Malaysian Journal of Applied Sciences

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1 Unverst Sultan Zanal Abdn eissn (Onlne) Malasan Journal of Appled Scences ORIGINAL ARTICLE A New Strateg of Handlng General Insurance Modellng Usng Appled Lnear Method *Wan Muhamad Amr W Ahmad a, Mohamad Arf Awang Naw b, and Mustafa Mamat b a School of Dental Scences, Unverst Sans Malasa, Health Campus, Kubang Keran, Kelantan, Malasa b Facult Informatcs and Computng, Unverst Sultan Zanal Abdn, Tembla Campus, Besut, Terengganu, Malasa *Correspondng author: wmamr@usm.m Receved: 04/12/2015, Accepted: 17/03/2016 Abstract Ths paper proposes the use of bootstrap, robust and fuzz multple lnear regressons method n handlng general nsurance n order to get mproved results. The man obectve of bootstrappng s to estmate the dstrbuton of an estmator or test statstc b resamplng one's data or a model estmated from the data under condtons that hold n a wde varet of econometrc applcatons. In addton, bootstrap also provdes appromatons to dstrbutons of statstcs, coverage probabltes of confdence ntervals, and reecton probabltes of hpothess tests that produce accurate results. In ths paper, we emphasze the combnng and modellng usng bootstrappng, robust and fuzz regresson methodolog. The results show that alternatve methods produce better results than multple lnear regressons (MLR) model. Kewords: Multple lnear regresson; MM estmaton; robust regresson; bootstrap method; fuzz regresson Introducton Multple lnear regresson modellng s a ver powerful technque n statstcs and s wdel used n numerous research felds ncludng fnance, economc, agrculture. Ths method estmates lnear relatonshp between dependent (response) and ndependent (eplanator) varables. The multple lnear regresson model s epressed as Y b0 b1x 1 b n X n where b s s parameters and s the error term assumed to be, followng a normal dstrbuton. The parameters are usuall estmated usng method of least squares. A good eplanaton of varous aspects of multple lnear regresson methodolog s gven n Draper and Smth (1998). The prmar goal of robust regresson s to provde resstant results n the presence of outlers. In pursut of ths stablt, robust regresson lmts the nfluence of outlers. Robust regresson analss provdes an alternatve to the least squares regresson when fundamental assumptons are unfulflled b the nature of the data (Marona et al., 2006). The propertes of effcenc, breakdown, and hgh leverage ponts are used to defne robust technques

2 performance n a theoretcal sense. One of the goals of robust estmator s a hgh fnte sample breakdown pont defned b Donoho and Huber (1983). Chrstmann (1994) and Rousseeuw and Lero (1987) state that the breakdown pont could be defned as the pont or lmtng percentage of contamnaton n the data at whch an test statstcs frst becomes swamped. Hence, the breakdown pont s smpl the ntal pont at whch an statstcal test becomes swamped due to contamnated data. Some regresson estmators have the smallest possble breakdown pont of 1/n or 0/n. In other words, onl one outler would cause the regresson equaton to be rendered useless. Other estmators have the hghest possble breakdown pont of n/2 or 50%. If robust estmaton technque has a 50% breakdown pont, then 50% of the data could contan outlers and the coeffcents would reman useable. MM estmaton s a specal tpe of M-estmaton developed b Yoha (1987). In hs paper, Stromberg (1993) states that MM-estmaton s a combnaton of hgh breakdown value and effcent estmatons. Yoha's MM estmator was the frst estmaton of a hgh breakdown pont and hgh effcenc under normal error. MM-estmators have three-stage procedures; 1. The frst stage nvolves the calculaton of S-estmate wth nfluence functon n 1 K ; K s a constant, obectve functon satsfes as: n s 1. s smmetrc and contnuousl dfferentable, and 0 0. such that s strctl ncreasng on,. There ests 0,. K and constant on 2. The second stage nvolves the calculaton of MM parameters that provde the mnmum n ' value of ˆ MM where s the nfluence functon used n the frst stage 1 ˆ 0 and ˆ 0 s the estmate of scale form the frst step (standard devaton of the resduals). 3. The fnal step computes the MM estmate of scale as the soluton to 1 n ' ˆ 0.5 n p 1 s Bootstrap s a technque for resamplng based on random sorts wth retreval n the data formng a sample. Addtonall, ths method provdes appromatons to dstrbutons of statstcs, coverage probabltes of confdence ntervals, and reecton probabltes of hpothess tests that produce accurate results (Hall, 1992; Efron and Tbshran, 1993). The theoretcal bootstrap model s as follows; * Y X ˆ u * (1) where * u s a random term obtaned from the resduals û of the ntal regresson. At each teraton ( b 1,..., B) model. * 1 b, a sample n of sze (n, 1), s created from the theoretcal bootstrap 46

3 Snce the OLS resduals are smaller than the errors the estmate, the random term of the theoretcal bootstrap model s constructed from the followng transform resduals whch have the same norm as the error term u : n u u u ~ ˆ 1 ˆ 1 h n h 1 1 The theoretcal bootstrap model s hence epressed as: where z u ~ * b * s resampled from b X ˆ u~ * b, = 1 n (2) u ~. Let us consder the random varable z, defned as ˆ, the standard confdence nterval of derves from the assumpton accordng s ˆ to whch z s dstrbuted accordng to a student's dstrbuton wth n-p degrees of freedom. Thus for a confdence level 1 2, ths confdence nterval takes the followng form: ˆ t, ˆ s ˆ t ] ˆ s (3) [ 1, n p, n p where t s the percentle values and 1 from the bootstrap-t percentles wth n p degrees of freedom. The bootstrap confdence ntervals are constructed from two percentle and percentle-t approaches. The frst method, based eclusvel on bootstrap estmatons, s the smplest one for obtanng confdence ntervals. For a level 1 2, the percentle confdence nterval for parameter s gven b: where B, ˆ 1 ˆ B (4) -th value (respectvel ˆ B s the B ˆ 1 B the 1 B -th value) of the ordered lst of the B bootstrap replcatons. The threshold values are hence selected so that % of the replcatons provde smaller (larger) ˆ than the lower (upper) bound of the percentle confdence nterval. A fuzz regresson model correspondng to multple lnear regresson equaton could be stated as; A0 A1 1 A2 2 A k k (5) Prevousl, eplanaton varables ' s are assumed to be precse. However, accordng to the equaton above, response varable Y s not crsp but s nstead fuzz n nature. That means the parameters are also fuzz n nature. Our obectve s to estmate these parameters. In further dscusson, A ' s are assumed as smmetrc fuzz numbers whch could be presented b nterval. For eample, A could be epressed as fuzz set gven b A a1 c, a1w where a c s centre and aw s radus or vagueness assocated. Fuzz set above reflects the confdence n the regresson coeffcents around a c n terms of smmetrc trangular membershps functon. Applcaton of ths method should be gven more attenton when the underlng phenomenon or the response varable s fuzz. So, the relatonshp s also consdered to be 47

4 fuzz. Ths A a1 c, a1w could be wrtten as A1 a 1 L, a1 R wth a1l a1c a1w and a1r a1c a1w (Kacprzk and Fedrzz, 1992). In fuzz regresson methodolog, parameters are estmated b mnmzng total vagueness n the model. A A A A k k (6) Usng A a a, we could wrte 1 c, 1 w a0c, a0w a1c, a1w 1 a a c, a w nc, a nw n. Thus w a c 0 w As a 0 c a equaton w w 1w a 1c 1 1 a a nw nc n n represent radus and could not be negatve, therefore on the rght-hand sde of a a a 0 w 1w 1 nw n, absolute values of are taken. Suppose there are m data pont, each comprsng an 1 row vector. Then parameters A are estmated b mnmzng the quantt, whch s total vagueness of the model-data set combnaton, subect to the constrant that each data pont must fall wthn estmated value of response varable. Materals and Methods A Case Stud of General Insurance Table 1. Descrpton of the varables Varables Y X1 X2 X3 X4 X5 Descrpton Proftablt of General Insurance Companes Net Investment Income Total Labltes and Assets Management Epenses Annual Premum Net Clams Pad b The Compan Source: (Naw, et al. 2012) /* Frst we do Multple lnear regresson */ procreg data= general; model = ; run; 48

5 Approach the MM-Estmaton Procedure for Robust Regresson /* Then we do robust regresson, n ths case, MM-estmaton */ ods graphcs on; procrobustreg method= MM fwls data= general plot=ftplot(nolmts) plots=all; model = / dagnostcs tprnt; output out=resds out=robout r=resdual weght=weght outler=outler sr=stdres; run; ods graphcs off; Procedure for Bootstrap wth Case Resamplng (n =100) /* And fnall we use a bootstrap wth case resamplng */ ods lstng close; procsurveselect data=general out=boot1 method=urs samprate=1outhts rep=100; run; Procedure for Bootstrap nto Fuzz Regresson Model /*Combnaton of Bootstrap Technque wth Fuzz Regresson*/ ods lstng close; procoptmodel; set = 1..30; Number {}, 1{}, 2{}, 3{}, 4{}, 5{}; read data boot1 nto [_n_] ; /*Prnt */ Prnt ; number n nt 30; /*Total of Observatons*/ /* Decson Varables bounded or not bounded*/ /*Theses three varables are bounded*/ var aw{1..6}>=0; /*These three varables are not bounded*/ var ac{1..6}; /* Obectve Functon*/ mn z1= aw[1] * n + sum{ n } 1[] * aw[2]+sum{ n } 2[] * aw[3]+sum{ n } 3[] * aw[4]+sum{ n } 4[] * aw[5]+sum{ n } 5[] * aw[6]; /*Lnear Constrants*/ con c{ n 1..n}: ac[1]+1[]*ac[2]+2[]*ac[3]+3[]*ac[4]+4[]*ac[5]+5[]*ac[6]-aw[1]- 1[]*aw[2]-2[]*aw[3]-3[]*aw[4]-4[]*aw[5]-5[]*aw[6]<=[]; con c1{ n 1..n}: ac[1]+1[]*ac[2]+2[]*ac[3]+3[]*ac[4]+4[]*ac[5]+5[]*ac[6]+aw[1]+1[ ]*aw[2]+2[]*aw[3]+3[]*aw[4]+4[]*aw[5]+5[]*aw[6]>=[]; epand;/* Ths provdes all equatons */ solve; prnt ac aw; qut; ods rtf close; 49

6 Data Collecton and Cleanng Data (General Insurance) Unvarate and Multvarate Analss Model Buldng wth Robust Regresson Yes Dagnostcs Test : Checkng for outlers Robust Procedure n SAS Algorthm Selecton of MM estmaton No Formaton of Lnear Regresson Model Addng Bootstrap to the Lnear Regresson wth Replcaton (n = 100) Constructon Algorthm Fuzz Regresson Model Data Analss Based on Alternatve Model Bootstrap Procedure In SAS Results Fgure 1. Flow chart of robust, bootstrap and fuzz regresson Results and Dscusson A hgher R-squared value shows how well the data ft the model and ndcates a better model. Table 2. Goodness-of-ft Statstc Value R-Square AICR BICR Devance Usng the method of Multple lnear regresson (MLR), we obtaned the result as shown n Table 3 usng bootstrappng method for fuzz regresson wth n = 100. The am of bootstrappng procedure s to appromate the entre samplng dstrbuton of some estmator b resamplng (smple random samplng wth replacement) from the orgnal data (Yaffee, 2002). 50

7 Parameter DF Estmate Table 3. Parameter estmates for fnal weghted least squares ft Standard Error 95% Confdence Lmts Ch-Square Pr > ChSq Intercept <.0001 X X X X <.0001 X <.0001 Scale Method of Fuzz Regresson (FR) (OPTMODEL) Table 4 summarzes the fndngs of the calculated parameter. When usng bootstrap procedure, we generate dfferent output whle usng AC or AW, where AC denotes the centre and AW denotes the radus,.e. half of the wdth of A. The net step s to compare the performance of multple lnear regresson and fuzz regresson. Table 4. Value of center (AC) and radus (AW) AC AW The Ftted Model for Multple Lnear Regressons Y X X (7) X X X Standard Error (0.0166) (0.0038) (0.0015) (0.0037) (0.002) (0.0025) The upper lmts of predcton nterval are computed b coeffcent plus standard error Y ( ) ( ) X ( ) X ( ) X 1 ( ) X 4 ( ) X5 The lower lmts of predcton nterval are computed b coeffcent mnus standard error Y ( ) ( ) X ( ) X ( ) X ( ) X 4 ( ) X

8 Table 5. Average wdth for former multple lnear regresson model and fuzz bootstrap regresson model Multple Lnear Regresson Model Fuzz Bootstrap Regresson Model Lower Lmt Upper Lmt Wdth Lower Lmt Upper Lmt Wdth Average 0.34 Average

9 The Ftted Model for Fuzz Bootstrap Regresson Y X X (8) X X X The upper lmts of predcton nterval are computed b coeffcent plus standard error Y [ ] [ ] X [ ] X [ ] X [ ] X [ ] X 5 1 The lower lmts of predcton nterval are computed b coeffcent mnus standard error Y [ ] [ ] X [ ] X [ ] X [ ] X [ ] X 5 1 The wdth of predcton ntervals n respect of multple lnear regresson model and fuzz regresson model correspondng to each set of observed eplanator varables were computed manuall. As shown n Table 5, the average wdth for former multple regresson was found to be 0.34 whle usng fuzz regresson, whle the average wdth for fuzz regresson s 0.13 whch ndcates the superort of fuzz regresson methodolog. From ths analss, the most effcent method to obtaned relatonshp between response and eplanator varable s to appl fuzz regresson method compared to lnear regresson method Concluson Ths paper dscusses the combnaton of an algorthm wth robust, fuzz regresson and bootstrap method. The reasons for usng a small sample sze were (a) to appl a bootstrap method n order to acheve an adequate sample sze; (b) to compare the effcenc of orgnal method and the bootstrap method; and (c) to gve a better understandng on how the algorthm works. Accordng to general nsurance data, three ndependent varables n ths case were sgnfcant to the proftablt of general nsurance companes. Wthout usng robust and bootstrappng, the result shows that onl one out of fve varables were sgnfcant. Interestngl when usng robust to detect outlers and to provde resstant results n the presence of outlers and bootstrappng method (wth n = 100), the entre sgnfcant varable are ncluded n the model. Ths algorthm provdes us wth mproved understandng of the modfed method and underlng relatve contrbutons. Further stud lookng at possblt to approach response surface methodolog for each of sgnfcant varables n sngle algorthm s warranted. References Chrstmann, A. (1994). Least medan of weghted squares n logstc regresson wth large strata. Bometrka, 81, Donoho, D. L., & Huber, P. J. (1983). The noton of breakdown pont. In Bckel P. J., Doksum K. A., & Hodges, J. L. (Eds.), A festschrft for Erch, L. Lehmann (pp ). Belmont: Wadsworth. Draper, N., & Smth, H. (1998). Appled regresson analss (3 rd ed.). New York: Wle. Efron B., & Tbshran, R.J. (1993). An ntroducton to the bootstrap. New-York: Chapman and Hall. Hall, P. (1992). The bootstrap and edgeworth epanson. New-York: Sprnger Verlag. Kacprzk, J., & Fedrzz, M. (1992). Fuzz regresson analss. Warsaw: Omntech Press. 53

10 Marona, R., Martn, R., & Yoha, V. J. (2006). Robust statstcs theor and methods. England: John Wle & Sons Ltd. Naw, M. A. A., Ahmad, W. M. A. W., & Aleng, N. A. (2012). Effcenc of general nsurance n Malasa usng stochastc fronter analss (SFA). Internatonal Journal of Modern Engneerng Research, 2(5), Rousseeuw, P. J., & Lero, A. M. (1987). Robust regresson and outler detecton. New York: Wle- Interscence. Stromberg, A. J. (1993). Computaton of hgh breakdown nonlner regresson parameters. Journal of the Amercan Statstcal Assocaton, 88(421), Yaffee, R. A. (2002). Robust Regresson Analss: Some Popular Statstcal Package Optons. ITS Statstcs, Socal Scence and Mappng Group, 23, Yoha, V.J. (1987). Hgh breakdown-pont and hgh effcenc robust estmates for regresson. The Annals of Statstcs, 15, How to cte ths paper: Ahmad, W.M.A.W., Naw, M.A.A. & Mamat, M. (2016). A new strateg of handlng general nsurance modellng usng appled lnear method. Malasan Journal of Appled Scences, 1(1),