THE BALANCED CREDIBILITY ESTIMATORS WITH MULTITUDE CONTRACTS OBTAINED UNDER LINEX LOSS FUNCTION

Transcription

1 Joural of Stattc: Advac Thory ad Applcato Volum 4 Numbr 5 Pag - Avalabl at DOI: THE BALANCED CREDIBILITY ESTIMATORS WITH MULTITUDE CONTRACTS OBTAINED UNDER LINEX LOSS FUNCTION QIANG ZHANG LIJUN WU ad QIANQIAN CUI Dpartmt of Appld Mathmatc Najg Uvrty of Scc ad Tchology Najg 94 P. R. Cha -mal: zhagqag899@63.com Collg of Mathmatc ad Sytm Scc Xjag Uvrty Urumq 8346 P. R. Cha Abtract Codrg th targt prmum w propo LINEX lo fucto to olv th problm of hgh prmum by ug a balacd lo fucto mot of clacal crdblty modl. Th homogou ad homogou crdblty tmator wth multtud cotract ar drvd udr LINEX lo fucto. Fally th mulato hav b troducd to how th cotcy of th crdblty tmator. Mathmatc Subjct Clafcato: 6P5. yword ad phra: LINEX lo fucto targt prmum crdblty tmator multtud cotract. Supportd by Th Natoal Natural Scc Foudato of P. R. Cha [3658]. Rcvd Novmbr Sctfc Advac Publhr

2 QIANG ZHANG t al.. Itroducto I urac practc crdblty thory a t of quattatv mthod whch allow a urr to adjut prmum bad o th polcyholdr xprc ad th xprc of th tr group of polcyholdr. It ha b wdly ud commrcal proprty of lablty urac ad group halth or lf urac. Th wll-kow crdblty formula obtad ar wrtt a a wghtd um of th avrag xprc of th polcyholdr ad th avrag of th tr collcto of polcyholdr. Th formula ar ay to udrtad ad mpl to apply du to thr lar proprt. Th modr crdblty thory blvd to b attrbutd to th rmarkabl cotrbuto by Bühlma [] whch th frt o that bad th crdblty thory o modr Bay tattc. For th rct dtald troducto Bühlma ad Glr [] whch dcrb modr crdblty thory comprhvly. I clacal dco thory th lo fucto uually focu o prco of tmato. Howvr good of ft alo a vry mportat crtro. Thu thr a d to provd of tmato formally. Zllr [] troducd a gral cla of balacd lo fucto of th form θ L ( x θ) w( δ ( x) θ) + ( w)( x ) (.) whr w ad δ ( x) a pr-dtrmd targt tmator of θ. Huag ad Wu [5] tudd th Bühlma ad Bühlma-Straub modl udr balacd lo fucto tablhd th crdblty prmum wth commo ffct. Furthrmor ug balacd lo fucto a gralzato of th crdblty xpro Bühlma [] udr th dtrbuto fr approach alo obtad. Howvr om tmato problm u of ymmtrc lo fucto may b -approprat for xampl Vara [9] Brgr [3] Frgudo [4] Promlow [6] Promlow ad Youg [7]. Crdblty tmator ug ymmtrc lo fucto uually lad to vry hgh

3 THE BALANCED CREDIBILITY ESTIMATORS WITH 3 malu. To ovrcom th problm ug aymmtrc lo fucto buldg crdblty tmator codrd. That for a polcyholdr payg too much mor rou tha ot payg ough. So aymmtrc lo fucto ovrcharg w hould b palzd mor tha udrcharg to atfy th polcyholdr. Vara [9] troducd LINEX (lar xpotal) lo fucto whch a uful aymmtrc lo fucto. a ( ) ( x θ x θ ) a( x θ) a. L > (.) Th LINEX lo fucto r approxmatly xpotally o o d of zro ad approxmatly larly o th othr d of zro th proprty of a LINEX fucto ufull buldg a crdblty tmator. Iprd by th papr w am at xtdg crdblty tmator wth multtud cotract udr LINEX lo fucto. Th rt of th papr arragd a follow. I Scto modl aumpto ar troducd ad om prlmar ar dcud. Scto 3 drv th crdblty tmator udr LINEX lo fucto. Fally th mulato hav b do to vtgat th cotcy of crdblty tmator udr LINEX lo fucto.. Modl Aumpto ad Prlmar Codr a portfolo of urd dvdual. I th portfolo ach dvdual I aocatd wth a clam xprc X j ovr tm prod j. Wrt X ( X X ). Our trt to prdct th futur clam X + for ach dvdual takg to accout all obrvd clam xprc X X X. It wll kow from tattcal thory that th bt LINEX prmum bad o all th obrvd clam X X X dotd by H( X + ) l a ( ax ( ) ) + E X X X problm th oluto of th mmzato

4 4 QIANG ZHANG t al. a( g( X X X ) X + ) m E[ a g X X X X + g ( ( ) ) ]. (.) I th clacal crdblty thory w aumd th rk qualty of a dvdual ca b charactrzd by a rk paramtr Θ whch a uobrvabl radom varabl. Gv Θ th clam X X X ar dpdt ad dtcally dtrbutd. Formally th + aumpto of th modl ar tatd a followg: Aumpto.. For fxd cotract gv Θ X ar codtoally j ( Θ ) dpdt wth E( X j Θ ) µ ( Θ ) ad Var( X j Θ ) whr m m j ar kow wght. W wll u th followg otato rgardg th wght: M dag( m m ) m M m m. Aumpto.. Th rk paramtr Θ Θ Θ ar dpdt ad dtcally dtrbutd a th am tructur dtrbuto fucto π ( θ). j Aumpto.3. Th radom vctor ( X Θ ) ar dpdt for. Udr th aumpto abov w ca df th rk prmum of X udr LINEX lo fucto for. + Dfto.. Th rk prmum ar gv by whr µ ( Θ ) E( Θ ). H ( X Θ ) l α > (.) a µ ( Θ ) ax + From th crdblty tmator of µ ( Θ ) th tmator of H( X Θ ) ca b aly drvd by ( HX Θ ) l ( ( ) ). So w frtly codr a µ Θ

5 THE BALANCED CREDIBILITY ESTIMATORS WITH 5 th homogou tmator of µ ( Θ ) by ma of crdblty da.. to olv th followg optmal problm: m E c cj R [ w( aδ ( X ) c axj cj ) + ( w)( µ ( Θ ) j ax c j cj ) ] (.3) j whr aδ ( X ) a pror cho targt prdctor of µ ( Θ ). For tattc ( ) aδ X troduc th followg mor otato: [ E aδ ( X ) ] µ Cov[ aδ ( ) X ax j dj d d d d ] ( ) λ m j λ + mjτ λ X ax j X d j dtλt t dtλt X t. t To mplfy w t δ ( Y ) aδ ( X ) axj Yj j ad Y ( Y Y Y ) hr Y ( ) Y Y Y. W ca gt th followg lmma: Lmma.. Udr th Aumpto.-.3 w hav () Th ma of Y ad Y ar gv by E( Y ) µ E( Y ) µ (.4) whr a -vctor wth all of th tr. () Th covarac btw µ ( Θ ) ad Y gv by Cov( µ ( Θ ) Y ) τ (.5) µ ( Θ ) Y whr a vctor wth th -th try ad th othr tr. Hr dcat th rockr product of matrc.

6 6 QIANG ZHANG t al. (3) Th covarac of Y gv by Cov( Y Y ) I dag( M + τ m m ). (.6) (4) Th vr of th varac matrx of Y gv by τ mm I ( ). M (.7) + mτ 3. Th Crdblty Etmator Udr LINEX Lo Fucto I th cto w procd to drv th crdblty tmator of H( X Θ ) udr th LINEX lo fucto. W tat th followg thorm: Thorm 3.. Udr Aumpto.-.3 th homogou crdblty tmator H( X Θ ) udr LINEX lo fucto ar gv by ( HX Θ ) l α Z X + ZXd + ( Z Z ) µ (3.) mτ whr Z Z. dtλt + m τ Proof. For ach fxd troduc Z Iδ ( Y ) + ( I ) µ ( Θ ) whr I a auxlary radom varabl tattcally dpdt of all th othr radom varabl th ytm ad dtrbutd a P ( I ) P ( I ) w th wght (.3). Thu w ca rwrt th optmzato problm a m E c c [ Z c c Y ) ] (3.3) whr c R c R. So th homogou crdblty tmator of µ ( Θ ) xactly th orthogoal projcto of Z o lar pac: L ( Y ) { c + c Y c R c R }.. µ ( Θ ) proj ( Z L ( Y ) ) (

7 THE BALANCED CREDIBILITY ESTIMATORS WITH 7 for xampl W t al. []). From th rlatohp btw orthogoal projcto ad crdblty tmator w hav ( ) E ( Z ) µ Θ + ZY ( Y E ( Y )). From th dfto of Z th ma E ( Z ) ca b computd by E ( Z ) µ. Morovr vw of th fact E ( Y I ) E( Y ) a cotat yldg th qualty Cov ( E( Z I ) E( Y I )). Th covarac matrx ZY ZY ca b computd by Cov( Z Y ) wd + ( w). (3.4) By Lmma. w d to gt th followg trm: ad µ ( Θ ) Y m t wd ( Y E( Y )) wdt ( X d µ ) (3.5) + m τ t t ( w) µ ( Θ ) Y ( Y E( Y )) ( w) m τ + mτ ( X µ ). (3.6) Th m t ( w) mτ µ ( Θ ) wdt Xd + X + m τ + m τ t t + ( w d mt ( w) mτ t + mt τ + mτ ) µ. If w cotra th tmator of µ ( Θ ) to b a homogou lar cla of Y w ca drv th homogou crdblty tmator. Hc w hould olv th followg problm: m E[ Z c Y ) ] wth E( Z ) E( c Y ). c c (3.7)

8 8 QIANG ZHANG t al. Th w obta th followg thorm: ar Thorm 3.. Th homogou crdblty tmator of H( X Θ ) ( HX Θ ) l a + + ( ) ZXd ZX Z Z X whr Z Z ar th am a Thorm 3. ad X X. Proof. Wrt L ( Y ) c Y wth E ( Z ) E( c Y ). Th th homogou crdblty tmator of µ ( Θ ) xactly th orthogoal projcto th lar pac L ( Y ).. µ ( Θ ) proj ( Z L ( Y ) ) ( W t al. []). Sc L( Y ) L( Y ) from th tratvly of projcto oprator ( Bühlma ad Glr []) o ca obta µ ( Θ ) proj ( proj ( Z L( Y ) L( Y) ) Z X Rcall th formula + Z Xd + ( Z Z ) proj ( µ L( Y )). µ ( Y ) Y proj( µ L( Y ). (3.8) E( Y ) E( Y ) Th proof ca b rfrrd to ( W t al. []). Irtg (.4) ad (.7) to (3.8) w gt proj( µ L( Y ) µ µ I I τ m m + mτ τ m m + mτ ( M ) Y ( M ) X.

9 THE BALANCED CREDIBILITY ESTIMATORS WITH 9 Rmark. For fxd f th X µ ( Θ ) a.. from th ctral lmt thorm. 4. Numrcal Exampl Hr w gv a xampl to how th crdblty ymator udr LINEX lo fucto ad chck th cotcy of crdblty tmator HX ( Θ ) gv a Thorm 3.. W aum that th clam of th -th cotract j-th yar X j dtrbutd a ( Θ ) ad th rk paramtr Θ xpotal θ µ varabl wth dty fucto π( θ). I ordr to compar th µ tmator Thorm 3. (3.) th followg mulato ar dd. Frt w tak a.5. 8 ad m j for j. I th mulato w aum that δ ( X ) l( X ) th E ( ( aδ X ) ) µ. W ca gt a dj Cov( aδ ( X ) ax jt aδ + j ) Cov( l ax jt τ ) l j ad d + τ X d X Th th crdblty tmator of H( X Θ ) ar gv by ( w + τ ( w) HX Θ ) l [ X + µ ]. a + τ + τ Th corrpodg quatt dfd Scto ca b drv a. µ.5a + aµ τ a + aµ a ( + aµ ) a + aµ a

10 QIANG ZHANG t al. ad H ( X Θ ) aθ +.5 a. N dffrt valu Θ ad thr wght w. w.5 w.7 ar codrd. For ach combato of valu of paramtr Θ ad w w carry out a mulato of tm. W drv th mulato rult ar ltd th followg tabl: Tabl. Th rult wth w. Θ H( X Θ ) HX ( Θ ) td Tabl. Th rult wth w. 5 Θ H( X Θ ) HX ( Θ ) td Tabl 3. Th rult wth w. 7 Θ H( X Θ ) HX ( Θ ) td whr td dcat th ma quar rror for th tmator HX ( Θ ). W ca from th tabl abov that HX ( Θ ) cott wth th prmum H( X Θ ).

11 THE BALANCED CREDIBILITY ESTIMATORS WITH Rfrc [] H. Bühlma Exprc ratg ad crdblty J. At Bullt 4 (967) [] H. Bühlma ad A. Glr A Cour Crdblty Thory ad t Applcato Sprgr Th Nthrlad 5. [3] J. O. Brgr Stattcal Dco Thory: Foudato Cocpt ad Mthod Acadmc Pr Nw York 98. [4] T. S. Frgudo A Dco Thortc Approach Acadmc Pr Nw York 967. [5] W. Z. Huag ad X. Y. Wu Th crdblty prmum wth commo ffct obtad udr balacd lo fucto Ch Joural of Appld Probablty ad Stattc 8() () 3-6. [6] S. D. Promlow Maurmt of quty Traacto of th Socty of Actuar 39 (987) [7] S. D. Promlow ad V. R. Youg Equty ad xact crdblty ASTIN Bullt 3() () 3-3. [8] R. Rao ad H. Toutburg Lar Modl Sprgr Nw York 995. [9] H. R. Vara A Baya approach to ral tat amt Stud Baya coomtrc ad tattc hoor of Loard J. Savag Amtrdam North- Hollad (975) [] L. M. W X. Y. Wu ad X. Zhou Th crdblty prmum for modl wth dpdc ducd by commo ffct Iurac: Mathmatc ad Ecoomc 44() (9) 9-5. [] A. Zllr Baya ad o-baya tmato ug balacd lo fucto Stattcal Dco Thory ad Rlatd Topc V (J. O. Brgr ad S. S. Gupta Ed.) Sprgr-Vrlag Nw York (994) g