Approximating the ruin probability of finite-time surplus process with Adaptive Moving Total Exponential Least Square

Transcription

1 WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee Approimatig the rui probability of fiite-time surplus process with Adaptive Movig Total Epoetial Least Square S. KHOTAMA, S. BOONTHEM, W. KLONGDEE * Departmet of Mathematics Kho Kae Uiversity THALAND * Correspodig author kwatch@ kku.ac.th Abstract: - The adaptive movig total least square (AMTLS) method has bee used for curve fittig. this study, AMTLS method is used for the rui probability fittig ad estimatio of the rui probability o a arbitrary iitial capital i fiite-time surplus process (or risk process). But i reality, it is difficult ad complicated to fid a fittig method for a appropriate estimate i order to obtai the best performace. So, a ew method is developed to estimate the rui probability of fiite-time surplus process. This ew method is called adaptive movig total epoetial least square (AMTELS) method that applies AMTLS method with least-square fittig epoetial. Claim data of motor isurace compay from Thailad has used i risk process for the rui probability fittig. Both AMTLS ad AMTELS methods cosider weighted fuctio for the distace betwee ode ad poit with a differet costat value. These methods are compared the performace by usig the mea squared error (MSE) ad the mea absolute error (MAE) that is, the error betwee the real rui probability value that is obtaied by the eplicit formula ad the rui probability fittig value. With these data, the rui probability approimatig eamples are give to prove that AMTELS method shows the better performace tha AMTLS method. Moreover, AMTELS method with the arrow value shows the better performace tha AMTELS method with the wide value. Key-Words: - Adaptive movig total least square, epoetial claim, least-square fittig epoetial, movig total least squares, the rui probability fittig, weighted fuctio 1 troductio Nowadays, the isurace is a importat part for esurig i life ad asset. Cosequetly, the study i isurace has may iterestig problems. For istace, the quatity of the risk is measured by the probability of isolvecy i busiess, it is so-called the rui probability. The rui probability has bee itroduced by [1]. the discrete-time risk process, rui probability of epoetial claim severities is easily calculated by usig eplicit formulas i []; see [3]. See also [4] for the fiite-time rui probability uder proportioal reisurace i discrete-time surplus process. Moreover, eplicit formulas for rui probability of a discrete-time risk process with proportioal reisurace ad ivestmet for epoetial ad Pareto distributio is preseted i [5]. But rui probability is difficult to fid the eplicit formulas for other distributio claim severities, such as Weibull, log-ormal, Pareto distributios ad etc. However, the rui probability is approimated by usig the simulatio techique whe parameters of claim size distributios are kow; see [6], [7]. Although the simulatio techique ca be used for the calculatio of approimatios for the fiite-time rui probability, the simulatio techique uses a log time. To solve this problem, may methods have bee proposed for a approach i regressio aalysis to approimate the fittig value provided by a model. Movig Total Least Squares (MTLS) as a approimatio method has bee used for data fittig; see [8]. Referece [9] has bee used MTLS method for geeratig a surface approimatig form the give data. The Adaptive Movig Total Least Squares (AMTLS) method is suggested by [10] for curve fittig. this paper, a rui probability fittig approach called adaptive movig total epoetial least square (AMTELS) method is proposed. AMTLS ad AMTELS method are compared to estimate rui probability of fiite-time surplus process. A cocise descriptio is give for the rui probability i Sectio. Sectio 3, both AMTLS ad AMTELS methods are preseted i detail. Rui probability approimatig eamples are give i Sectio 4 for comparig betwee AMTLS ad AMTELS methods ad coclusios are show i Sectio 5. E-SSN: Volume 15, 018

2 WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee Rui probabilities with epoetial claims We assume that the claim size process { X : } is idepedet ad idetically distributed (i.i.d.), ad we cosider the rui probability of fiite-time surplus process with epoetial claim size distributio. The epoetial distributio with itesity parameter 0 0 has the probability desity fuctio is give by 0 f ; e, 0. (1) X 0 0 The cumulative distributio fuctio of the epoetial distributio is obtaied by: Referece [] proposed the eplicit formulas for fiite-time rui probability of epoetial claim severities with itesity is obtaied by k 1 [ 0 ( u kc)] 0 ( ukc) ( u c) e (3) k 1 ( k 1)! ( u kc) ( u), where u is the iitial surplus ad 0,1,,.... c 0 is the costat premium rate ad ca be calculated i terms of the epected value priciple, i.e., c (1 ) E[ X1] where 0 is the safety loadig of isurer ad EX [ 1] 1/ 0. 0 F ; 1 e, 0. () X 0 Fig.1 Probability desity fuctio of epoetial distributio Fig. Cumulative distributio fuctio of epoetial distributio E-SSN: Volume 15, 018

3 WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee 3 Descriptio of the estimators this sectio, we are describe AMTLS ad AMTELS methods 3.1 Adaptive movig total least squares (AMTLS) method AMTLS approimatio, a fuctio for the local approimatio at is defied as a 1 1 a 1 J ( a, b ) w( ) ( a b y ) (4) where (, y) : 1,,3,..., is the set of discrete poit to be fitted ad w( ) is a weighted fuctio. a ad b are the coefficiet of (4) i the sese of the total least squares by miimizig the weighted sum of squared distaces. The parameters of local approimats, a ad b ca be obtaied by solvig a system of oliear equatios as follows: J a ( a a a ) w 4 1 (3 a b a b a b b ) w 4 1 ( a 3a a 1) w y 1 1 ( a b a b ) wy ( a a ) wy ( a b a b ) w 0. 1 (5) Fig. 3 Local approimats of AMTLS ad J a w b w wy 0. b (6) AMTLS method, a parameter associated with the directio of local approimats is itroduced to assig the directio local approimats (see Fig. 3). case of 0, we see that (4) is called the movig total least squares (MTLS) method ad is a fuctio for local approimats as show i (7): ( a b y ) J ( a, b ) w( ). (7) 1 a 1 case of 1, we see that (4) is called the movig least squares (MLS) method, defie a fuctio for local approimats as: E-SSN: Volume 15, 018

4 WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee (8) 1 J ( a, b ) w( )( a b y ). 3. Adaptive Movig Total Epoetial Least Square (AMTELS) method This paper develops the least-square fittig epoetial with AMTLS method ad is called Adaptive Movig Total Epoetial Least Square (AMTELS) method which its algorithm is described as follows: The least-square fittig epoetial is i the form BX Y Ae. (9) The, take logarithm of both sides, we obtai ly l A BX. (10) For coveiet, let y l Y, X, b l A ad a B. Now, the model is chaged ito the form y a b. Applyig the least square criterio (10) to this situatio requires the miimizatio of a 1 (, ) ( ) J a b w ( a ) b y a 1 1 (11) where y l Y ad X. The, Yˆ ep( y), where Ŷ is the rui probability fittig value i AMTELS method. The weighted fuctio is importat framework of fuctio for local approimats of both AMTLS ad AMTELS methods. Throughout this research, we use the followig weighted fuctio: ( / d ) 1/ e e if, ( ) 1/ d w 1 e (1) 0 elsewhere, which give by [11], where is a parameter determiig the weighted fuctio s shape. The value of d has to be large eough to iclude a miimum umber of odes as metioed above. 4 Simulatio study This eperimet uses data of motor isurace compay i Thailad. The purpose of this paper is to compare the performace of approimatio of the rui probability betwee AMTLS method ad AMTELS method for epoetial claim. The first operatio, we cosider the data ad histogram of the claim size as show i Fig.4, Fig.5, respectively. We foud that scale parameter of epoetial claim 6 6 as , i.e., X 1 ep( ). After that, we determie 356, c (1 0.1)(1/ ), ad the calculate the fiite-time rui probability y, 1,,...,55, with epoetial claim size by (3) for iitial capital ( 0,0.1,0.,...,5). Net, we approimate the rui probability of iterested iitial capital as 0.00,0.01,0.0,...,5.00 by AMTLS method with weighted fuctio for the distace betwee ode ad poit with a costat value d 0.5, 0.5 ad 0. The rui probability fuctio of fiite-time surplus process with epoetial claims is plotted i Fig.6. 6 E-SSN: Volume 15, 018

5 WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee Fig.4 The claim size of motor isurace i Thailad Fig.5 Histogram of claim data Fig.6 The approimatio rui probability E-SSN: Volume 15, 018

6 WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee Fig.7 The rui probability fittig of AMTLS method Fig.8 The rui probability fittig of AMTELS method From Fig.6, we foud that the rui probability fuctio is also a decreasig fuctio o the iitial capital, ad has a very clear epoetial distributio fuctio feature. Therefore, least square fittig epoetial is developed to estimate rui probability, which is AMTELS method. We ow approimate of the rui probability by AMTLS method ad AMTELS method with weighted fuctio that differet costat value d ( d 0.1, ,0.7, ad 0.9), ad 0.5. These results are preseted i Fig.7 ad Fig.8, respectively. Net, we compare the performace of both methods by usig the mea squared error (MSE) ad the mea absolute error (MAE) of the real rui probability value that are obtaied by the eplicit formula (3) ad the rui probability fittig value: 1 MSE y y ( i ˆi), (13) i MAE yi yˆ i, (14) 501 i1 E-SSN: Volume 15, 018

7 WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee where y i is the real rui probability value ad y ˆi is the rui probability fittig value for measured data. The eperimet result is preseted i Table 1. For the same costat value d, it is clear that both MSE ad MAE values of AMTELS method are less tha AMTLS method. the same method, both MSE ad MAE values of costat value d 0.1are less tha d 0.3,0.5,0.7,0.9, respectively. d Table 1. Eperimet result of rui probability estimators MSE 1 MAE 6 ( 10 ) ( 10 ) AMTLS AMTELS AMTLS AMTELS Coclusio this study, AMTELS method is developed ad represeted to approimate the rui probability. ts superior features are show over AMTLS method. Suitability of both methods is judged base o MSE ad MAE values for the weighted fuctio with differet costat value d. The rui probability fittig of AMTELS method is give the better performace tha the rui probability fittig of AMTLS method. Furthermore, AMTELS method with the arrow value d shows the better performace tha AMTELS method with the wide value d. Therefore, i this study, the appropriate method for approimatig the rui probability of fiite-time surplus process is AMTELS method with the arrow value d ad it does ot take log. Although AMTELS method fit the rui probability tha AMTLS method, two methods do t have the eplicit formulas of approimatios for the rui probability. These methods deped o regressio aalysis to approimate the fittig value. Ackowledgmet This research is supported by Sciece Achievemet Scholarship of Thailad (SAST). Refereces: [1] J. Gradell, Aspects of Risk Theory, New York: Spriger-Verlag, [] W. S. Cha ad L. Zhag, Direct derivatio of fiite-time rui probabilities i the discrete risk model with epoetial or geometric claims, North America Actuarial Joural, Vol.10, No.4, 006, pp [3] P. Sattayatham, K. Sagaroo, ad W. Klogdee, Rui Probability-Based itial Capital of the Discrete-Time Surplus Process, Variace, Vol.7, No.1, 013, pp [4] S. Khotama, K. Sagaroo, ad W. Klogdee, A sufficiet coditio for reducig the fiitetime rui probability uder proportioal reisurace i discrete-time surplus process, Far East Joural of Mathematical Scieces (FJMS), Vol.96, No.5, 015, pp [5] H. Jasiulewicz ad W. Kordecki, Rui probability of a discrete-time risk process with proportioal reisurace ad ivestmet for epoetial ad Pareto distributios, Operatios Research ad Decisios, Vol.5, No.3, 015, pp [6] S. Khotama, T. Thogjuthug, K. Sagaroo, ad W. Klogdee, O Approimatig the Miimum itial Capital of Fire surace with the Fiite-time Rui Probability usig a Simulatio Approach, Asia-Pacific Joural of Sciece ad Techology (APST), Vol.0, No.3, 015, pp [7] W. Klogdee ad S. Khotama, Miimizig the itial Capital for the Discrete-time Surplus Process with vestmet Cotrol uder Alpharegulatio (Published Coferece Proceedigs E-SSN: Volume 15, 018

8 WSEAS TRANSACTONS o BUSNESS ad ECONOMCS S. Khotama, S. Boothiem, W. Klogdee style), i the teratioal MultiCoferece of Egieers ad Computer Scietists 018, Hog Kog, 018, pp [8] R. Scitovski, S. Ugar, D. Juki, ad M. Crjac, Movig Total Least Squares for Parameter detificatio i Mathematical Model, Operatios Research Proceedigs, Spriger, Berli, Vol. 1995, pp [9] R. Scitovski,. Ugar, ad D. Juki, Approimatig surfaces by movig total least squares method, Applied Mathematics ad Computatio, Vol. 93, No.1-, 1998, pp [10] Z. Lei, G. Tiaqi, Z. Ji, J. Shiju, S. Qigzhou, ad H. Mig, A adaptive movig total least squares method for curve fittig, Measuremet, Vol. 49, 014, pp [11] U. H. Combe ad C. Kor, A adaptive approach with the Elemet-Free-Galerki method, Computer methods i applied mechaics ad egieerig, Vol.16, No.1-4, 1998, pp E-SSN: Volume 15, 018