Estimating Total Claim Size in the Auto Insurance Industry: a Comparison between Tweedie and ZeroAdjusted Inverse Gaussian Distribution


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1 Avalable onlne at BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar Estmatng Total Clam Sze n the Auto Insurance Industry: a Comparson between Tweede and ZeroAdjusted Inverse Gaussan Dstrbuton Adrana Bruscato Bortoluzzo * Emal address: Ibmec São Paulo São Paulo, SP, Brazl. Danny Pmentel Claro Emal address: Ibmec São Paulo São Paulo, SP, Brazl. Marco Antono Leonel Caetano Emal address: Ibmec São Paulo São Paulo, SP, Brazl. Rnaldo Artes Emal address: Ibmec São Paulo São Paulo, SP, Brazl. * Correspondng author: Adrana Bruscato Bortoluzzo Rua Quatá, 300, São Paulo, SP, , Brazl. Copyrght 2011 Brazlan Admnstraton Revew. All rghts reserved, ncludng rghts for translaton. Parts of ths work may be quoted wthout pror knowledge on the condton that the source s dentfed.
2 A. B. Bortoluzzo, D. P. Claro, M. A. L. Caetano, R. Artes 38 Abstract The objectve of ths artcle s to estmate nsurance clams from an auto dataset usng the Tweede and zeroadjusted nverse Gaussan (ZAIG) methods. We dentfy factors that nfluence clam sze and probablty, and compare the results of these methods whch both forecast outcomes accurately. Vehcle characterstcs lke terrtory, age, orgn and type dstnctly nfluence clam sze and probablty. Ths dstnct mpact s not always present n the Tweede estmated model. Auto nsurers should consder estmatng total clam sze usng both the Tweede and ZAIG methods. Ths allows for an estmaton of confdence nterval based on emprcal quantles usng bootstrap smulaton. Furthermore, the ftted models may be useful n developng a strategy to obtan premum prcng. Key words: auto nsurance; clam sze; regresson; Tweede; ZAIG mater. BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
3 Estmatng Total Clam Sze n the Auto Insurance Industry 39 Introducton There s a well known problem n the nsurance ndustry concernng the proper prcng of an nsurance polcy. An nsurance company s pure premum for an nsured ndvdual s made up of two components: clam probablty and expected clam sze. The clam probablty for any ndvdual s related to the number of clams expected to occur n a gven year. The clam sze s smply the dollar cost assocated wth each clam. The dffculty of estmatng the sze and probablty of clams n the nsurance ndustry has been extensvely reported n the lterature (e.g. Jong & Heller, 2008). In the past, the man dffculty was related to the credblty of the nsurance company datasets (Wesberg & Tomberln, 1982). Insurance datasets were typcally very large, contanng from tens of thousands to mllons of cases. Problems such as mssng values and nconsstent or nvald records arose. As current nformaton technology systems have become more sophstcated over the years, the processng of nformaton has become more credble than ever before. The challenge then s to employ a proper statstcal technque to analyze nsurance data. Clams and rsks have long been estmated usng a pure algorthmc technque or a smple stochastc technque (Wüthrch & Merz, 2008). These methods result n poor estmatons. Huang, Zhao and Tang (2009) consder a rsk model n whch the clam number process s treated as a Posson model and the ndvdual clam sze s assumed to be a fuzzy random varable. Jørgensen and Souza (1994) suggested a Posson sum of Gamma random varables called Tweede to estmate nsurance rsk. Accordng to Smyth and Jørgensen (2002), there s also another problem n that the proposed Tweede model does not permt the separate estmaton of probablty and clam sze. Recent studes have perceved that a zeroadjusted Inverse Gaussan (ZAIG) dstrbuton may be approprate to estmate clam and rsk n nsurance data (Heller, Stasnopoulos, & Rgby, 2006). A mxed dscretecontnuous model, wth a probablty mass of zero and an Inverse Gaussan contnuous component, appears to estmate accurately n extreme rght skewness dstrbutons. Ths suggests that probabltes can be calculated from datasets wth a large number of zero clams. The ZAIG model explctly specfes a logtlnear model for the occurrence of a clam (.e. clam probablty). When a clam has been made, the ZAIG model also specfes loglnear models for the mean clam sze and the dsperson of clam szes. It s mportant to measure the probablty and sze of clams separately because t s possble for the probablty to depend on a set of ndependent varables whch s dfferent from those that nfluence clam sze. Therefore, ZAIG estmaton appears to be more approprate for estmatng the prce of nsurance polces. Once an estmaton method has been defned, the challenge s to dentfy potental explanatory varables. Typcally, polcy holders are dvded nto dscrete classes on the bass of certan measurable characterstcs predctve of ther propensty to generate losses. We evaluate clams by consderng vehcle varables that are frequently used n the lterature. In addton to terrtory, clams have also been studed n relaton to the car manufacturer and vehcle s characterstcs: age, type and orgn. Based on prevous research, all of these varables must be used n the estmaton. Our objectve s to present the ZAIG method of estmaton to determne probablty clams and the expected clam sze n the nsurance ndustry and to formally test the results wth an estmaton based on a Tweede regresson model usng an nsurance dataset. Insurance data were collected to analyze the mpact of factors estmated by the Tweede and ZAIG methods. Ths work s dvded nto fve sectons. In the next secton, we wll dscuss the theoretcal background based on prevous research n nsurance clam estmates. We also present the Tweede and ZAIG methods n the next secton. The thrd secton dscusses the methodology and the dataset. The fourth secton presents the analyss of the results and a comparson of the fndngs from the two methods. Fnally, we present our concludng remarks and hghlght the major contrbutons of our study. BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
4 A. B. Bortoluzzo, D. P. Claro, M. A. L. Caetano, R. Artes 40 Theoretcal Background Insurance: mportance of predctons and predctors The probablty and clam sze forecast s very mportant, snce an nsurance company can use these estmates to offer or not offer premum dscounts dependng on an ndvdual clent s characterstcs or create strateges for detectng fraudulent clams (Vaene, Ayuso, Gullen, Van Gheel, & Dedene, 2007). An nsurance company can also estmate total clam sze usng vehcles characterstcs to get an dea of how much wll be spent on the clam over a certan perod and for a specfc clent portfolo. Insurance companes are constantly lookng for ways to better predct clams. Overall, nsurance nvolves the sum of a large number of ndvdual rsks of whch very few wll result n nsurance clams beng made. Meulbroek (2001) argues that nsurance companes need to treat rsk management as a seres of related factors and events. Boland (2007) suggests that, n order to handle clams arsng from ncdents that have already occurred, nsurers must employ predctve methods to deal wth the extent of ths lablty. Therefore, an nsurance company has to fnd ways to predct clams and approprately charge a premum to cover ths rsk. The predcton problem has to be consdered n the lght of compettve market nsurance (Wesberg, Tomberln, & Chartterjee, 1984). It s possble for an nsurer to beneft at least temporarly by dentfyng segments of the market that are currently beng overcharged and offerng coverage at lower rates or by avodng segments that are beng undercharged (Doherty, 1981). Regulators are usually concerned about the possblty of rate structures that severely penalze ndvduals wth some characterstcs (e.g. where they lve, model of vehcle). Therefore, nsurers look for better ways to capture the characterstcs of ndvduals that affect clam sze and probablty, and consequently dentfy nsured drvers that have a hgher propensty for generatng losses. Insurance companes attempt to estmate reasonable prces for nsurance polces based on the losses reported for certan knds of polcy holders. Ths estmate has to consder past data n order to grasp the trends that have occurred (Wesberg & Tomberln, 1982). Informaton avalable to predct the prce for a perod n the future usually conssts of the clam experence for a populaton or a large sample from the populaton over a perod n the past. Accurate estmaton may consder a large number of exposures n a dataset and a stable clam generaton process over tme. The predctors for estmatng the approprate prce for nsurance polces were selected from the automoble ndustry. In our study, we consder the ssue of prce predcton n the context of the automoble ndustry because the most sophstcated proposals have been developed n ths ndustry (Jong & Heller, 2008). Prevous research n the automoble settng has used predctors such as terrtory (e.g. Chang & Farley, 1979) and car manufacturer (e.g. Heller et al., 2006). Wesberg et al. (1984) suggest ncludng varables assocated wth the status of the vehcle such as age, type and orgn. Prevous studes have recognzed the utlty of the Tweede method n estmatng auto nsurance clams (Smyth & Jørgensen, 2002) and recent studes have shown that the ZAIG method may produce accurate models of estmaton (Heller et al., 2006). In order to estmate, t s necessary to let y be the sze expended on clams for clent and to let x be a vector of ndependent varables related to ths clent. One may represent the varable y as y 0, wth probablty (1 π ) = W, wth probablty π where W s a postve rght skewed dstrbuton. Ths type of varable belongs to the class of the zero nflated probablty dstrbutons (e.g. Gan, 2000). The parameter π s the clam probablty and W represents the clam sze related to clent. BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
5 Estmatng Total Clam Sze n the Auto Insurance Industry 41 It s mportant to note that a clam s, n general, a rare event. A small proporton of clams n a sample may lead to problems n predctng clam occurrence by a logstc model because, n ths case, the predcted probabltes tend to be small. Kng and Zeng (2001) proposed a correcton to be used n these stuatons. They used the fact that, n the presence of rare events, the ndependent varable coeffcents are consstent, but the ntercept may not be. Tweede regresson models A Tweede dstrbuton (Jørgensen, 1987, 1997) s a member of the class of exponental dsperson models. It s defned as a dstrbuton of the exponental famly (e.g. Jong & Heller, 2008) wth mean μ and varance φ μ p ; n ths work, as n Jørgensen and Souza (1994) and Smyth and Jørgensen (2002), we consder the case 1<p<2. It s possble to wrte N 0, f N = 0 y =, W = X j, W, f N > 0 j= (1) where N s a Posson random varable that represents the number of clams that have occurred for the clent and X 1, L, X N are ndependent dentcally dstrbuted Gamma random varables (contnuous varables). As a consequence W also follows a Gamma dstrbuton, whch has a postve and rght skewed densty probablty functon. In ths work, we use a loglnear Tweede regresson model, gven by T x γ μ = e, where x s a matrx of ndependent varables and γ s the parametrc vector. ZAIG regresson model The varable y follows a ZAIG dstrbuton (Heller et al., 2006) f W s a Gaussan nverse random varable. The Gaussan nverse s a postve and hghly skewed dstrbuton wth two parameters: the mean (μ ) and a dsperson parameter (λ ). It may be proved that 2 2 E(y ) = π μ and Var(y ) = π μ ( 1 π + μ λ ). In the context of ths work, μ s the expected clam sze and λ s a parameter related to clam sze dsperson. It s possble to propose regresson models for π, μ and λ as π = T h 1( x β), μ = h ( z T γ), λ h ( w T δ) 2 = 3, where h 1, h 2 and h 2 are contnuous twce dfferentable nvertble functons, β, γ and δ are parametrc vectors, and x, z and w are vectors of ndependent varables for clent. In an nsurance context, t s hghly convenent to use dfferent sets of ndependent varables to model these three parameters. Consder, for nstance, a varable that ndcates the locaton of a car owner s resdence. It s well known that robbery rates vary wthn a cty, but the prce of a vehcle does not. Snce t s expected that the locaton wll be mportant when t comes to explanng π, but not μ, then one may nclude the varable n the probablty model but not n the expected clam sze model. Ths example llustrates the statement by Heller et al. (2006, p. 227) that A problem wth the Tweede dstrbuton model s that probabltes at zero cannot be modeled explctly as a functon of explanatory varables BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
6 A. B. Bortoluzzo, D. P. Claro, M. A. L. Caetano, R. Artes 42 The followng models are adjusted: T x β e T T x γ x δ π = T, μ x β = e and λ = e. 1+ e (2) In short, ths modelng opton assures that μ and λ are, as expected, always postve and that π s modeled as a logstc regresson. It s mportant to remember the bad performance of logstc models n predctng clams, when the frequency of clams n the sample s small. Results and Dscusson Dataset and sample summary statstcs A sample was collected from a major automoble nsurance company resultng n a dataset of 32,783 passenger vehcle records belongng to a corporate fleet. As all corporaton employees could drve the vehcle, t makes no sense to use ndvdual drver characterstcs as explanatory varables for explanng probablty and clam sze. The dataset was processed to remove mssng values and generate a selecton of relevant varables. We have focused the analyss on yearly clams nvolvng robbery or accdents wth clam szes whch were greater than the vehcle s value. Clam sze refers to the dollar cost pad as a lablty of a clam. Clam probablty refers to the percentage of clams over the perod of a year. The average annual clam probablty s 1.17%, and the average clam sze s $ When the event occurs, the average clam sze ncreases to $21, For every nsurance polcy holder, twenty explanatory varables were employed. The varables correspond to vehcle characterstcs and are coded by means of bnary varables, as descrbed n Table 1. Table 2 shows the descrptve statstcs of the varables. Table 1 Lst of Explanatory Varables Varable Vehcle Age (II) Descrpton Equals 1 f the nsured vehcle s one or two years old (n relaton to contract year), otherwse 0 Vehcle Age (III) Equals 1 f the nsured vehcle s three or four years old (n relaton to contract year), otherwse 0 Vehcle Age (IV) Equals 1 f the nsured vehcle s fve or sx years old (n relaton to contract year), otherwse 0 Vehcle Age (V) Equals 1 f the nsured vehcle s seven to nne years old (n relaton to contract year), otherwse 0 Vehcle Age (VI) Equals 1 f the nsured vehcle s ten or more years old (n relaton to contract year), otherwse 0 Orgn Equals 1 f the nsured vehcle s mported and 0 f t s domestc Model/Manufac. Terrtory A combnaton of dfferent models and manufacturers n the dataset. Groups were assgned on the bass of a CHAID analyss. Clusters of terrtory were assgned based on Herarchcal Cluster Analyss Method for clam sze. It dvdes the regon nto a set of exclusve areas thought to be relatvely homogeneous n terms of clams. Vehcles are assgned to terrtores accordng to where they were usually garaged. Contnues BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
7 Estmatng Total Clam Sze n the Auto Insurance Industry 43 Table 1 (contnued) Varable Descrpton Vehcle Type (II) Equals 1 for a mdsze vehcle, otherwse 0 Vehcle Type (III) Equals 1 for a luxury vehcle, otherwse 0 Intercept New Vehcle (zero years old n relaton to contract year  Vehcle s Age (I)), Domestc, Model/Manufacturer (I), Terrtory (I) and small vehcle (Vehcle Type (I)) Table 2 Descrptve Statstcs for Proporton of Clams and for Clam Sze Varable Value Percentage of Clam Total Sample Clam sze Clam Sze>0 Mean SD Mean SD Sample Sze Vehcle Age I , , , ,854 II , , , ,559 III , , , ,589 IV , , , ,286 V , , , ,497 VI , , , Orgn Domestc , , , ,320 Imported , , , ,463 Model/Manuf. I , , , ,462 II , , , ,100 III , , , ,861 IV , , , ,331 V , , , VI , , , VII , , , ,789 VIII , , , ,879 IX , , , ,684 X , , , ,581 Terrtory I , , , ,895 II , , , ,888 Vehcle Type Small , , , ,218 Mdsze , , , ,290 Luxury , , , ,275 Complete Sample , , , ,783 Based on Table 2 one can perceve that clams occur more often wth older cars, but the cost of the clam reduces as the car s age ncreases. Domestc and mported vehcles have approxmately the same percentage of clams and clam szes. There are dfferences n the frequency and cost of clams BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
8 A. B. Bortoluzzo, D. P. Claro, M. A. L. Caetano, R. Artes 44 dependng on the model and the manufacturer (Model/Manuf). Regon I has the largest percentage of clams as well as the hghest cost for these clams. In terms of vehcle sze, most of the clams are for small and mdsze cars, whle the costs ncrease n percentage accordng to the sze of the vehcle. Inferental analyss The ZAIG model was estmated by the GAMLSS package (Stasnopoulos & Rgby, 2007; Stasnopoulos, Rgby, & Akantzlotou, 2008) for the R system (R Development Core Team, 2007). The Tweede model was estmated usng the SPSS package (verson 16.0). In ths secton we dvde the sample nto two parts: a subsample of 22,783 to ft the models, and a subsample of 10,000 to forecast the total clam sze. Table 3 shows the results of the estmates. The dependent varable s the clam sze and refers to robberes or accdents wth repar szes greater than the vehcle s value. Table 3 Tweede and ZAIG Model Results Varable Tweede Equaton 1 Equaton 2: ν=1π (Clam Probablty) ZAIG Equaton 3: μ (Clam Sze) Equaton 4: λ Estmate (SE) Estmate (SE) Estmate (SE) Estmate (SE) Intercept 6.54** (0.29) 3.70** (0.20) 10.20** (0.12) 5.32** (0.11) Vehcle s Age (II) (0.22) (0.18) 0.33** (0.11) 0.24* (0.12) Vehcle s Age (III) 0.48* (0.27) 0.01 (0.20) ) 3.22** (0.14) Vehcle s Age (IV) 0.56* (0.34) (0.23) 0.94** (0.12) (0.16) Vehcle s Age (V) (0.35) 0.86** (0.21) 1.02** (0.11) 0.52** (0.14) Vehcle s Age (VI) (0.52) 0.86** (0.30) 1.28** (0.12) 0.38** (0.21) Model/Manuf (II) 1.01** (0.40) (0.31) 0.66** (0.12) Model/Manuf (III) 0.89* (0.49) 0.79* (0.43) 0.29** (0.11) Model/Manuf (IV) 0.74** (0.26) 0.49** (0.17) (0.08) Model/Manuf (V) 0.28 (0.68) 0.89** (0.36) 0.22** (0.1) Model/Manuf (VI) 0.04 (0.65) 0.33 (0.52) 0.21 (0.50) Model/Manuf (VII) 1.24** (0.35) 1.27** (0.28) 0.20 (0.14) Model/Manuf (VIII) 0.69* (0.35) 0.98** (0.30) 0.74** (0.12) Model/Manuf (IX) 0.24 (0.37) 0.23 (0.30) 0.30** (0.11) Model/Manuf (X) (0.37) 0.98** (0.33) 0.36** (0.16) Orgn (0.40) 0.09 (0.31) 0.13 (0.09) Terrtory (II) 0.84** (0.22) 0.61** (0.15) 0.10* (0.06) Vehcle Type (II) 0.35 (0.22) (0.15) 0.12 (0.08) 0.22** (0.09) Vehcle Type (III) (0.30) 0.87** (0.25) 0.33** (0.07) 1.37** (0.15) Scale (107.38) Note. *p<0.10; **p<0.05. Regresson coeffcents are standardzed coeffcents (β) and standard error wthn parentheses (SE). BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
9 Estmatng Total Clam Sze n the Auto Insurance Industry 45 Several explanatory varables were sgnfcantly related to dependent varables. Consderng all vehcle age varables, we can say that there s a sgnfcant ncrease n the expected clam probablty as the vehcle becomes older. On the other hand, the expected clam sze decreases for older vehcles. Ths s n lne wth ntuton and descrptve analyss, because old vehcles are less expensve to replace and there s also the fact that old vehcles are more attractve targets. One mght suggest that old vehcles are more attractve targets because there s a great auto part replacement market that gets flooded wth stolen parts for these old cars. Older cars also tend to be poorly mantaned, and ths ncreases the probablty of accdents. The varable model/manufacturer s related to clam probablty and sze. In general, the model/manufacturer s more closely related to clam probablty than clam sze. It s noteworthy that there s no way of clearly dentfyng whether clam sze or probablty s causng the sgnfcance of the Tweede coeffcents. The varable for vehcle orgn does not nfluence the clam probablty or sze. Ths suggests, ceters parbus, that domestc and mported vehcles tend to have the same clam sze. Terrtory s generally related to clam probablty and sze; n ths case there are some regons that have more carjackngs than others. Vehcle type s related to clam sze and probablty. The clam probablty decreases for luxury vehcles. However luxury cars lead to hgher clam szes compared to small and mdsze cars. Lookng at the Tweede results, the dffculty n accurately predctng clams becomes obvous gven the nonsgnfcance of the Tweede coeffcent for vehcle type. One mght suggest that the nonsgnfcant coeffcent s due to a negatve clam probablty effect and a postve clam sze effect, as found n the ZAIG coeffcents. The total clam sze forecast was made by addng together the ndvdual forecast clam szes based on the Tweede and ZAIG models. Usng parametrc bootstrap smulaton, we obtaned a 95% confdence nterval, based on emprcal quantles of 5,000 bootstrap estmates. For further detals, see Efron and Tbshran (1986). Table 4 shows the estmated and true total clam sze and the 95% confdence nterval. The ZAIG model was better than the Tweede model when t came to forecastng the total clam sze, and both models showed negatve bas. We notce that the forecasts le wthn the confdence bands for both models, ndcatng good estmaton results. Usng nferor and superor lmts, the nsurance company can begn to pcture total clam sze dsperson. We also calculated the mean squared error (MSE) and the mean absolute error (MAE) for the resduals. The results are very smlar for both the Tweede and ZAIG models. Table 4 Total clam sze, nferor and superor lmts, mean squared error and mean absolute error for Tweede and ZAIG models Tweede ZAIG True Total Clam Sze $ 2,089,845 $ 2,213,629 $ 2,432,513 LI $ 1,407,114 $ 1,429,470 LS $ 4,407,782 $ 3,230,824 MSE x x10 12 MAE 2,075,580 2,184,292 BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
10 A. B. Bortoluzzo, D. P. Claro, M. A. L. Caetano, R. Artes 46 Concludng Remarks In ths work we have tackled a wellknown problem n the nsurance ndustry, whch s the proper prcng of an nsurance polcy. Employng the ZAIG estmaton method for clams and rsks n the nsurance ndustry, we found dstnct factors that nfluence clam sze and probablty. Factors such as terrtory, a vehcle s advanced age, orgn and type dstnctly nfluence clam sze and probablty. The dstnct mpact s not always present n the Tweede estmated model. The ZAIG estmaton method also allows nsurance companes to create a score system to predct clams, based on the logstc model. Ths score system dentfes polcy holders who tend to be more rsky. These estmated models thus may be employed to develop a strategy for premum prcng. Moreover, nsurance companes can use vehcle characterstcs to estmate total clam sze and thus get an dea of how much they wll have to spend on a clam over a certan perod of tme and for a specfc clent portfolo. Some lmtatons to ths study should be ponted out. Frst, the methods requre a hgh computatonal effort that may preclude the use of larger datasets. Second, there s room for developng sutable methods for longtudnal data analyss. Future work may consder the use of estmatng equaton technques or multvarate ZAIG dstrbutons. We concentrated our research on the auto nsurance ndustry and specfc vehcle varables. Further studes may address other nsurance ndustres and nclude customer related varables. Receved 03 February 2010; receved n revsed form 28 June References Boland, P. J. (2007). Statstcal and probablstc methods n actuaral scence. Boca Raton: Chapman & Hall/CRC. Chang, L., & Farley, W. B. (1979). Prcng automoble nsurance under multvarate classfcaton of rsks: addtve versus multplcatve. The Journal of Rsk and Insurance, 46(2), Doherty, N. A. (1981). Is rate classfcaton proftable? The Journal of Rsk and Insurance, 48(2), Efron, B., & Tbshran R. (1986). Bootstrap methods for standard errors, confdence ntervals, and other measures of statstcal accuracy. Statstcal Scence, 1(1), do: /ss/ Gan, N. (2000). General zeronflated models and ther applcatons. Unpublshed doctoral dssertaton, North Carolna State Unversty, North Carolna, Unted States of Amerca. Heller, G., Stasnopoulos, M., & Rgby, B. (2006, July). The zeroadjusted nverse Gaussan dstrbuton as a model for nsurance clams. Proceedngs of the Internatonal Workshop on Statstcal Modellng, Galway, Ireland, 21. Huang, T., Zhao, R., & Tang, W. (2009). Rsk model wth fuzzy random ndvdual clam amount. European Journal of Operatonal Research, 192(3), do: /j.ejor Jong, P., & Heller, G. Z. (2008). Generalzed lnear models for nsurance data. Cambrdge: Cambrdge Unversty Press. Jørgensen, B. (1987). Exponental dsperson models. Journal of the Royal Statstcal Socety, 49(2), BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
11 Estmatng Total Clam Sze n the Auto Insurance Industry 47 Jørgensen, B. (1997). Theory of dsperson models. London: Chapman & Hall. Jørgensen, B., & Souza, M. C. P. de (1994). Fttng Tweede s compound Posson model to nsurance clams data. Scandnavan Actuaral Journal, 1(1), Kng, G., & Zeng, L. (2001). Logstc regresson n rare events data. Poltcal Analyss, 9(2), Meulbroek, L. (2001). A better way to manage rsk. Harvard Busness Revew, 79(2), R Development Core Team (2007). R: A language and envronment for statstcal computng. R foundaton for statstcal computng, Venna, Austra. Retreved January 12, 2008, from Smyth, G. K., & Jørgensen, B. (2002). Fttng tweede s compound posson model to nsurance clams data: dsperson modelng. Actuaral Studes n Nonlfe nsurance (ASTIN) Bulletn, 32(1), do: /AST Stasnopoulos, D. M., & Rgby, R. A. (2007). Generalzed addtve models for locaton scale and shape (GAMLSS). Journal of Statstcal Software, 23(7), Stasnopoulos, D. M., Rgby R. A., & Akantzlotou, C. (2006). Instructons on how to use the GAMLSS package n R (Techncal Report 01/06), London, UK, STORM Research Centre, London Metropoltan Unversty. Vaene, S., Ayuso, M., Gullen, M., Van Gheel, D., & Dedene, G. (2007). Strateges for detectng fraudulent clams n the automoble nsurance ndustry. European Journal of Operatonal Research, 176(1), do: /j.ejor Wesberg, H. I., & Tomberln, T. J. (1982). A statstcal perspectve on actuaral methods for estmatng pure premums from crossclassfed data. The Journal of Rsk and Insurance, 49(4), Wesberg, H. I., Tomberln, T. J., & Chartterjee, S. (1984). Predctng nsurance losses under crossclassfcaton: a comparson of alternatve approaches. Journal of Busness & Economc Statstcs, 2(2), Wüthrch, M. V., & Merz, M. (2008). Stochastc clams reservng methods n nsurance. West Sussex: John Wley & Sons. BAR, Curtba, v. 8, n. 1, art. 3, pp , Jan./Mar. 2011
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