A Statistical Analysis of Intensities Estimation on the Modeling of Non-Life Insurance Claim Counting Process

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1 Applied Mahemaics, 1, 3, 1-16 hp://dx.doi.org/1.436/am Published Online January 1 (hp:// A Saisical Analysis of Inensiies Esimaion on he Modeling of Non-Life Insurance Claim Couning Process Uraiwan Jaroengeraikun, Winai Bodhisuwan, Ampai Thongeeraparp Deparmen of Saisics, Kasesar Universiy, Bangkok, Thailand ur_jaroen@yahoo.com, {fsciwnb, fsciamu}@ku.ac.h Received Ocober 1, 11; revised November 19, 11; acceped November 8, 11 ABSTRACT This sudy presens an esimaion approach o non-life insurance claim couns relaing o a specified ime. The objecive of his sudy is o esimae he parameers in non-life insurance claim couning process, including he homogeneous Poisson process (HPP) and he non-homogeneous Poisson process (NHPP) wih a bell-shaped inensiy. We use he esimaing funcion, he zero mean maringale (ZMM) as a procedure of parameer esimaion in he insurance claim couning process. Then, Λ (), he compensaor of N () is proposed for he number of claims in he ime inerval (, ]. We presen siuaions hrough a simulaion sudy of boh processes on he ime inerval (, ]. Some examples of he siuaions in he simulaion sudy are depiced by a sample pah relaing N () o is compensaor Λ (). In addiion, an example of he claim couning process illusraes he resul of he compensaor esimae misspecificaion. Keywords: Esimaing Funcion; Zero Mean Maringale; Non-Life Insurance Claim Couning Process; Poisson Process; Bell-Shaped Inensiy 1. Inroducion Nowadays, insurance is a common way of managing risks and he insurance indusry has grown rapidly over ime. Insurance indusry owners, especially, consider he componens of risk managemen, such as he premiums which are he main income of insurance businesses, reserves, underwriing, invesmen planning, reinsurance planning, ec. Also, esimaing claims play an imporan par in each componen in he non-life insurance field. In he pas four decades, a few researchers have sudied he claim couns model for non-life insurance. Klugman e al. [1] and Denui e al. [] were ineresed in sudying he frequency disribuion of insurance claims, including he parameer esimaion mehods. Bühlmann [3,4] presened he credibiliy approach in he form of a linear funcion o esimae and predic he expeced claim couns in upcoming periods, using pas experience of claims as a risk class or relaed risk classes. Bühlmann s credibiliy approach is ineresing and can be exended o oher approaches, such as he Bühlmann-Sraub model, Jewell s model or he Exac credibiliy approach, ec., (see Klugman e al. [1]). Calculaing he expeced claim couns using he credibiliy approach only depends on he informaion from prior experience of claim couns, and does no consider he occurrence behavior of claim couns over ime. Some auhors have found an alernaive approach o claim couns relaing o a specified ime or heir behavior over ime, for example, Mikosch [5] viewed he claim couning process as a homogeneous Poisson process (HPP) in he Cramér-Lundberg model, one of he mos popular and useful risk models in non-life insurance, and Masui and Mikosch [6] also considered a Poisson cluser model for he modeling of a oal claims amoun by a poin of claim couns as an HPP wih a consan rae of occurrence called he consan inensiy. For some non-life insurance porfolios, he claim couns during a ime period are caused by periodic phenomena or seasonaliy. These claim couns are modeled in erms of a non-homogeneous Poisson process (NHPP) wih a period ime-dependen inensiy rae. Morales [7] presened he periodic risk model consising of he claim couning process wih a bell-shaped inensiy funcion (called he Gaussian inensiy) of he form 1 1 exp ; 1 1 π,1, s, where s is an iniial season, s =, 1,,, σ and are parameers, and is he sandard normal disribuion funcion. He esimaed he unknown parameers of he periodic model inensiy by using he maximum likelihood esimaion (MLE), and he Copyrigh 1 SciRes.

2 U. JAROENGERATIKUN ET AL. 11 also considered evaluaing he ruin probabiliy hrough a simulaion sudy. Furhermore, Lu and Garrido [8] explored he periodic NHPP model wih a Bea-shaped inensiy funcion. The precision of claim coun esimaion is a key o running he insurance business successfully. In his sudy, we will presen an esimaion approach o non-life insurance claim couns relaed o a specificaion of he wo differen claim couning processes, i.e., HPP, and NHPP wih a bell-shaped inensiy funcion, hrough a simulaion sudy. Our purpose is o esimae he parameers in he non-life insurance claim couning process. The parameers in he insurance claim couning process, inensiy funcion ( ) in erms of mean value funcion u du, makes a complicaed disribuion func- ion of insurance claim couns. An esimaing funcion, such as he zero mean maringale (ZMM), is used here as a procedure of parameer esimaion of an insurance claim couns model, and he parameers of model inensiy are esimaed by he MLE mehod.. A Definiion of he Non-Life Insurance Claim Couning Process We define he insurance claim couning process N # i 1: Ti ;, and he insurance claim couns which have occurred in he ime inerval (,] where Tn W1 Wn ; n 1 is a claim arrival ime and Wi is independen and idenically disribued (iid) Exponenial wih he parameer w i, called he inensiy rae, N N ; is a couning process which is non-decreasing, N can be wrien as N dn u where dn is an incremen of in a small fracion period. The Poisson disribuion is ofen considered as a common disribuion modeling of insurance claim couns, and our main ineres in he process of insurance claim couns is he Poisson process, i.e., HPP, and NHPP wih he bell-shaped inensiy funcion. This ineres lies in he inensiy rae, in which he insurance claim couns occur, and wheher hese change over ime. In an HPP, he inensiy rae is consan for a given ime, and he process is called an NHPP, if i changes as a funcion of ime [,5,9]. On a probabiliy space,,p, N is Poisson disribued wih he parameer u N du, wih a mean value funcion E N. As k ; is called he muliplicaive inensiy, where and ensiy rae and he exposure risk, respecively. We con- k are defined as he in- sider N as a non-decreasing righ coninuous sep funcion a ime = and jumps of size 1, and n exp P Pr N n ; n,1,, n! and Prd N 1 d EdN. In his sudy we consider he insurance claim couning process which are he HPP wih, a consan inensiy, and he NHPP wih a bell-shaped inensiy funcion as an iniial season, s = [7], 1 1 exp 1 1 π where (an average number of claims over a period) and are he parameers,,. 3. Parameer Esimaion in he Non-Life Insurance Claim Couning Process In his secion, we inroduce he mehods which are useful for parameer esimaion in he non-life insurance claim couning process, including he esimaing funcion, he maringale mehod, and he MLE Esimaing Funcion On a probabiliy space Ω,,P, where, is an open inerval on he real line, P pn ;. Sup - pose ha he observaion N n, he esimaing funcions, g N;, are funcions of N and he parameer. By solving g N ;, a so-called esimaing equaion, an esimae of is obained. Then g N; is an unbiased esimaing funcion if E g N; for all [1]. In his sudy, he esimaing funcion for parameer esimaion in he insurance claim couning process is provided by he maringale mehod. 3.. The Maringale Mehod The maringales are random processes relaing o ime. On a probabiliy space Ω,,P, we suppose he increasing family ;, a filraion or hisory, which is he available daa a he ime. The process M M ; is a maringale wih respec o s if E M exis, and EM M for all s. As a resul of he properies of he maringale, EM EM for all, hen EM for a ZMM [11,1]. This sudy of he maringale mehod is useful for con- srucing an esimaing funcion for a parameer esima- (1) Copyrigh 1 SciRes.

3 1 U. JAROENGERATIKUN ET AL. ion in he insurance claim couning process. The process akes place over a small ime inerval (, d], EdN d and as a resul of he meaning of maringale propery, he maringale can be wrien as dn d dm () which is a maringale-difference. Then, he following maringale is dnu udu dm u, or i is rewrien in he form of N M ZMM. Based on ZMM, we obain E M E N., is a Thus, N is an esimaing equaion for he parameer esimaion in he insurance claim couning process. Also, as a resul of he parameer esimae in he process, his can be inerpreed as an N esimae or, in oher words, is called he compensaor of N, and his esimae is useful for predicing he imes of occurrence of insurance claim couns [1]. We can depic he sysemaic par of he process of insurance claim couns, N, relaed o is compensaor,, and he associaed maringale N M in Figures 1(a) and (b), respecively, b ased on a sample of 15 independen random imes of claims occurrence in he NHPP wih an inensiy of 1 48exp A Maximum Likelihood Esimaion of he Model Inensiy In order o ge he esimae of he compensaor of N, ˆ, on he modeling of he non-life insurance claim couning process, boh he HPP and NHPP, he parameers of he inensiy funcion are esimaed by he MLE mehod. Given N n, we suppose ha 1,, 3,, N () are he arrival imes of he claims in he ime inerval (, ] wih a cumulaive disribuion funcion (a general order saisics model) F 1exp. The likelihood funcion [5,13] is given by l n θ; iexp n i1 (3) where θ is a vecor of he parameers of he model inensiy, denoes he se of arrival imes, he inensiy in he HPP is as θ and he inensiy in he NHPP is given in Equaion (1) as θ,. The esimae of θ can be simply found if we ake he logarihm of he likelihood funcion and we seek a value of θ ha maximizes he log likelihood funcion. The following parameer esimae of he HPP model is: ˆ ˆ n θ. For he NHPP, he calculaion of he N () n (a) (b) Figure 1. In a sample of 15 independen random imes of claims occurrence wih he inensiy (a) Non-life insurance claim couning process M N. N relaed o is compensaor ; and (b) Maringale 1 48exp 1, Copyrigh 1 SciRes.

4 U. JAROENGERATIKUN ET AL. 13 MLE esimaor of he model inensiy, which is a comdure, i.e. he Newon-Raphson algorihm, o solve hese plicaed sysem of equaions, requires an ieraive proceequaions []. 4. Simulaion Sudy In his sudy, a simulaion sudy is used o invesigae how he observaion of he non-life insurance claim couning process can be used o esimae is model parameer, i.e. inensiy or in erm, using he esimaing funcion provided by he maringale mehod wih ZMM. In paricular, he HPP of he insurance claim couns, wih as a consan inensiy and he NHPP of he insurance claim couns, wih given in Equaion (1) as a bell-shaped inensiy, we mus firs consider he simulaion sudy of he HPP of he insurance claim couns in he ime inerval (,] in which he observaion involves he claim arrival imes, 1,, 3,, N (), generaed by applying an exponenial law wih he inensiy, i.e. wih a mean 1 as λ =.1 and 1. The second simulaion sudy of he insurance claim couning process, in which we consider he NHPP wih a bell-shaped inensiy funcion, or as he general form of mean value funcion [].5 1 [],, where [] is he greaes ineger funcion, he claim arrival imes, 1,, 3,, N ( ), can be simulaed by using he mean value funcion as a claim arrival ime of he HPP wih mean one [7]. I implies ha E1, E, E3,, En are independen and exponenially disribued wih mean one, where Ei i i1, for al l i 1,, 3,, n. So, in his sudy, he n h claim arrival ime, N () n s generaed by, i 1 N () n E1EE3 En (4). 5 is he inveribl e 1 1 where D 1 funcion of, D D, 1 1 D and 1 is he quanile func- ion of a sandard normal disribuion, wih.1,.5,.1, 5, 1,.5 and 1, 5. In his simulaion sudy of boh he HPP and he NHPP of he insurance claim co uns in he ime inerval (,], he number of observaions, N n, is composed of 5, 1, 15 and. The HPP and he NHPP are carried ou wih 5 sample pahs. In each sample pah, he pa- inensiy is compued us- ing he MLE meh od and he esimaing funcion, such as rameer esimae of he model he ZMM which is used o esimae he parameer in he process (or he compensaor of N ), i.e. fiing he compensaor esimae ˆ o N. Also, he mean squared error (MSE) is provided o measure hings, fiing ˆ o N as he following form, MSE ˆ i i u N u du p, i1 p where p denoes he number of sample pahs. Noice ha he MSE of he compensaor esimae ˆ of N for boh processes, as shown in Tables 1 and, depends on he parameers of he model inensiy as he following deails, for he HPP wih a consan inensiy.1, a small inensiy rae, and he MSE of he compensaor esimae ˆ of N increases exponenially as he number of observaions increases. On he oher hand, he same process wih 1, he MSE of he compensaor esimae ˆ of N decreases exponenially, while he observaion number increases unil he observed 15 imes of claims occurrence, and hen he MSE value begins o increase as he observaion number increases. For he NHPP wih a bell-shaped inen siy, he Table 1. MSE of he compensaor esimae in he HPP of non-life insurance claim couns. ˆ of N λ N() MSE λ N() MSE Table. MSE of he compensaor es imae ˆ of in he NHPP of non-life insurance claim couns. λ σ N() M SE λ σ N( ) MSE.1.5 N Copyrigh 1 SciRes.

5 14 U. JAROENGERATIKUN ET AL. parameers of is model inensi y.1,.5 and.1, 5 (a small average number of a claims over a pe riod), he MSE of he compensaor esi- mae ˆ of N increases as he observaion number increases. When we consider he NHPP beween he parameers of he model in ensiy.1,.5 and.1, 5, we found ha in he process wih h e parameers of he model inensiy.1, 5, he MSE of he compensaor esimae ˆ of N is much lower. In he same NHPP wih he parameers of model inensiy 1,.5, he MSE of h e compensaor esimae ˆ of N decreases while is observaion number increases unil he observed 15 imes of claims occurrence, and hen is MSE v alues be- gins o increase as he observaion number increases. The MSE of he compensaor esimae ˆ of N in he same NHPP wih he parameers 1, 5 decreases exponenially as is observaion number increases. When we consider he NHPP beween he para meers of he model inensiy 1,.5 and 1, 5, we found ha in he process wih he parameers of he model inensiy 1, 5, he MS E of he compensaor esimae ˆ of N is much lower han he oher one. Some examples in hese siuaions of boh he HPP and he NHPP wih a bell-shaped inensiy of non-life insur- ance cla im couns based on a sample of 5, 1, 15 and imes of claims occurrence are illusraed in Figures and 3, including he N and is compensaor. Figure shows a sample pah of he HPP wih a con san inensiy 1. The N and is compensaor are characerized by he inensiy, i.e. 1, he compensaor fis well wih N, as he observaion number is 15 and (slighly larger han he inensiy 1 ). Similarly, he N and is compen- saor in he NHPP are characerized by he parameers of he model bell-shaped inensiy 1, 5 in Figure 3. The compensaor fis wih N, as he obser vaion number is 15 and (slighly larger han he parameer of model inensiy 1 ). Figure 4 illus raes a sample pah of NHPP, and we can see he difference wih he compensaor esimae which (a) (b) N (c) Figure. and is compensaor claims; (c) 15 claims; and (d) claims. in he HPP wih he inensiy λ = 1 based on a sample of (a) 5 claims; (b) 1 (d) Copyrigh 1 SciRes.

6 U. JAROENGERATIKUN ET AL. 15 (a) (b) Figure 3. N (c) and is compensaor in he NHPP wih he parameers of a bell-shaped inensiy based on a sample of (a) 5 claims; (b) 1 claims; (c) 15 claims; and (d) claims. (d) = 1, σ = 5 uses he esimaion mehod of NHPP wih a period imedependen inensiy, ˆ NHPP, fis well wih N by MSE = 1.48 shown wih a dashed line and he M SE = 5.35 along 5 sample pahs. I is noable ha if, insead, he compensaor esimae misspecificaion ˆ is cal- NHPP culaed more easily by using he esimaion mehod of HPP wih a consan inensiy, and he compensaor esimae misspecificaion ˆ flucuaes a lo from, HPP N shown on he doed line in Figure 4. T he MSE of fiing he compensaor esimae misspecificaion ˆ HPP o N is.7, and he MSE = 5.53 along 5 sample pahs are larger han he fiing of he compensaor esi- mae ˆ N. NHPP o Figure 4. N, non-life insurance claim couns. ˆ HPP, and ˆ NHPP in he NHPP of 5. Conclusion This simulaion sudy of he non-life insurance claim couning process, of boh he HPP and he NHPP wih a bell-shaped inensiy, demonsraes ha he fiing of he Copyrigh 1 SciRes.

7 16 U. JAROENGERATIKUN ET AL. compensaor esimae ˆ o N in he ime inerval (, ] depends on he parameers of model inensiy as in he following deails, firsly, regarding he HPP wih a small inensiy rae, wih almos no claim occurrences, while he number of observaions is very small, he compensaor esimae ˆ is a good fi o N wih less of a MSE. In he same process wih a consan inensiy rae, he claims occurrence rae, when he number of observaions is slighly larger han he consan inensiy, he MSE of he compensaor esimae ˆ of N is much less. Secondly, as regards o he NHPP wih he parameers of he model inensiy, a has a very small average number of claims over a period, almos no claim occurrences over a period, and any, as he number of observaions is very small, he compensaor esimae ˆ is a good fi o N wih less of a MSE. Using he same process wih he parameers o f he model inensiy, wih an average number of claim s over a period is no less han one and any, while he number of observaions is slighly larger han he value of, he MSE of he compensaor esimae ˆ of N is m uch less. Some examples of he siuaions in he simulaion sudy are also depiced by a sample pah relaing N and is compensaor. Furhermore, he resul of he compensaor esimae misspecificaion ˆ of N is illusraed by a sample pah of he NHPP so ha he MSE of fiing he compensaor esimae misspecificaion ˆ o N is much larger han he fiing of he compensaor o N. REFERENCES [1] S. A. Klugman, H. H. Panjer and G. E. Willmo, Loss Models from Daa o Decisions, 3rd Ediion, John Wiley & Sons, Hoboken, 8. [] M. Denui, X. Maréchal, S. Pirebois and J. F. Walhin, Ac- uarial Modelling of Claim Couns, John Wiley & Sons, Hoboken, 7. doi:1.1/ [3] H. Bühlmann, Inroducion Repor Experience Raing and Credibiliy, ASTIN Bullein, Vol. 4, No. 3, 1967, pp [4] H. Bühlmann, Credibiliy Procedures, Proceedings of he Sixh Berkeley Symposium on Mahemaical Saisics and Probabiliy, Universiy of California Press, Berkeley, Vol. 1, 197, pp [5] T. Mikosch, Non-Life Insurance Mahemaics, nd Ediion, Springer-Verlag, Berlin, 9. doi:1.17/ [6] M. Masui and T. Mik osch, Predicion in a Poisson Cluser Model, Journal of Applied Probabiliy, Vol. 47, No., 1, pp doi:1.139/jap/ [7] M. Morales, On a Surplus Process under a Per iodic Environmen: A Simulaion Approach, Norh American Acuarial Journal, Vol. 8, No., 4, pp [8] Y. Lu and J. Garrido, On Double Periodic Non-Homogeneous Poisson Processes, Bullein of he Swiss Associaion of Acuaries Swiss Associaion of Acuaries-Bern, Bern, 4, pp [9] S. M. Ross, Inroducion o Probabiliy Models, 5h Ediion, Academic Press, Inc., San Diego, [1] P. Mukhopadhyay, An Inroducion o Esimaing Funcions, Alpha Science Inernaional Ld., Harrow, 4. [11] P. K. Andersen, O. Borgan, R. D. Gill and N. Keiding, Saisical Models Based on Couning Processes, Springer- Verlag, New York, [1] P. Yip, Esimaing he Number of Error in a Sysem Using a Maringale Approach, IEEE Transacions on Reliabiliy, Vol. 44, No., pp doi:1.119/ [13] J. E. R. Cid and J. A. Achcar, Bayesian Inference for Nonhomogeneous Poisson Processed in Sofware Reliabiliy Models Assuming Nonmonoonic Inensiy Funcions, Compuaional Saisics & Daa Analysis, Vol. 3, No., 1999, pp doi:1.116/s (99)8-6 Copyrigh 1 SciRes.