Possibilistic Modeling for Loss Distribution and Premium Calculation

Transcription

1 Possbstc Modeg fo Loss Dstbuto ad Pemum Cacuato Zhe Huag Ottebe Coege Westeve, OH 4308 Lja Guo Agothmcs, Ftch Goup Newto, MA ABSTRICT Ths pape uses the possbty dstbuto appoach to estmate the suace oss amout A speca cass of paametc possbty dstbutos s used to mode suace oss vaabes The paametes of the possbty dstbuto ae estmated by combg statstca aayss of sampe data ad doma kowedge povded by actuaa expets Isuace pemums ae cacuated usg possbstc mea ad possbstc vaato Estmato of possbty dstbuto of aggegate oss amout s aso dscussed Key wod: suace oss dstbuto, pemum, possbty dstbuto, possbstc mea, possbstc vaato, ad aggegate oss amout Itoducto Covetoay the ucetaty of poteta suace oss amout s modeed by pobabty theoy The suace pemum s the cacuated by usg the estmated pobabty dstbuto of pobabstc oss vaabe Possbty theoy povdes ateatve way fo modeg ucetaty [Zmmema (200)] Possbty theoy, ogated fom fuzzy sets theoy [Zadeh (965)], was fst poposed by Zadeh (978) ad was futhe deveoped by Dubos ad Pade (988) The dea of usg possbty dstbuto to mode ucetaty has bee apped to dffeet pobems Oe of such appcatos s to mode the ates of etu potfoo seecto pobems by possbty dstbutos, athe tha covetoa pobabty dstbutos [Guo ad Huag (996), Iuguch ad Tao (2000), Vechea, Bemúdeza ad Seguab (2007)] Some of the appcatos of possbty theoy ad fuzzy ogc suace ad faca sk maagemet wee evewed by Shapo (2004) I ths pape, we suggest usg possbty dstbuto to mode suace oss amout Ou pape s ogazed as foows I Secto 2 we befy toduce some cocepts o possbty dstbutos I Secto 3 we gve a hybd pocedue of estmatg possbty dstbuto of oss vaabe by usg sampe data ad expets kowedge I Secto 4 we povde fomuas

2 fo measug possbstc mea ad possbstc vaato Some pemum cacuato methods fo possbstc oss vaabes ae poposed Dscusso of possbty dstbuto, possbstc mea, ad possbstc vaato of aggegate oss cam amout s gve Secto 5 Cocusos ae gve Secto 6 2 Possbty Dstbuto Defto 2 [Dubos ad Pade (980), Zmmema (200)] Possbty dstbuto of a vaabe o ea umbe set s a mappg fom to teva [0, ], ad thee s at east oe c0 so that ( c0)= Possbty dstbuto ca be epeseted by fuzzy umbe usg membeshp fucto as possbty fucto Exampe : A tagua possbty dstbuto = ( ms,, s) has possbty fucto LR ( m x)/ s f x [ m s, m), ( x) = f x= m, ( x m)/ s f x ( m, m + s] Defto 22 A LR-possbty dstbuto = ( m, m, s, s ) has possbty fucto LR m x L f x [ m s, m), s ( x) = f x [ m, m], x m R f x ( m, m + s], s whee the eft efeece fuctos L :[0,] [0, + ) ad the ght efeece fucto R :[0,] [0, + ) satsfy the foowg codtos: () L(0) = R(0) =, (2) L(x) ad R(x) ae stcty deceasg, (3) L(x) ad R(x) ae uppe sem-cotuous o bouded supp( A ) = { x : ( x) > 0} The teva [ m, m ] s peak of, m ad m ae eft-mode ad ght-mode, ad s ad s ae eft-spead ad ght-spead 2

3 Tagua possbty dstbuto Exampe s a LR-possbty dstbuto wth m = m = m, ad the eft ad ght efeece fuctos Lu ( ) = Ru ( ) = u, 0 u A cass of specay costucted LR-possbty dstbuto that ca be used fo modeg suace oss vaabes s gve Huag ad Guo (2007) 3 Possbstc Modeg of Loss Vaabe I the foowg, we use possbty dstbuto to mode suace oss vaabe athe tha the covetoa pobabty dstbuto appoach Fo evey x, ( x ) s the possbty that x ca possby be assged to oss vaabe The possbty dstbuto ad pobabty dstbuto fo same vaabe must satsfy the Pobabty/Possbty Cosstecy Pcpe poposed by Zadeh (978): The possbty of a evet s aways geate tha o equa to ts pobabty Mathematcay, et be the possbty dstbuto ad p be the pobabty desty fucto fo the same vaabe, the the possbty/pobabty cosstecy pcpe s sup ( x) = Possbty( D) Pobabty( D) = p ( x) dx, x D fo ay uo D of dsjot tevas [Dubos ad Pade (980)] I suace, oss vaabes typcay have ght o heavy ght tas We popose a cass of paametc fuctos, caed powe-expoeta mxed fuctos, to ft hstogam of oss amout sampe data Defto 3 The cass of powe-expoeta mxed fuctos s defed as p m x a f x [ m s, m), s a f x [ m, m], ( x) = x m a exp p f x ( m, m + s], s 0 othewse whee a > 0, p, ad p > 0 D 3

4 Note that ( x) has seve paametes: heght paamete a; age paametes m, m, s, ad s ; ad shap paametes p ad p If we omaze ( x) though dvdg ( x) by sup, the we obta a powe-expoeta mxed LR-possbty dstbuto ( x) ( x) = ( m, m, s, s) LR = =, fo x sup a Sce ( x) has seve paametes, by choosg dffeet paamete vaues ( x) ca be used to ft a age cass of oss vaabes ethe wth ght o heavy ght tas Moe mpotaty, we have poved that the possbty dstbuto ad the pobabty dstbuto duce fom ( x) satsfy the Possbty/Pobabty Cosstecy Pcpe [Huag ad Guo (2007)]: ( x) ( x) The possbty dstbuto ( x) = =, fo x, ad the pobabty desty sup a fucto p ( x) = ( x)/ ( x) dx satsfy the Possbty-Pobabty Cosstecy Pcpe, amey fo ay uo D of dsjot tevas sup ( x) Possbty( D) Pobabty( D) p ( x) dx x D = = Ou pocedue uses sampe data combg wth actuaa expets kowedge to estmate seve paametes of ( x) The the possbty dstbuto ( x ) fo oss vaabe s duced fom ( x) A detaed descpto of the hybd pocedue ad umeca exampes ae gve by Huag ad Guo (2007) D 4 Pemum Cacuato fo Possbstc Loss Vaabe Dubos ad Pade (987) fst toduced the mea vaue of a fuzzy umbe as a cosed teva bouded by the expectatos cacuated fom ts uppe ad owe dstbuto fuctos Casso ad Fu'e (200) defed csp possbstc mea vaue ad vaace of fuzzy umbes Majede ad Fu e (2003) toduced weghted possbstc mea ad vaace of fuzzy umbes Sce possbty dstbutos ca be epeseted as fuzzy umbes, the cocepts of possbstc mea vaue ad vaace fo fuzzy umbes ca be atuay tasated to possbty dstbutos I ths secto a possbty dstbutos used ae assumed to be LR-possbty dstbutos We eed to toduce the cocept of α-eve cut fo possbty dstbuto befoe defg possbstc mea Defto 4 α-eve cut of a possbty dstbuto s defe by 4

5 ( ) α = { u ( u) α}, α > 0 It s easy to see that fo LR-possbty dstbuto, ( ) = [f( ),sup( ) ] Defto 42 (Casso ad Fu'e, 200) (a) The possbstc mea of s defed by ( ) = 0 α + M ( ) α sup( ) f( ) α dα (b) The possbstc vaace of s defed by 2 Va( ) = α( sup( ) α f( ) α) dα 2 0 (c) The possbstc stadad devato of s defed by 2 σ = Va( ) ( sup( ) f( ) ) = 2 α 0 α α dα α α α Possbstc stadad devato σ defed above ca be tepeted as a measue of vaato fo the possbty dstbuto Sce ( sup( ) α f( ) α ) defto of σ s aways geate o equa to zeo fo ay α [0,], we popose aothe way to measue the vaato of possbty dstbuto Ou foowg defto of possbstc vaato s sma to the absoute devato pobabty theoy Defto 43 Possbstc vaato of s defed by V( ) = α(sup( ) f( ) ) dα 0 α α Sce ( ) α = [f( ) α,sup( ) α], the dffeece (sup( ) α f( ) α ) ca be vewed as vaato of ( ) α, theα-eve cut of It foows that the possbstc vaato V ( ) s the expected vaue of vaatos of ts eve cuts V ( ) ca be used as a measue fo vaato of Exampe 4 Let = ( ms,, s) LR be a tagua possbty dstbuto, the (a) ( ) α = [ m s( α), m+ s( α) ], fo α [0,] (b) The possbstc mea vaue of s s s M( ) = m+ 6 (c) The possbstc vaato of s 5

6 s + s V ( ) = 6 Remaks: () If s = s (possbty fucto s symmetc) the M ( ) = m s + s age (2) The age of = ( ms,, s) LR s s + s, t meas that V ( ) = = 6 6 Ths s sma to the estmato fomua fo stadad devato of adom vaabe age statstcs σ 6 Ou pemum cacuato fo possbstc oss vaabe s aaogous to the covetoa vaace o stadad devato oadg pemum cacuato method whe the oss vaabe s modeed by pobabty dstbuto [Czek, Hade, ad Weo (2005)] We poposed the foowg sk oadg pemum cacuato method P ( θ) = M( ) + θv( ), θ 0, whee θ s the sk oadg facto RL A detaed dscusso of othe pemum cacuato methods fo possbstc oss vaabes ae gve by Huag ad Guo (2007) 5 Possbstc Aggegate Loss Vaabe Addto ad scaa mutpcato of possbstc vaabes dstbuto ca be defed by the sup-m exteso pcpe as addto ad scaa mutpcato of LR-fuzzy umbes [Dubos ad Pade (980)] Addto of possbstc vaabes wth LR-possbty dstbutos Let, =, 2, be two possbstc vaabes wth LR-possbty dstbutos = ( m,,, ),,2 m s s LR = The the possbty dstbuto of + 2 s a LRpossbty dstbuto + = ( m + m, m + m, s + s, s + s ) LR Scaa mutpcato Let be a possbstc vaabe wth LR-possbty dstbuto = ( m, m, t, t) LR, ad w a oegatve ea umbe The the possbty dstbuto of scaa mutpcato, w, s a LR-possbty dstbuto = ( wm, wm, wt, wt ) Poposto 5 w LR 6

7 Let, 2,, be possbstc vaabes ad w, w 2,, w be oegatve ea umbes Each has LR-possbty dstbuto = ( m,,, ) m s s LR wth the eft efeece fucto L ad the ght efeece fucto R, =,2,, The the possbty dstbuto of the ea combato dstbuto = wm, wm, ws, ws = = = = LR efeece fuctos L ad R Poposto 52 Let () = = wm = = w, the M( ) ( ) (2) V wv ( ) ( ) = = = = w s aso a LR-possbty wth some eft ad ght Futhemoe f a, =, 2,,, have the same the eft efeece fucto L ad the same ght efeece fucto R, the the eft ad ght efeece fuctos of ae the same as L ad R of Combg ths wth Poposto 5, we gve a compete descpto of possbty dstbuto of the foowg Poposto 53 Let, =,2,,, be possbstc vaabes wth LR-possbty dstbuto = ( m,,, ) m s s LR ad w, =,2,,, be oegatve ea umbes If a, =,2,,, have the same eft efeece fucto L ad the same ght efeece fucto R, the the possbty dstbuto of the ea combato aso a LR-possbty dstbuto eft ad ght efeece fuctos L ad R = wm, wm, ws, ws = = = = LR = w s = wth the Note that athough, =,2,,, have the same eft efeece fucto L ad the same ght efeece fucto R, they may st have dffeet possbty dstbutos due to dffeet m, m, s, ad s 7

8 Now f, =,2,,, ae possbstc oss vaabes wth LR-possbty dstbutos = ( m, m, t, t ) of the same eft efeece fucto L ad the same ght efeece LR fucto R, the the aggegate oss () (2) = = = = = = LR = has the foowg popetes: has LR-possbty dstbuto wm, wm, wt, wt M( ) = M( ) = (3) V V ( ) ( ) = = The possbty dstbuto of possbstc aggegate oss vaabe ca be expcty descbed by the possbstc dstbutos of dvdua possbstc oss vaabes Ths s ot tue geea fo the pobabty dstbuto of pobabstc aggegate oss vaabe 6 Cocusos Ths pape vestgated modeg suace oss evets ad the aggegate cam amouts usg possbty theoy whch has bee apped successfuy modeg ucetates fo othe pactca pobems Ou possbty dstbuto mode copoates actuaa expets kowedge to estmatg the possbty dstbuto fo oss vaabe A expct aaytca fom of possbty dstbuto of aggegate oss vaabe s deved Usg possbty dstbutos appoach to mode ucetates suace aea s a vey eay stage Futhe appcatos of possbty methods pcg suace poducts ad sk maagemet ae eeded 8

9 Refeeces [] Casso, C, ad Fu'e, R (200) O possbstc mea vaue ad vaace of fuzzy umbes, Fuzzy Sets ad Systems, 22, [2] Czek, P, Hade, W, ad Weo, R (2005) Statstca Toos fo Face ad Isuace, Spge [3] Dubos, D, ad Pade, H (980) Fuzzy Sets ad Systems: Theoy ad Appcatos, Academc Pess, New Yok [4] Dubos D, ad Pade H (987) The mea vaue of a fuzzy umbe, Fuzzy Sets ad Systems, 24, [5] Dubos D, ad Pade H (988) Possbty Theoy, Peum Pess, New Yok [6] Dubos, D ad Pade, H (2000) Fudametas of Fuzzy Sets, Kuwe Academc, Dodecht [7] Guo, LJ ad Huag, Z (996) A Possbstc Lea Pogammg Method fo Asset Aocato, Joua of Actuaa Pactce, 2, [8] Huag, Z ad Guo, LJ (2007) Possbstc modeg fo oss dstbutos, Wokg pape [9] Iuguch, M ad Tao, T (2000) Potfoo seecto ude depedet possbstc fomato, Fuzzy Sets ad Systems, 5, [0] Majede, P ad Fu e, R (2003) O weghted possbstc mea ad vaace of fuzzy umbes, Fuzzy Sets ad Systems, 36, [] Shapo, A (2004) Isuace appcatos of fuzzy ogc: the fst 20 yeas, 39 th Actuaa Reseach Cofeece [2] Vechea, E, Bemúdeza, JD ad Seguab, JV (2007) Fuzzy potfoo optmzato ude dowsde sk measues, Fuzzy Sets ad Systems, 58, [3] Zadeh, LA (965) Fuzzy Sets, Ifomato ad Coto, 8, [4] Zadeh, LA (978) Fuzzy sets as a bass fo a theoy of possbty, Fuzzy Sets ad Systems,, 3-28 [5] Zmmema, H-J (200) Fuzzy Set Theoy-ad Its Appcatos, Fouth Edto, Kuwe Academc Pubshes 9