A Statistical Model for the Costs of Passenger Car Traffic Accidents

8/6/2019 A Statistical Model for the Costs of Passenger Car Traffic Accidents

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A Statistical Model for the Costs of Passenger Car Traffic Accidents

Author(s): B. S. Van Der Laan and A. S. LouterSource: Journal of the Royal Statistical Society. Series D (The Statistician), Vol. 35, No. 2,Special Issue: Statistical Modelling (1986), pp. 163-174Published by: Blackwell Publishing for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2987520 .

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The tatistician1986) 35,pp. 163-174A statisticalmodel for he costs of passenger car trafficccidentsB. S. VAN DER LAAN& A.S. LOUTEREconometricnstitute,rasmusUniversityotterdam,ostbus 738, 000DRRotterdam,heNetherlands

Abstract.he total osts fdamage ausedbypassengerar rafficccidentserdistancentervalf nindividualmotoristanbe split ntotwo components:henumber f accidentswhich ccur o themotorist,nd the mount f damage ssociatedwith hese ccidents.In ourstudywe present probability odel or henumberf accidentsf n individualmotorist.We also discussmodels, n empirical esults, or heamount f damage.The integrationf bothmodels esultsn a generalmodel or he osts f damage.From hisgeneralmodelwe derive modelfor he costs f ndemnityo be paidby car nsurancecompanies.The models reapplied o Dutch ar nsurance ata.1 IntroductionIn this aperweconstructstatistical odel or he osts fdamage fpassengerartrafficccidents fDutchmotorists.n the iteratureoncerningisks faccidents,models f his ype redescribedy, or xample,eal 1969),Beard t l. (1977), ndPanjer& Willmot1983), nd applied y, or xample,Weber1970).We remarkhattheoriginalevelopmentfa model f this ype atesbacktoLundberg1909).Asdistinct romheusualapproach, hich tarts romhenumberfaccidentserunitoftime,we considerhenumberfaccidents erdistanceriven.Weanalysehefollowingypes f nsurancesnddamages:(i) Third-partynsurancenconnectionith hird-partyamage,nd(ii)All-risknsurance,hichs a third-partylusa casco nsurance,ith ifferentamountsf dditionalxcess,nconnectionith llrisk amage, hichsthe um fthird-partyluscascodamage, xcluding amage esultingrom articularypes faccidents,uch s collisionwith irds r animals, s well s damage esultingromfire,heft,torm, indscreenrack,tc.We donot nalyse articularypes f ccidents,ecause hese ypesf ccidentsonotoccur requently,heir mountsfdamage reusuallyow, ndthedrivers, ngeneral,otresponsibleor hedamage.Moreover,claim or amage f particulartype faccident oesnot nfluenceheno-claimiscount,or s it,usually,ubjectoan excess.2 Ageneralmodel or he osts f ar ccidents2.1 AgeneralmodelFor the onstructionf model oexplainhe otal osts fdamage romccidentsfindividualmotoristser distance nitwefollowheapproach f,for xample,eal(1969,p. 12), nd Beard t al. (1977,pp.21-22).Wemake he ollowingssumptions:


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164 van derLaan & Louter(i) The drivingf a certain istancentervalyan individualmotorists a randomexperimentith ccident s outcomes.(ii) The amount fdamage, ssociated ithn accident,s the utcome f randomexperiment.(iii) The probabilityistributionsf thenumber f accidents, hich ccur o anindividualmotorist,nd that f the mount f damage, ave specific orms,uttheparameteraluesvary rommotoristomotorist,ue to factors hich nfluenceherisk hemotorists exposed o.(iv) The number foutcomes, ,of he xperimentiven nder i) doesnotdependon theoutcomes f the xperimentiven nder ii); theoutcomes fthe xperimentgiven nder ii) are mutuallyndependent,hey re subject o the same probabilitylaw, nd they re ndependentfthedistanceriven.On thebasisofthese ssumptionse are able to constructgeneralmodel or he

costs fdamage froad ccidents or n individualmotorist. edefinehefollowingrandom ariables.N=the number f accidents hich ccur o an individualmotorist rivingomedistancenterval.etp(n) bethemassfunctionfN,with 0, 1, 2,....Y=the amount fdamage f given ccident. et F(y) be thedistributionunctionof his andom ariable, ith >O.X=the total ostsofdamage f accidents fan individualmotoristrivingomedistancenterval.etG(x)be thedistributionunctionfX,with >- .Wethen etX-0 ? forN=0XI YI+ +Y' forN=n>0Assumptioniv) can nowbe written ore xactlynterms fN and YJ s follows:(iv) The sequence Y1, Y2) ... . Yn ... is a sequenceof independentdenticallydistributedandom ariables,atisfying[IYI < oo, = 1,2, . ., n, . ., andN is aninteger-valuedandom ariable, atisfying[N] oo, whose aluesn does notdependon thevalues f heYj's.The distributionunctionfX can bewrittens

G(x) Pr[X?x] = Pr[Y1 . . + YN


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Costsofcar trafficccidents 165The expected ostsofdamage or n individualmotoristriving unitdistanceequals

E[X] =E[N] E[Y]andthevariancesgiven yvar(X) E[N] var(Y) E[Y}2 var(N)

(cf Feller 1971,p. 167)).Now we have obtained general rameworkorthe total costsof damageofaccidents or n individualmotorist. hen he distributionunction(y) andthemass function(n) areknown, r estimated, e areable, nprinciple,o derive hedistributionunction (x).This mplies hatwe areable, nprinciple,o estimatehetotal osts fdamage f ccidents f nindividualmotorist.An insuranceompany oes not always ay ndemnityor heentire mount fdamage esultingromn accident.tpays ndemnityor he mountf he laim,.e.the mount fthedamage educedythe mountf he xcess, ndmoreover,tpaysat most the amount f the suminsured, educed y the amount f the excess.Therefore,e want o adjust hegeneralmodel o a modelfor hecostsof claims,suitableor pplicationsithnsurancelaimdata.Given ome rbitraryntervalu= [a,b],where representshe mountf he xcessand b the um nsured, e definehefollowingandom ariables.K=the number f claims, .e., thenumber f accidents ith mountsf damageequal to or exceeding . Letp(k;u)be the massfunction fK,withn= 0, 1,2, ...Z=the size ofthe laim fa given ccident, here

I ifYbLetH(z;,u) e thedistributionunctionfZ with >0.U=the otal osts fclaims faccidentsfan individualnsureduringhedrivingof omedistancenterval.etG*(u;u)ethedistributionunctionfU,with >-O.The probabilityistributionsf Z, K and U can be related o theprobabilitydistributionsf, respectively,, N andX. In thiscaseK can be understoods a

random sum of N stochastic ernoulli-variables,.., RN, withparameterp= Pr[Y-a]= 1-F(a). The massfunctionfK isgiven yp(k;,u) Pr[K= k] E Pr[K= AIN=] Pr[N= i]

i=n

00

= E (k)pk (1 -p)ik Pr[N= ]i=kIt caneasily e derivedhat hemathematicalxpectationfK equals

E[K]=p E[N]and that tsvariancesequaltovar(K)=p2 var(N)+p (1-p) E[N]


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166 vander aan & LouterThe probabilityistributionsfZ can be expresseds a functionf hedistributionfunctionf Y as follows

0 for


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Costs f ar trafficccidents 167Many uthorsmphasisehat henumber f miles riven eryear s an importantriskfactor.We mentionMehring1962), Jorgenson1969), Weber 1970), Ferreira

(1971),White1976),Foldvary1975,1976, 977, 978 nd 1979), utt t l., 1977),and Van der Laan (1979). To incorporateherelation etween henumber faccidentsndthe nnualmileage, ecan nclude his actors anexplanatoryariablefor hePoisson arameter, s suggestedyWeber1970). nouropiniont smoresatisfactoryo considerhenumber faccidents s a linear unctionf thedistancecoveredatherhen s a linear unctionf he ime eriod,s already roposedyVanderLaan 1971).It is oftenrgued hat he oisson istributions not proper nefor henumberfaccidents,ecause t can be expectedhat heparameter s not constant. owever,thenwe canconsiderhe ifferentalues fAas the utcomesf randomariable .We choose s prior istributionhegamma, he ognormal,nd the runcatedormaldistribution. e remarkhat hePoisson istributionith hegamma istributionstheprior orA, esultsna negativeinomial osterioristribution.There s no theoreticalerivationf distributionor he mount fdamage. eardet al. (1977)state hat the xistencef thedistributionunction(z) [of he ize ofthe laims,ndthereforef the mount fthedamageaddedbythe uthors)]s anaxiomnthe heoryfrisk" p. 18).Anextensiveeviewfdistributions,hich anbeconsidereds distributionsor he mount fdamage f ccidents,sgiven yKupper(1962).Biihlmann Hartman1956),Leimkuhler1963),and Van derLaan& Boermans(1970) found hat (truncated)ognormal istributionitted erywellto data ofaccidents.Weber1970) foundhat mixtureftwo xponentialrobabilityistribu-tions nd that compound xponentialistributionrovided reasonableit ohissample f claim izes.We must eep nmind hat hese our tudies avebeenbasedonsampleswhichreprobablyotdrawn rom omogeneousopulations.ewhouseet al. (1980) suggestheBox-Coxfamily f distributions.heyfit hatfamilyfdistributionsithuccess o a setofdata nvolving edical xpenses.As there renotheoreticalonsiderationshicheadto a probability odel or heamountfdamage, eshall etour hoice epend nempiricalonsiderations.n thenext ection ive istributionsreconsidered:he ognormalistribution,hegammadistribution,he sum of twoexponentialistributions,compound xponentialdistributionwhere heparameterf theexponentialistributions assumed o begammadistributed),ndthenormal istribution,fter heobservationsretrans-formedy heBox-Coxransformation.his ransformationeads s follows. etYbesome continuous andom ariable.ThenBox & Cox (1964) state hatfor omedistributionsfY,theresa valueof i,such hat herandom ariablep-iZ- yl- I for 0

=log Y for =Ois normallyistributed.3 Anapplicationf the model oDutch car insurance ata3.1 ThedataWe have available nsurancend claimdataconcerning1981passengerar nsur-anceswith especto theyears1971and 1972,whichwe obtained rom nationalDutch nsuranceompany. his companyouldnotprovide ata on thenumberf


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168 vander aan & Loutermiles he nsurants rive eryear.They ould onlygive nformationegardingheinsurantstatementfhe drivesmore r essthan 0,000kilometres12,427-5miles)per year.Thedata, herefore,ad to be supplementedith ata on thenumber fmilesthe nsurants riveper year.These data wereprovided y the NetherlandsCentral ureau f Statisticsrom n investigationn thepossession ndtheuse ofpassengerars ntheNetherlands.ow both ets fdatawere ombineds describedby Van der Laan & Louter 1985).They lso give detailed escriptionfthedataused n thepresent aper.Van der Laan & Louter 1985) explorewhichof the variables, bout whichinformations available,make significantontributiono the explanation'fthenumber f claimsper mile ravelled,nd of the laim ize. t appears hat hemostimportantiskfactor or he number f claims s the numberf claim-freeears.There ppears o be a weak ssociationetweenhe laim ize and factorsuch stheprice f the ar, he reaofcoverage fthe nsurancend thedegree furbanisation.For a selectionfmore r esshomogeneousroups fthe nsurantse fit robabilitydistributionsothenumber fclaims ermile ravelledndto the mount fdamage.3.2 Fitting istributionso the ataIn this ubsectionegive, or achof he wo ategoriesf nsurances,he esult f heprocess f fittingf four robabilityistributionso thenumber f claims ermiletravelled,nd offive robabilityistributionso the ize ofclaim.It is a matter f course hat here rebig differencesetween henumber fmilesthe nsurantsrove eryear. he average umber fmiles f ll nsuredonsideredsabout10,500miles ndthe standard eviations about4500miles.We consider,subsequently,he numberfclaims fthose,more r less,homogeneousroups finsurants, hodrove t leastm milesduringheperiod onsidered,ntheir irstmmiles,wherem equals5000,10,000 nd15,000, espectively.The parametersf the distributionsre estimated n thebasis of the maximumlikelihoodstimationrocedure.he goodnessffit s compared ymeans f he hi-square est tatistic, 2, lthought s known hatwhen his est tatistics basedonmaximumikelihood stimators,t is notX2-distributedcf Chernoff Lehmann,1954).Thus, he hi-squarealues ivenmust econsideredarefully.We remarkhat hevariance f thenegative inomial istribution,s well s thevariance f the Poisson ognormal istribution,lways xceeds hecorrespondingmathematicalxpectation.n thecasesthatthe samplemeanexceeds hesamplevariance,hefit fthese wodistributionsends oan unconditionaloisson istribu-tion.ThePoisson runcatedormal istributionhows similaresult.nsuch aseswewillnotfithe hreeompoundoisson istributions.his mplieshatwedo nothave n alternativeor hePoisson istributionnthese ases.Because f he ow verage umber fclaims ermmiles,nthe nehand, ndtherelativelymall izeof thesample he nsurants,n theother and, henumberfinsurantsaving wo r more laims erm miles s relativelymall. herefore,t s nsomecases mpossibleo test hegoodness f fitwith hehelpofa chi-squareest,because henumber fdegreesffreedoms lessthan ne.Table1 shows heresultsfthefits. roup1 is thegroup fthird-partynsurants,group that f all-risknsurants.heresultsfbothgroupsreof nterestor hemodel f the osts fclaims. heresults fgroup canalsobe used oestimatehevaluesof theparametersfthemodel or he osts f accidents.tappears hat hePoisson istributionits ometimesather ell ndsometimeserywell nall cases fthird-partylaims,nd also nthe asesof ll-risklaims or hensurantsaving or1 claimfreeyear. he compound oissondistributionsre not pplied n six of the


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Costs f artrafficccidents 169Table1. Frequencyistributionsf thenumberfclaims er m miles

PoissonNumber f Observed Poisson truncated Poissonclaims frequency Poisson gamma normal lognormalGroup1:third-partylaims, ll insurants, =5000 miles, 276 nsurants0 6965 6961-0 6965 3 6965.4 6965 51 301 308 1 2998 2996 299-52 9 6-9 10.9 110 11 0>2 1X2 154 008 10 11Group1: third-partylaims, ll insurants, = 10000miles, 449 nsurants0 1332 13311 13318 13319 1331 81 111 1130 1115 1114 111.6>1 6 49 56 57 56X2 027 003 002 004Group 1:third-partylaims, ll insurants, = 15000miles, 25 insurants0 660 6574 6598 6597 65951 59 644 598 600 604>1 6 3.3 54 53 5.1X2 277 009 0 11 020

Group1: third-partylaims,nsurants ith or 1claimfree ear,m=5000 miles, 523 nsurants0 1319 13260 * * *1 197 183.72 7 127>2 0 06A2 4 18Group1: third-partylaims,nsurants ith or I claimfree ear,m= 10000miles, 05 insurants0 147 1500 * * *1 52 4682 6 73>2 0 08X2 169Group1:third-partylaims,nsurants ith or1 claimfree ear,m= 15000miles, 7 insurants0 67 669 * * *1 24 248>1 6 52A? 0 14

Group : all-risklaims, ll insurants, = 5000miles, 061 nsurants0 2888 28815 28878 28863 288721 161 174.2 1622 164.9 163.42 12 103 9.3 9.5>2 0 0.7 0.5 0 8A? 9 19 1.02 1 44 1 49Group2: all-risklaims, ll insurants, = 10000miles, 879 nsurants0 1689 16687 1688.5 1677.2 1686.91 160 1980 1632 1804 166.72 27 122 230 192 206

>2 3 4.3 22 4.7A? 3337 1 14 586 288twelve ases,where hesamplemeanexceeds hesamplevariance. he fit fthenegative inomial istributions good n the other ix cases,althought mustbe


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170 van derLaan & LouterTable 1. (continued)

Group2: all-risklaims, ll insurants, = 15000miles, 47 insurants0 560 5518 5598 5577 55921 73 878 739 772 7522 12 112 10-6 10.3>2 2 2.1 1 6 22A, 8-57 008 051 036Group2: all-risklaims, nsurants ith or 1 claimfree ear,m=5000 miles, 77 insurants0 625 6292 * * *1 140 13282 12 14.0>2 0 1.0A, 175Group2: all-risklaims, nsurants ith or1 claimfree ear,

m= 10000miles, 68 insurants0 249 2516 * * *1 101 9572 15 182>2 3 25X2 0096Group : all-risklaims, nsurants ith or1 claimfree ear,m= 15000miles, 45 nsurants0 87 907 * * *1 50 42 52 6 100>2 2 18A2 308

* The samplemean xceeds he ample ariancenthese ases.remarkedhat here rerelativelyew lasseswith ver neobservation.he resultsfthe fits f the other wocompound oisson distributionsndicate hat hey recomparable ith r nferioro thefit fthenegativeinomial istribution.oreover,we point utthat hese its akemuch omputerime.Next,we consider hefive istributionf the claim ize.The parametersf thedistributionsre estimated ith hemethod fmaximumikelihood,ndthe oodnessoffits tested ymeans f he hi-squareest tatistic,2.Wecomputeor achfit heP-value elated o a valueobtained y thevalueofAP, s ifAP sx2- istributedithk-r-degrees ffreedom,here is thenumber f classes ndr is thenumber funknownarameters.hernoff Lehmann1954) proved hat herealvalueof theP-values higherhan heonebasedon the 2(k-r-l)-distribution,ut ower han heone basedon theX2(k-1)-distribution.ence,we must ake nto ccount hat he ealP-values xceed he orrespondingomputedalues.The third-partylaimsizes are, n general, qual to the amounts fthird-partydamage ssociatedwith heaccidents.herefore,e can fit nconditionalistribu-tions o the hird-partylaim ize. With especto thegroup f ll-risknsurants,erestricthe nalysisf he laim izeto thosensured, hohave n excess fDfl.100orDfl.150. We fit onditionalistributionso the ll-risklaim ize.Firstly,e takeas point f runcationfl.150, econdly, etakeDfl.500. The reason or hishigherpoint f runcations,thatweexpecthat laimswith ow izes reunder-representedinthe ample. or, tcanbe profitableor n insurantotto claim lowamount fdamage, o thathedoesnot ose his no-claim iscount.Moreover,laims or ascodamageof accidents ausedby drivers f 24 yearsold or youngerre under-represented,ecause hese rivers ad an additionalxcess fDfl.150for ach laim.


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Costs f artrafficccidents 171Table 2 givestheapproximating-values or selectionf groups. rom heseresultst s clear hat he ognormalistributionnd theBox-Cox amilyit erywell

to the et ofthird-partylaimdata. Forall-risklaim izes, nd point ftruncationDif.150, nly heBox-Cox amilyhows goodfit. o get cceptableits fthe therdistributions,t s clear hatwemust ot runcatehedistributionlose o the mountofthe xcess.Moreover, e seethat hegamma istributionives oodfitsnlyn thecasesof ll-risklaimswith high oint ftruncation.Table 2. P-values fthefits f five istributionso the laim ize

Mixed Compound Box-CoxLognormal Gamma exponential exponential familyGroup1: third-partylaimsCar nsurances ith arprice


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172 vander aan & Louter40

30-

U_

10_

0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 3900 4200 4500 4800Size ofdamageFig. 1. Empirical requencyunctionnd density unctions or ll-risk amages: lognormal

densityunction. .......ox-Cox amilyensityunction.

120-

100-C 80LL

60-

40-

20-

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800Size ofdamage

Fig.2. Empirical requencyunctionnddensityunctionsor hird arty amages:lognormalensityunction, lognormalensityunction, ?---Box-Coxv amilyle.nsityuvnction. Boxv-Cox%am"1iyest uc Ion2


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Costs f artrafficccidents 1734 ConclusionsIn thispaperwe derived statistical odel forthecosts of claimsofinsurants,deduced rom model or he osts fpassengerar rafficccidents. e showedhat,given ome ssumptions,t spossible oestimateheparametersf hedistributionsof henumber f ccidentsndof he mount fdamage nthe asis f laim ataofinsurants.Theapplicationfthemodelwashamperedythe acthatwehadonly vailablelimited etofdatarelatingo 1or2 years. he numberf hosensured ith claimsormore ppeared obe,relatively,mall. herefore,ecouldnot xpectatisfactoryresultswhenwe fitdistributionso thenumberfclaims.Nevertheless,heresultsobtained ndicate hat the Poissondistributionits erywell to the number faccidents ccurringo an individualmotoristrivingomedistancenterval. hevalueofthePoisson arameteran beestimated ore ccuratelyromnsurancendclaimdata,whenwe considert as a functionf setofcharacteristicselatingo theinsurant,iscar, ndthe ype f nsurance,nspite ftheno-claimule.Wecannot erive, ntheoreticalrounds, distributionor he mount fdamage,however,wodistributionshow very oodfit o our et f laim ata: he ognormaldistributionnd theBox-Cox amily. heestimationftheparameter oftheBox-Coxfamilys hamperedythefact hat tsmaximumikelihoodstimatorannot edeterminednalytically.AcknowledgementThis articles a shortenedersion freport 530/A f theEconometricnstitute,Erasmus niversity,otterdam.heauthors ish o thank rofessor.Koerts or isvaluable ontributions.he authors ish lso to thankAGOVerzekeringenaftermergerow alledAEGONVerzekeringen)tGroningen,national utch nsurancecompany, hich rovideds withhe nsurancendclaim ataon whichwebased heempiricalart f thestudy,ndtheNetherlandsentral ureau fStatisticst theHague,which rovided s with ataconcerningumber f miles riven eryear yDutchmotorists.ReferencesBEARD, R.E.,PENTIKXINENT. & PESONEN, E. (1977) Risktheory,he tochasticasisof nsurance,nded (London, Chapman & Hall).Box, G.E.P. & Cox, D.R. (1964) An analysis oftransformationswithdiscussion), Journal ftheRoyalStatisticalSociety Series B, 26, pp. 211-252.BUHLMANN,H. & HARTMANN,W. (1956) Xnderungen n derGrundgesamtlichkeit er Betriebsunfall-kosten,Mitteilungener Vereinigungchweizerischenersicherungsmathematiker,and 56, pp.303-320.CHERNOFF, H. & LEHMANN, E.L. (1954) The use of maximum likelihood estimatesin X2 testsforgoodness of fit,TheAnnals fMathematicaltatistics,5, pp. 579-586.DUTT, A.D., REINFURT, D.W. & STUTTS, J.C. (1977) Accidentnvolvementndcrashnjuryates; ninvestigation y make,model and year ofcar, AccidentAnalysis and Prevention, , pp. 275-283.ERLANDER, S. (1971) A review of some statisticalmodels used in automobile insurance and roadaccidentstudies,AccidentAnalysis and Prevention, , pp. 45-75.FELLER, W. (1971) AnIntroductionoProbabilityheory nd itsApplications,nded,vol. I, (NewYork, JohnWiley).FERREIRA, J. (1971) Some analytical aspects of driver icencing and insuranceregulation,TechnicalReport No. 58 OperationResearch Center, Massachusetts Institute of Technology,Cambridge,U.S.A).


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174 vanderLaan & LouterFOLDVARY, .A. (1975) Road accident nvolvement er milestravelled-I, Accident nalysis ndPrevention,, pp. 191-205.FOLDvARY,L.A. (1976) Road accidentnvolvementer milestravelled-I1,Accident nalysis ndPrevention,, pp. 97-127.FOLDVARY, .A. (1977) Road accidentnvolvementer miles ravelled-IV,Accident nalysisndPrevention,, pp. 22-54.FOLDVARY, .A. (1978) Road accidentnvolvementer miles ravelled-IV,Accident nalysis ndPrevention,0,pp. 143-176.FOLDVARY, .A. (1979) Road accidentnvolvementer milestravelled-V,Accident nalysis ndPrevention,1,pp. 75-99.JORGENSEN,.O. (1969) A modelfor orecastingrafficccidents, ccident nalysis ndPrevention,1, pp. 261-266.KUPPER,J. 1960) Wahrscheinlichkeitstheoretischeodelle n derSchadenversicherung.eil I: DieSchadenzahl, ldtter erdeutschenesellschaftfurersicherungsmatematik,, pp. 451-503.KUPPER,J. (1962) Wahrscheinlichkeitstheoretischeodelle in der Schadenversicherung.eil II:Schadenhohe ndTotalschaden, ldttererdeutschen esellschafturVersicherungsmatematik,,pp. 95-130.LEIMKUHLER, . (1963) Truckingf RadioactiveMaterials: afety s economyn highwayransport(Baltimore,John Hopkins Press).LUNDBERG, . (1909) Uber die Theorie erRuickversicherung,ransactionsftheVIth nternationalCongressfActuaries, , pp.877-955.MEHRING, . 1962) Die SchadenstrukturnderKraftfahrtHaftpflichtversicherungonPersonenwagen,Bldtter erDeutschen esellschaftiirVersicherungsmathematik,, pp. 23-41.NEWHOUSE, .P.,ROLPH,J.E. MORI, B. & MURPHY,M. (1980) The effectfdeductiblesn thedemandformedical are ervices, ournalftheAmerican tatisticalociety,5, pp. 525-533.PANJER, .H. & WILLMOT,G.E. (1983)Compound Poisson models in actuarial risktheory,Journal fEconometrics,3, pp. 63-76.SEAL,H.L. (1969) Stochasticheory f RiskBusinessNew York,JohnWiley).VANDERLAAN, B.S. (1971) A note on the probability istributions f thenumber nd theamountofthe claims of motor car accidents, Report 7103 ofthe Econometric nstituteRotterdam,ErasmusUniversity).VAN DER LAAN,B.S. (1979) Motorists nd accidents an empirical tudy),Report7901 of theEconometric nstituteRotterdam,ErasmusUniversity).VANDER LAAN,B.S. & BOERMANS,.J.J.1970) Someprobabilityistributionsnthefield fmotorardamages an empirical tudy),Report7001 of the EconometricnstituteRotterdam,rasmusUniversity).VANDERLAAN,B.S. & LOUTER,A.S. 1985)On the numberofclaims and the claim size ofpassengercar trafficccidentsn theNetherlands, eport 529/A ftheEconometricnstituteRotterdam,Erasmus niversity).WEBER,D.C. (1970) A stochasticmodel for utomobile ccident xperience,nstitutef Statistics,MimeographeriesNo. 651 Raleigh, orth arolina,North arolina tateUniversity).WHITE, S.B. (1976) On the use of annual vehiclemiles of travel stimates rom ehicle wners,AccidentnalysisndPrevention,, pp.257-261.

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A Statistical Model for the Costs of Passenger Car Traffic Accidents