1 Applied Matheatical Sciences, Vol. 8, 4, no. 6, HIKARI td, Constuction Actuaial Model fo Aggegate oss unde Eponentiated Inveted Weibull Distibution Osaa Hanafy Mahoud Depatent of Matheatics, Statistics and Insuance Sadat Acadey fo Manageent Sciences, Egypt Copyight 4 Osaa Hanafy Mahoud. This is an open access aticle distibuted unde the Ceative Coons Attibution icense, which peits unesticted use, distibution, and epoduction in any ediu, povided the oiginal wok is popely cited. Abstact The poble of planning einsuance policies in popety and causality insuance copanies and how to estiate the loss eseve play an ipotant ole fo the esults of the insuance copany. Also, they have past o pesent data fo nube of clais and its aounts which want to use in pediction futue clai fequency and clai seveity. Many statistical distibution ae fitting to find an appopiate distibution to epesent ou data. In this pape, we intoduce a statistical distibution known as Eponentiated Inveted Weibull (EIW distibution to epesent the clai aount and its chaacteistics fo applying it in actuaial studies. Second, we test the tail weight of the distibution and aiu likelihood estiation fo its paaetes. we pesent ou aggegate loss odel unde collective isk theoy when the clai fequency distibution is Poisson o egative Binoial distibution. Also, we pesent how to calculate the einsuance pue peiu in case of stop loss einsuance. Finally, The siulation nueical eaple is given to epesent ou esults. Keywods: Eponentiated Inveted Weibull Distibution, Tail Weight of Distibution, Maiu ikelihood Estiation, Aggegate oss Model
2 886 Osaa Hanafy Mahoud. Intoduction Insuance copanies need to investigate clais epeience and apply atheatical techniques fo any puposes such as ateaking, eseving, einsuance aangeents and solvency. Many papes have been pesented to aggegate losses as: Heckan and Meyes (983 discussed aggegate loss distibutions fo the pespective of collective isk theoy fo seveity and count distibutions. They include eaples fo calculating the pue peiu fo a policy with an aggegate liit, calculating the pue peiu of an aggegate stop-loss policy fo goup life insuance; and calculating the insuance chage fo a ulti-line etospective ating plan, including a line which is itself subject to an aggegate liit. Vente (983, Distibution functions ae intoduced based on powe tansfoations of beta and gaa distibutions, and popeties of these distibutions ae discussed. The gaa, beta, F, Paeto, Bu, Weibull and loglogistic distibutions ae special cases. The tansfoed gaa is used to odel aggegate distibutions by atching oents. The tansfoed beta is used to account fo paaete uncetainty in this odel. Robetson (99, Povided an application of the fast Fouie tansfo to the coputation of aggegate loss distibutions fo abitay fequency and seveity distibutions. Papush el al (, addessed the question what type of oal, ognoal and Gaa distibutions is the ost appopiate to use to appoiate aggegate loss distibution. Vila et al. (8, descibed a nonpaaetic appoach to ake infeence fo aggegate loss odels in the insuance faewok by assuing that an insuance copany povides a histoical saple of clais given by clai occuence ties and clai sizes. Botoluzzo et al. (9, aied estiating clai size in the auto insuance by using zeo adjusted Invese Gaussian distibution. Shevchenko ( eviewed nueical algoiths that can be successfully used to calculate the aggegate loss distibutions. In paticula Monte Calo, Panje ecusion and Fouie tansfoation ethods ae pesented and copaed. Also, seveal closed-fo appoiations based on oent atching and asyptotic esult fo heavy-tailed distibutions ae eviewed. One of the ost significant goals of any insuance isk activity is to achieve a satisfactoy odel fo the pobability distibution of the total clai aount. In this pape, we intoduce a statistical distibution known as Eponentiated Inveted Weibull (EIW distibution to epesent the clai aount and its chaacteistics fo applying it in actuaial studies. This pape is oganized as follows: Section we pesentsthe Model clai seveity unde Eponentiated Inveted Weibull Distibution and test the tail weight
3 Constuction actuaial odel fo aggegate loss 887 of a distibution. In Section 3 we discuss the poble of estiating the paaetes of distibution by using aiu likelihood ethod. Section 4 we pesent ou aggegate loss odel unde collective isk theoy when the clai fequency distibution is Poisson o egative Binoial distibution. Section 5, how to calculate the einsuance pue peiu in case of stop loss einsuance. Finally, The siulation nueical eaple is given to epesent ou esults.. The Model unde Eponentiated Inveted Weibull Distibution Recently any studies in pobability distibutions and its applications pesented the Eponentiated Inveted Weibull distibution as: Flaih et al (, Consideed the standad eponentiated inveted weibull distibution (EIW that genealizes the standad inveted weibull distibution (IW, the new distibution has two shape paaetes. The oents, edian, suvival function, hazad function, aiu likelihood estiatos, least-squaes estiatos, fishe infoation ati and asyptotic confidence intevals have been discussed. A eal data set is analyzed and it is obseved that the (EIW distibution can povide a bette fitting than (IW distibution. Aljouhaah Aljuaid, (3, pesented Bayes and classical estiatos have been obtained fo two paaetes eponentiated inveted Weibull distibution when saple is available fo coplete and type II censoing schee. Hassan (3, dealt with the optial designing of failue step- stess patially acceleated life tests with two stess levels unde type-i censoing. The lifetie of the test ites is assued to follow eponentiated inveted Weibull distibution. Hassan et al. (4, pesented estiation of population paaetes fo the eponentiated inveted Weibull distibution based on gouped data with equi and unequi-spaced gouping. Seveal altenative estiation schees, such as, the ethod of aiu likelihood, least squaes, iniu chi-squae, and odified iniu chi-squae ae consideed. If ou clais aount of insuance potfolio,, 3,, follow the n Eponentiated Inveted Weibull (EIW distibution with paaetes and in the following fo fo the pobability density function (pdf: f ( ( e (,, ( Theefoe, its cuulative pobability function (cpf can be witten in the fo: F( ( e,, ( The (ight- tail of a distibution is the potion of the distibution coesponding to
4 888 Osaa Hanafy Mahoud lage values of the ando vaiable. A distibution is said to be a heavy-tailed distibution if it significantly puts oe pobability on lage values of the ando vaiable. We also say that the distibution has a lage tail weight. In contast, a distibution that puts less and less pobability fo lage values of the ando vaiable is said to be light-tailed distibution. To test the tail weight of a distibution, we can use the Eistence of Moents ethod as follows: A distibution f ( is said to be light-tailed if E ( fo all and the distibution f ( is said to be heavy-tailed if eithe does not eist fo all o the oents eist only up to a cetain value of a positive intege, Finan (4. The th oents of the eponentiated inveted weibull distibution is given as follows: ( ( e d,, This can be witten as: (,, (3 Poof: The pdf of the EIW distibution is: f ( ( e (,, The th oents function can be witten in the fo: f ( d ( e ( d By taking tansfoation H We can wite the th oents function as: ( H H H H e By siplification of the above equation,we can get H e H dh dh This integal known as gaa function, theefoe the th oents function is:
5 Constuction actuaial odel fo aggegate loss 889 Fo the above equation, we can find to obtain the th oent ust the value of geate than to be eist. Since the oents ae not finite fo all positive ; the eponentiated inveted weibull distibution is heavy-tailed. Fo the above equation, we can find the ean and the vaiance of EIW distibution as follows: By putting = (4 (,, And the second oent by putting = in the fo: (,, (5 Thus the vaiance is: V( ( ( ( 3. Maiu ikelihood Estiation (ME fo paaetes Suppose that we have postulated a pobability odel, such as the Eponented Inveted Weibull distibution, to descibe a given loss aount distibution. The net step in ou pocedue should be to estiate values fo the paaetes of the odel. We use the aiu likelihood ethod (ME fo estiating the unknown paaetes and of Eponented Inveted Weibull distibution, as follows:- (, f ( i i Then the likelihood function is as follows, (, i ( e ( e ( i ( i i ( ( i i by taking the natual logaith fo the likelihood function, we get (6
6 89 Osaa Hanafy Mahoud i i ( ln i i ln (, ln ln (7 So, we need to estiate the two paaetes and. The fist deivatives fo the natual logaith of the likelihood function with espect to and, ae given by ln (, i (8 i ln (, ( i ln i ln (9 i i The aiu likelihood estiatos of and could be obtained by equating the equations (8 and (9 by zeo, and solving the siultaneously using an iteative technique. We obtain the appoiate vaiance covaiance ati by eplacing epected values by thei aiu likelihood estiatos and inveting the Fishe infoation ati, defined by: ln I ln ln ln Whee, the second deivatives of the natual logaith of likelihood function defined in equation (6 ae given as follows: i ln (, ( ln (, ( i ln i ( ln (, i i ( i (ln i The ME ˆ and ˆ have an asyptotic vaiance covaiance ati obtained by inveting the Fishe infoation ati. 4. Aggegate loss odel unde collective Risk Theoy Suppose that potfolio has clais in the past peiod of tie in ou epeience and each unit has i is the clai size which is independent identical distibuted (
7 Constuction actuaial odel fo aggegate loss 89 eponented inveted Weibull with paaetes and its p.d.f in the equation ( and cuulative pobability function in the equation (. Then the aggegate losses is S whee the su of clai aounts as: S 3 Suppose also that the individual loss aounts i ae independent on the annual loss fequency. Then it follows that: The pobability density function (pdf of aggegate losses is k * k f ( S p( k f ( s (3 s * Whee f k ( s is called the k th fold convolution of 3 the k th fold convolutions ae often eteely difficult to copute in pactice and theefoe one encountes difficulties dealing with the pobability distibution of S: An altenative appoach is to use vaious appoiation techniques. We conside a technique known as the Panje ecusive foula. The ean and vaiance of aggegate loss distibution can get as: s V ( s V ( V ( (4 The picing poble usually educes to finding oent of S. A coon picing foula is pice s k v( s Whee the pice is the epected payout plus a isk loading which k ties the vaiance of the payout fo soe k. The epected payout E (s is also known as the pue peiu and it can be shown to be E (. Estiation the Mean and the Vaiance of Aggegate losses Distibution: We will conside Poisson and egative Binoial distibutions fo the fequency distibution of losses as follows: I. Poisson Distibution Suppose that the annual fequency of losses fo a potfolio follows a Poisson distibution with paaete. In this case the ean of the loss distibution is:
8 89 Osaa Hanafy Mahoud E ( s ( (5 V ( s ( (6 II. egative Binoial Distibution Suppose that the annual fequency of losses fo a potfolio follows a egative Binoial distibution with paaetes and p. In this case the ean of the loss distibution is: p E ( s ( (7 p p p V ( s ( ( p (8 p 5. Stop oss Reinsuance Gauge and Hosking (8 and Finan (4 pesented the stop loss einsuance as: When a deductible D is applied to the aggegate loss S ove a definite peiod, then the insuance payent will be, S D S D MaS D, S S^ D S D, S D the einsue will pay the insue an aount equal to S D. The insue's etained loss is thus S^ D. The ain poble is how to calculate the einsuance pue peiu?. the einsuance pue peiu is payent as the stop-loss e insuance. Its epected cost is called the net stop-loss peiu and can be coputed as: E S D F ( d d f ( d (9 6. ueical Results D S S D In this section, we will pesent a nueical investigation of the aiu likelihood estiation fo the paaetes of and. We need to estiate the two paaetes and by using the aiu likelihood ethod. So, we will need to solve the thee non-linea equations of
9 Constuction actuaial odel fo aggegate loss 893 logaith likelihood function (8 and (9 siultaneously using ewton-raphson ethod.the iteative technique, can be applied as follows: A C whee and ln ln ln ln C Assuing initial values fo each of and, the ewton-raphson iteative pocedue is continued until eithe the nube of iteations will be ( o when X X+ < 5-5. In the following table, the estiates of unknown paaetes, the elative bias which is the absolute diffeence between the estiated paaete and its tue value divided by its tue value. - ˆ Bais Re ltive And the ean squae eo (MSE which is the ean squae of the diffeence between the estiated paaete ae pesented fo all the estiated paaetes consideing diffeent initial points of the paaetes. ˆ M SE whee is the nube of epeients cay out. A ln ln, ˆ ˆ, ˆ ˆ
10 894 Osaa Hanafy Mahoud Table ( Estiatos fo paaetes of EIW distibution, Relative Bias and MSE Paaetes Estiato Relative Bias MSE Table (, shows The Estiatos of the paaetes and of the odel, Relative Bias, and MSE. We can notice that the absolute value of the diffeence between the tue value of the paaete and its estiato is sall value conveges to zeo, so these estiatos ae said to be consistent estiatos. Estiate the ean and the vaiance of EIW distibution By substituting the estiated values of the paaetes ˆ and ˆ in equations (4 and (5, we can get the ean and the vaiance of EIW distibution as shown in Table (: Table ( The estiated ean and vaiance of EIW distibution ˆ ˆ E ( E ( V (
11 Constuction actuaial odel fo aggegate loss 895 Fo Table (, we can notice thee is diect elationship between the value of ˆ and the value of the ean and the vaiance of distibution. Also, thee is invese elationship between the value of ˆ and the value of the ean and the vaiance of distibution. Estiation the ean and the vaiance of Aggegate losses distibution: When the annual fequency of losses fo a potfolio follows a Poisson distibution with paaete 8 by substituting in equations (5 and (6, o the egative Binoial distibution with paaetes and p. 6 by substituting in equations (7 and (8 in table (3 as follows: Table (3 Estiation the ean and the vaiance of Aggegate losses distibution ˆ ˆ Poisson distibution egative Binoial distibution E (s V (s E (s V (s Concluding eaks: In this study, we addess the eponentaited inveted weubil distibution issue and its epiical application of aggegate losses. By testing the tail of the EIW distibution, we find it has heavy tail. The aiu likelihood ethod was applied fo estiating the paaetes of distibution. Unde collective isk odel, we estiate the ean and the vaiance of the aggegate losses distibution whee the fequency distibution fo clai counts is Poisson o egative Binoial. If we identify the aggegate losses distibution, we can depend on it to ateaking, aangeent fo stop of loss einsuance and estiate the needed loss eseve.
12 896 Osaa Hanafy Mahoud Refeences  A. Aljuaid, Estiating the Paaetes of an Eponentiated Inveted Weibull Distibution unde Type-II Censoing, Applied Matheatical Sciences, 7 (3, 35,  A. B. Botoluzzo, D. P. Clao, M. A. Caetano and R. Ates, Estiating Clai Size and Pobability in the Auto-insuance Industy: the Zeo adjusted Invese Gaussian (ZAIG Distibution, Inspe Woking Pape WPE: 75 (9.  M. B. Finan M. B., An Intoductoy Guide in the Constuction of Actuaial Models, Akansas Tech Univesity, 4.  A. Flaih, H. Elsalloukh, E. Mendi and M. Milanova. The Eponentiated Inveted Weibull Distibution, Applied Matheatics & Infoation Sciences, 6 (,,  M. Gauge and M. Hosking, Constuction of Actuaial Models, Bpp Pofessional Education, Inc, (8.  A. S. Hassan, On the Optial Design of Failue Step-Stess Patially Acceleated ife Tests fo Eponentiated Inveted Weibull with Censoing, Austalian Jounal of Basic and Applied Sciences, 7 (3,,  A. Hassan, A. Mawa,H. Zahe and E. Elshepiny, Copaison of Estiatos fo Eponentiated Inveted Weibull Distibution Based on Gouped Data, Int. Jounal of Engineeing Reseach and Applications, 4 (4, 4,  Heckan, Philip E. Meyes, Glenn G. (983, "The Calculation of Aggegate oss Distibutions fo Clai Seveity and Clai Count Distibutions", Poceedings of the Casualty Actuaial Society Casualty Actuaial Society - Alington, Viginia 983: XX - 6.  Papush D. E., Pateik, G. S. and Podgaits F., (, "Appoiations of the aggegate distibutions", CAS Fou, pp  Robetson John, (99, "The Coputation of Aggegation loss distibutions", PCAS, pp  Shevchenko P. V., (, "Calculation of aggegate loss distibutions", The Jounal of Opeational Risk 5 (, pp. 3-4.
13 Constuction actuaial odel fo aggegate loss 897  Vente G. (983, "Tansfoed Beta and Gaa distibutions and the aggegate losses" PCAS, XX, pp  Vila J., Cao R., Ausín M. C. and C. González-Fagueio C., (8, "onpaaetic analysis of aggegate loss odels", nd Intenational Wokshop on Coputational and Financial Econoetics (CFE'8, euchatel (Suiza, 9 a de junio de 8. Received: Octobe 7, 4; Published: ovebe 8, 4