise h he emium o coc ivolves esuls om ohe cocs. A close look his ssumio evels h his is so becuse he ohe cocs ovide ddiiol iomio o he sucue disibuio. Fo his eso he clim igues o ohe cocs co be igoed whe esimig he mees eig i he semi-lie cedibiliy esime o coc. I his icle he semi-lie cedibiliy model is eied by he ioducio o he isoled coc i collecive o cocs ll ovidig ideede iomio o he sucue disibuio. Bu sice he cocs e embedded i collecive o ideicl cocs we ow hve moe h oe obsevio vilble o he isk mee θ so we c esime hese sucul mees i he semilie cedibiliy model usig he sisics o he diee cocs. The bove wo heoems show h i is ossible o give ubised esimos o hese uiies he oolio chceisics i we embed he see coc i collecive o ideicl cocs. The icle cois desciio o he semilie cedibiliy model behid heeogeeous oolio ivolvig udelyig isk mee o he idividul isks. Sice hese isks c ow o loge be ssumed o be ideede mhemicl oeies o codiiol execios d o codiiol covices become useul. The oigil model ivolvig oly oe coc cois he bsics o ll uhe semi-lie cedibiliy models. I he eied semi-lie cedibiliy model oolio o cocs is sudied o be ble o use he semi-lie cedibiliy esuls. Theeoe he mi uose o his e is o ge ubised esimos o he oolio chceisics. The mhemicl heoy ovides he mes o clcule useul esimos o he sucue mees. Fom he cicl oi o view he oey o ubisedess o hese esimos is vey elig d vey +

12 4 Vigii Asiu cive. The c h i is bsed o comliced mhemics ivolvig codiiol execios codiiol covices d viiol clculus eeds o bohe he use moe h i does whe he lies sisicl ools like discimioy lysis scoig models SAS d GLIM. These echiues c be lied by ybody o his ow ield o edevo be i ecoomics medicie o isuce. R E F E R E N C E S []. Asiu V. Coibuios o he cedibiliy heoy Docol Disseio Uivesiy o Buches-Fculy o Mhemics. []. Asiu V. U model de cedibilie Revis Sudii şi Ceceăi de Clcul Ecoomic şi Cibeeică Ecoomică II [3]. Gooves M.J. Ks R. V Heewde Buwelickx T. Isuce Seies volume 3 Eecive Acuil Mehods Uivesiy o Amsedm The Nehelds 99. [4]. Peikäie T. Dyki C.D.Pesoe M. Pcicl Risk Theoy o Acuies Uivesié Pieé e Mie Cuie 99. [5]. Sud B. A Ioducio o No-Lie Isuce Mhemics Veöelichuge des Isius ü Vesicheugswissesch de Uivesiä Mheim Bd 8 VVW Klsuhe 99.