Credibility Modeling via Spline Nonparametrie Regression

7/29/2019 Credibility Modeling via Spline Nonparametrie Regression

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Credibility M odelingvia Spline Nonparametrie RegressionAshis Gangopadhyay and W u-Chyuan Gau

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Credibil ity M odeling via Spline Nonparam etric RegressionDr. Ashis Gangopadhyay

Associate Professor,Mathematics and Statistics,

Boston University,111 Cummington Street,

MA 02215, U.S.A.Phone: 617-353-2560.

Email: [email protected]. Wu- Chyuan Gau

Assistant Professor,Statistics and Actuarial Science,

University of Central Florida,Orlando, FL 32816,

U.S.A.Phone: 407-823-5530.

Email: [email protected].

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Cred ibility M odeling via Spline N onparam etric RegressionA b s t r a c t

Cred ib i l i t y mode l ing i s a ra t e making p rocess wh ich a l l ows ac tua r ie s t o ad jus t fu -tu re p rem iums acco rd ing to t he pas t expe r ience o f a r i sk o r g roup o f ri sks . Cu r ren tme thods in c red ib i l i t y t heo ry o f t en re ly on pa ram e t r i c m ode l s. B l ih lm ann (1967)deve loped an app roach based on the bes t l inea r app roxima t ion , wh ich l eads t o anes t im a to r t ha t i s a l inea r comb ina t ion o f cu r ren t obse rva t ions and pas t r eco rds . Dur -ing the l a s t decade , t he ex i s t ence o f h igh speed com pu te rs and s t a t i s t i ca l so f twa repackages al l owed the in t roduc t ion o f more soph is t i ca t ed m e thodo log ie s . Som e o ft h e s e t e c h n i qu e s a r e ba s e d o n M a r ko v C h a i n M o n t e C a r l o ( M C M C ) a p p r o a c h t oBayes ian in fe rence , wh ich requ i re s ex t ens ive com pu ta t ions . How eve r , ve ry few o fthese me tho ds m ade use o f t he add i t iona l cova r ia t e in fo rm a t ion re l a t ed t o t he r i sk ,o r g roup o f r isks; and a t t he sam e t ime accoun t fo r t he co r re l a t ed s t ruc tu re in t heda t a . In t h is pape r , we cons ide r a Bayes ian nonpa ra m e t r i c app roach to t he p rob l emof r i sk mode l ing . Th e m ode l inco rpo ra t e s pas t and p re sen t obse rva t ions re l a t ed t othe r i sk , a s we l l a s re l evan t cova r ia t e info rma t ion . Th e Bayes ian m ode l ing is ca r r iedou t by samp l ing f rom a m u l t iva r ia t e Gauss ian p r io r, w he re t he cova r iance s t ruc tu rei s based on a t h in -p l a t e sp line (Wahba , 1990). Th e m ode l u ses M CM C t echn iqueto com pu te t he p re d ic t ive d i s t r ibu t ion o f t he fu tu re c l a ims based on the ava i l ab l eda t a . Ex t ens ive da t a ana lys i s is conduc t ed t o s tudy the p rope r t i e s o f t he p roposedes t ima to r , and com pare aga ins t t he ex i s t ing t echn iques .Keywords : Cr ed ib i l i t y Mode l ing , Th in -p l a t e Sp l ine , M CM C, RK HS .

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1 I n t r o d u c t i o n

Th e d ic t iona ry de f in i t ion o f a sp l ine i s "a t h in s t r ip o f wood used in bu i ld ing con -s t ruc t ion . " Th is in fac t g ives in s igh t in to t he m a them a t ica l de f in i tion o f spl ines .His tor ica l ly , engineering draf t smen used long th in s t r ips of wood ca l led sp l ines tod raw a sm oo th cu rve be tw een speci f ied po in ts . A m a them a t ica l sp l ine is t he so lu t iont o a cons t ra ined op t imiza t ion p robl em.

In t he c red ib i l i t y con t ex t , suppose we wish t o de t e rmine how the cu r ren t c l a imloss, Yij, de pen ds on th e pa st losses, say Yi, j-1 and Yi, j-2. O ur app roa ch is to conside rt h e n o n p a r a m e t r i c r e g r e s s io n m o d e l

y , j = g ( y , , ~ - l , y , , j - 2 ) + , , , ~ = 1 , . . ., n , ( 1 )where g is a sm oo th fun c t ion o f i t s argumen t s . Our ob j ec t ive is t o m ode l t he de -pend enc y be tw een th e cur ren t o bserv at ions Yi~ for a ll po l icyho lders i = 1 , . .. , n , andthose pas t losses Y~, j-1 and Y~,3-2 throu gh a nonp aram et r ic regress ion a t occasion j .Th e concep t i s s imi l a r t o mu l t ip l e l inea r reg re ss ions . He re , t he dependen t va r iab l ehappe ns to be Y~j wh ile the pas t losses Y~,3-1 and Y~, j-2 are t re a te d as covaria tes .Fo r no t a t iona l conven ience , we l e t Y~ s t ands fo r t he depend en t va r iab l e (cu r ren t ob-serva t ion) and use s i o r t i for covaria tes (pas t losses). Th e ke y proble m is to f ind agood app roxim a t ion ~ o f g . Th is i s a trac t ab l e p robl em , and the r e a re m any d i f fe ren tso lu t ions t o t h i s p robl em. Th e pu rpose o f t h i s pape r i s t o deve lop a me thodo logy toes t im a t e t he func t ion g g iven the da t a . We use a nonpa ram e t r i c Bayes ian app roacht o es t im a t e t he m u l t iva r ia t e reg re ssion mo de l wi th Gauss ian e r ro rs. In t h i s approach ,ve ry l i t t le i s a s sumed rega rd ing the unde r ly ing mode l ( s igna l ); and we a l low the da t ato "speak fo r i t s e l f " .

Rep roduc ing ke rne l Hi lbe r t space (RKHS) mode l s have been in u se fo r a t l ea s t

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n i n e t y- f i v e y e a rs . T h e s y s t e m a t i c d e v e l o p m e n t o f r e p r o d u c i n g k e r n e l H i l b e r t s p a c et h e o r y i s g i v e n b y A r o n s z a j n ( 1 95 0 ). F o r f u r t h e r b a c k g r o u n d t h e r e a d e r m a y r e fe rt o W e i n e r t ( 1 9 8 2) a n d W a h b a ( 1 99 0 ). A r e c e n t p a p e r g i v e n b y E v g e n i 0 u (2 0 0 0)c o n t a i n s a n i n t r o d u c t i o n t o R K H S , w h i c h w e f o u n d t o b e u s e f u l fo r r e a d e r s i n t e r e s t e di n f u r t h e r r e a d i n g .

A r e p r o d u c i n g k e r n e l H i l b e r t s p a c e i s a H i l b e r t f u n c t i o n s p a c e c h a r a c t e r i z e d b yt h e f a c t t h a t i t c o n t a in s a k e r n e l t h a t r e p r o d u c e s ( t h r o u g h a in n e r p r o d u c t ) e v e r yf u n c t i o n in t h e s p a c e , o r , e q u iv a l e n tl y , b y t h e f a c t t h a t e v e r y p o i n t e v a l u a t i o n f un c -t i o n a l i s b o u n d e d . R K H S m o d e l s a r e us e f ul i n e s t i m a t i o n p r o b l e m s b e c a u s e e v e r yc o v a r i an c e f u n c t i o n is a l so a r e p r o d u c i n g k e r n e l f o r s o m e R K H S . A s a c o n s eq u e n c e ,t h e r e i s a c l os e c o n n e c t i o n b e t w e e n a r a n d o m p r o c e s s a n d t h e R K H S d e t e r m i n e d b yi t s c o v a r i a n c e f u n c t i o n . T h e s e e s t i m a t i o n p r o b l e m s c a n t h e n b e s o l v e d b y e v a l u a t i n ga c e r t a i n R K H S i n n e r p r o d u c t . T h u s i t is n e c e s s a r y t o b e a b l e t o d e t e r m i n e t h e f o rmo f i n n e r p r o d u c t c o r r e s p o n d i n g t o a g i v e n r e p r o d u c i n g k e r n e l.

I n o p t i m a l c u r v e a n d s u r f a c e f i tt i n g p ro b l e m s , i n w h i c h o n e is r e c o n s t r u c t i n g a nu n k n o w n f u n c t i o n b a s e d o n t h e s a m p l e d a t a , i t is i n e v i t ab l e t h a t t h e p o i n t e v a l u a ti o nf u n c t i o n a l s b e b o u n d e d . T h e r e f o r e , o n e i s f o r ce d t o e x p r e s s t h e p r o b l e m i n a R K H Sw h o s e i n n e r p r o d u c t i s d e t e r m i n e d b y t h e q u a d r a t i c c o s t f u n c t i o n a l t h a t n e e d s t ob e m i n i m i z e d . T o s o l v e t h e s e p r o b l e m s , o n e m u s t f i n d a b a s i s f o r t h e r a n g e o f ac e r t a i n p r o j e c t i o n o p e r a t o r . O n e w a y t o d o t h i s is t o d e t e r m i n e t h e r e p r o d u c i n gk e r n e l c o r r e s p o n d i n g t o t h e g i v e n i n n e r p r o d u c t .

C o n s i d e r a n u n i v a r i a t e m o d e l

y , = : ( t i ) + e,, i = 1, 2, . . .n (2 )

whe re E = ( c l , . .. , sn ) ' " N (0 , a 2 I ) a n d f i s o n l y k n o w n t o b e s m o o t h . I f f h a s m - 1

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c o n t i n u o u s d e r i v a t i v e s a n d i s m t h d e r i v a t i v e i s s q u a r e i n t e g r a b l e , a n e s t i m a t e o f fc a n b e f o u n d b y m i n i m i z i n g

1 _ _( y , _ S ( t~ ) ) 2 + A r /b (S (m t ) ) 2 d t , (3 )d ai =1

f o r s o m e s > 0 . T h e s m o o t h i n g p a r a m e t e r A c o n t r o l s t h e t r a d e - o f f b e t w e e n s m o o t h -ness and accu racy . A d i sc r e t e v e r s ion o f p rob le ms such a s ( 3 ) was cons ide r ed int h e a c t u a r i a l l i t e r a t u r e b y W h i t t a k e r ( 1 92 3 ), w h o c o n s i d e r e d s m o o t h i n g Y l, ...,Y nd i sc r e t e ly b y f i nd ing f = ( f l , . .. , f n ) to m in imize

n n--3l ~ ( y i _ f ~ ) 2 + A ~ - ' ] ( S ~ + 3 - 3 f ~ + 2 + 3 S i + l - S ~ ) 2 . ( 4 )

i =1 i =1

I f m = 2 , ( 3 ) b e c o m e s t h e p e n a l i z e d r e s i d u a l s u m o f s q u a r e s

'~ .X f b! V ' n ~ ( y , - S ( t , ) ) 2 + ( S " ( t ) ) 2 ~ x ( 5 )i =1wh ere A i s a f ixed co nst an t , an d a ~< t l


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e r ro r s . S ec t i on 5 gene ra l i ze s the r e su l t s i n sec t i on 4 to th e t r i v a r i a t e ca se . Sec t i on 6i n t r o d u c e s t h e a p p l i c a t i o n s o f t h e r e s u l t s d e v e l o p e d i n s e c t i o n 4 a n d s e c t i o n 5. A l lc o m p u t a t i o n s a r e c a r r i e d o u t u s i n g G i b b s s a m p l e r . A l l f u n c t i o n s , f u nc t i o n a ls , r a n d o mv ar i ab le s , and func t i on spaces i n th i s pape r w i l l b e r ea l v a lued un le ss spec i f i ca l lyn o t e d o t h e r w i se .

2 T h e C r e d i b i l i t y P r o b l e mT h e c l a s s ic a l d a t a t y p e i n t h i s a r e a i n v o lv e s r e a l i z a t io n s f r o m p r e s e n t a n d p a s t e x p e r i-ence o f i nd iv idu a l po l icyho lde r s . Suppo se we hav e n d i f f e r en t r i sk s ( o r po li cyho lde r s )w i t h a c l a i m s r e c o r d o v e r a c e r t a i n n u m b e r o f y e a r s, s a y T ;

Y,~l, Y, ,2, . . . , Y,~T.

T h e d a t a c a n b e t h e a m o u n t o f l o s s e s , t h e n u m b e r o f c l a i m s , o r t h e l o s s r a t i o f r o mi n s u r a n c e p o rt f ol i os . O u r g o a l i s t o e s t i m a t e t h e a m o u n t o r n u m b e r o f c l a im s t o b ep a i d o n a p a r t i c u l a r i n s u r a n c e p o l ic y i n a f u t u r e c o v e r a g e p e r i o d . T h e p r o b l e m o fi n t e r e s t is t o m o d e l t h e r e l a t i o n s h i p o f Y T + I t o t i m e a n d t h e p a s t o b s e r v e d v a lu e s o fY1 , Y2 , .. ., YT , i .e . , t o e s t a b l i sh the r e l a t i onsh ip :

y ~j = f ( t , y ~ l , y ~ 2 , . . . , y ~ j _ l ) + e ~ j f o r i = l , 2 . . .. . n ; j = l , 2 , . . . , T (6 )w h e r e f i s a n u n k n o w n f u n c t i o n a n d e i~ i s a r a n d o m e r r o r t e r m .

W e can a l so hav e a more gene ra l f o rm o f da ta con f igu ra t i on . Le t {Yi j : j =

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1 , 2 , . . . , T i ; i = 1 1 2 , . . . , n } b e a t i m e s e r i e s o f l e n g t h T i o v e r w h i c h t h e m e a s u r e m e n t so f t h e i th r i sk ( o r g r ou p o f r i sks ) we r e obse r ve d a t t i m e p o i n t s { ti 3 : j = 1 , 2 , . . , T i } .I n a d d i t i o n , l e t X ~ j b e t h e o b s e r v e d c o v a r i at e s , s u c h a s g e n d e r o f a p o l ic y h o l d e r , o ri n d u s t r y t y p e o f i n s u r e d s e t c . , f o r t h e i th r i s k ( o r g r o u p o f r i s k s ) a t t i m e t i j . I n e a c hr i sk ( o r g r o u p o f r i sk s ) , t h e d a t a h a s t h e f o r m

( y ~ j , X ~ j , t ~ j ) , j = 1,2, . . .,T~; i = 1 ,2 . . . . . n, (7)

wh e r e X i j = ( X i j l , X~ j2 , . .. , X~ ja) a r e the d c ova r i a t e v a r i a b l e s m e a su r e d a t t i m e t~ 3.I n t h i s ca s e , o f i n t e r e s t i s t o s t u d y t h e a s s o c i a t i o n b e t w e e n t h e c u r r e n t r e s p o n s e Y ijan d th e pa s t r es pon ses Y~5-1 = (Y~,j -1 , Y~,j -2 , . .. , Yi t) a s we l l a s th e cov ar ia te s an d t oe x a m i n e h o w t h e a s s o c ia t i o n v a ri e s w i t h t i m e . T a b l e 1 p r o v i d e s t h e d a t a l a y - o u ta s sum i ng j = 1 , 2 , . . . , T .

O c c a s i o nS u b j e c t 1 . . . T

1 Y l l , X l l , I , X l l , 2 , . . . l X l l , d l ~ l l " 9 " Y l T l X l T , 1 1 X l T , 2 , . .. l X l T , d ~ t l T: : . . . !

i Y i l 1 X i l , 1 ~ X i l , 2 , " " " , X i l , d ~ ~ i l " " " Y i T ~ X i l , T , X i l , T , . .. ~ X i T , d l t i T: : :

n Y n l , ~ n l , l , 2 ~ n l, 2 , . . .I X n l , d ~ n l ' ' " Y n T l E n T , 1 , T n T , 2 ~ .. . ~ T n T , d l ~ n TT a b l e 1 : D a t a C o n f i g u r a t i o n .

T h e r e f o r e , w e p r o p o s e t h e f o l lo w i n g m o d i f i e d m o d e l , w h i c h i s m o r e g e n e r a l t h a n ( 7 ),

Y i j = f ( t ~ j , X i 3 , y ~ , j _ l ) + e i j for i = 1, 2 . . . . . n; j = 1, 2, . . . , T~. (8)

I n t h i s p a p e r , a n e w B a y e s i a n a p p r o a c h i s p r e s e n t e d f o r n o n p a r a m e t r i c m u l t i v a r i -a t e r e g r e s s i o n w i t h G a u s s i a n e r r o r s . A s m o o t h n e s s p r i o r b a s e d o n t h i n - p l a t e s p l i n e si s a s s u m e d f o r e a c h c o m p o n e n t o f t h e m o d e l . W e u s e t h e r e p r o d u c i n g k e r n e l f o r a

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t h in -p l a t e sp l ine fo r an unknown m u l t iva r ia t e func t ion a s in W ahba (1990). A l l t hecom pu ta t ions a re ca r r i ed ou t u s ing the G ibbs samp l ing schemes (Wood e t . a l. , 2000).W i th a bu rn - in pe r iod , i t i s a s sumed tha t i t e ra t ions have conve rged to d raws f rompos t e r io r d i st r ibu t ions . A rando m sam ple f rom the conve rgence pe r iod axe u sed toes t im a t e cha rac t e r i s t i c s o f t he pos t e r io r d is t r ibu t ion . Th is m ode l i s u sed fo r e s t ima-t ion o f func t ion f and to p red ic t fo r t he fu tu re va lues . W e ana lyze a rea l da t a f romone Ta iwan based in su rance company . A compar i son i s be ing ca r r i ed ou t be tweenthe p roposed app roach aga ins t o the r ex i s t ing t echn iques .

3 T h e T h i n - P l a t e S p l i n eRK HS m e thods have be en success ft fl ly app l ied t o a w ide va r ie t i e s o f p robl ems in t hef ield o f op t im a l app roxim a t ion , wh ich inc lude in t e rpo la t ion and sm oo th ing v ia sp linefunc t ion in one o r m ore d imens ions . Th e one d im ens iona l ca se is gene ra li zed t o t hemu l t id ime ns iona l ca se by D uchon (1977) . Duc hon ' s su r face sp l ine i s ca l l ed " th inp l a t e " sp l ine , because t hey app roxim a te t he equ i l ib r ium pos i t ion o f a t h in p l a t ede f l ec ted a t s ca t t e r po in ts . Fo r an app l ica t ion o f t h in -p l a t e sp l ines t o m e t eo ro log ica lp robl ems see W ahba and W ende lbe rge r (1980) .

3 . 1 T h e T h i n - P l a t e S p l i n e o n E dTh e theo re t i ca l founda t ions fo r t he t h in -p l a t e sp l ine we re f rom Duc hon (1975, 1976,1977) and M eingue t (1979) , and some fu r the r re su l t s and app l ica t ions t o me t eo ro -log ica l p robl em s were g iven in Wahba and W ende lbe rge r (1980) and Wood e t . a l.( 2 0 0 0 ) .

Le t u s de f ine t he pena l ty func t iona l

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f 01(fr . ( f ) = ( ~ ) ( ~ ) ) 2 d ~ .I t i s a s s u m e d t h a t d a t a y = ( y l , ' 9" , y .) ~ f o l l o w s t h e m o d e l

y , = f ( x , ) + ~ , , i = l , 2 , . . . , n ,w h e r e X a E f / a n d f l is a g e n e r al i n d e x se t . T h e f u n c t i o n f i s a s s u m e d t o b e as m o o t h f u n c t i o n i n a r e p r o d u c i n g k e r n e l H i l b e r t s p a c e H o f a re a l - v a l u e d fu n c t i o n so n f~. T h e { ei} a r e i n d e p e n d e n t z e r o m e a n e r ro r s w i t h c o m m o n u n k n o w n v a ri an c e .I t is d e s i r e d t o f i n d a n e s t i m a t e o f f g i v e n y = ( Y l , ' " , Yn ) . T h e e s t i m a t e f a o f fw i l l b e t a k e n a s t h e m i n i m i z e r in H o f

t n~ - ~ ( y , - f ( X , ) ) 2 + A J m ( f ) , (9 )i= 1

w h e r e J ,n ( f ) i s a s e m i n o r m o n H w i t h M - d i m e n s i o n a l n u ll s p a c e s p a n n e d b y r , C M,M < n . T h e s e m i n a o r m o n t h e v e c t o r s p a c e H i s a m a p p i n g p : H ~ R s a t is f y i ngIla] l > 0, I laall = la] l la l l , an d Ila+bll


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W e w a n t X e n d o w e d w i t h t h e s e m i n o r m j d ( f ) t o b e a n R K H S ( t h a t i s, fo r t h ee v a l u a t io n f u n c t io n a l s i n X t o b e b o u n d e d w i t h r e s p e c t t o j d ( f ) ) . T h e n , a t h i n -p l a t e s m o o t h i n g s p l i n e i s t h e s o l u t i o n t o t h e f o ll o w in g v a r i a t i o n a l p r o b l e m . F i n df E X t o m i n i m i z e

I II~ ~-~(Yl - - f ( x l ( i ) , ' " , Xd( i) )) 2 + ~kJ d( f) . (12)i = l

L e t u s u s e t h e n o t a t i o n t = ( X l , . . . ,Xd) ' a n d t i = ( x l ( i ) , . . . , xd( i ) ) ' . T h e n u l ls p a c e o f t h e p e n a l t y f u n c t i o n a l j d ( f ) i s t h e M - d i m e n s i o n a l s p a c e s p a n n e d b y t h epo lynom ia l s i n d v a r i ab l e s o f to t a l d eg r ee ~< m - 1 , wh e r e

I n t h e s p a c e H = { f : J ~ ( f ) < o o } w i t h J ~ ( f ) a s a s q u a r e s e m i n o r m , i t i sn e c e s s a r y t h a t 2 m - d > 0 fo r t h e e v a l u a t i o n f u n c t i o n a l L t f = f ( t ) t o b e c o n t i n u o u s ;s e e D u c h o n ( 1 9 7 7 ), M e i n g u e t ( 1 9 79 ) , a n d W a h b a a n d W e n d e l b e r g e r (1 9 8 0) . F o rm = 2 , d = 2 ,

w i t h M - - 3 , a n d t h e n u ll s p a c e i s s p a n n e d b y r r 1 6 2 g i v e n b y( 1 4 )

r x 2 ) = 1 , r ~ ) = ~ 1 , r ~ 2 ) = ~ .B e f o r e w e g o f u rt h e r , w e n e e d a d d i t i o n a l n o t a t i o n s . L e t s , t E E d , s = ( s l , . . . ,Sdya n d t = ( t l , . . . , Q ) ' , t h e n

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W e c a n d e f i n e

dI ~ - t I = ( ~ ( ~ , - t i ) 2 ) ' 1 2 .

i = l

w h e r e

E ( r ) = O ~ r 2 m- d l og r i f d e v e n~d 2m-d= o m r i f d o d d ,

(15)

( - 1 ) d / 2 + 1i f d e v e n ( 1 6 )od = 22m- ' Trd / 2 (m - - 1 ) ! ( m - - d / 2 ) !

= ( - 1 ) m r ( d / 2 - m ) i f d o d d .22mTcd/2(m - 1) !

W e c a n a l s o d e f i n e

Em(s , t ) = E( I s - t [ ) . ( 1 7 )D u c h o n ( 19 7 7 ) s h o w e d t h a t , i f t l , . . . , t n a r e s u c h t h a t l e a s t s q u a r e s r e g r e s si o n o nr , C M is u n i q u e , t h e n ( 12 ) h a s a n u n i q u e m i n i m i z e r f x w i t h r e p r e s e n t a t i o n

M nI ~( t) = ~ d . r + E c i E m ( t , t, ) . (18)

~ = I i = I

N o t e t h a t 0 ~ c a n b e a b s o r b e d i n t o c / i n ( 1 8 ) . L e t u l , . . . , U M b e a n y f i x e d p o i n t si n E d s u c h t h a t l e a s t s q u a r e s r e g r e s s i o n n t h e M - d i m e n s i o n a l s p a c e o f p o l y n o m i a l so f t o t a l d e g r e e l e s s t h a n m a t t h e p o i n t s u l ,. . . , U M i s u n i q u e . L e t P l , " " , P M

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b e t h e p o l y n o m i a l s o f t o t a l d e g r e e l e ss t h a n m s a t i s f y in g p i (u # ) = 1 , i = j a n dp i ( u j ) = 0 , i C j . A n d l et

MK I ( s , t ) = E . ~ ( s , t ) - ~ p , ( t ) E . ~ ( u i , s )i=1

M- ~ p j ( s ) E m ( t , ~ )j = lM M

+ ~ _ _ _ ~ p i ( t ) p j ( s ) E m ( u , , u S ) .i=l j~l

(19)

I t c a n b e s h o w n t h a t K I is p o s it i v e s e m i d e fi n it e a n d i s a r e p r o d u c i n g k e r n e l fo r H Ka n d f x h a s a r e p r e s e n t a t i o n ( W a h b a , 1 9 9 0 )

w h e r e

M nS~ = Z d~ o. + Z ~g,~,( t ) , (20),-'=1 i=l

K ~ ( . ) = K l ( t , . ) .T h e r e s u l t f ro m ( 2 0) c a n b e s h o w n t o b e t h e s a m e a s ( 1 8 ).

3 .2 B a y e s M o d e l B e h in d T h e T h i n - P l a t e S p l in eL e t u s n o w t a k e a l o o k a t t h e B a y e s e s t i m a t e s b e h i n d t h e t h i n - p l a t e s p li n e . I t i sk n o w n t h a t c e r t a i n B a y e s e s t i m a t e s a r e s o l u t i o n s t o v a r i a t i o n a l p r o b l e m s , a n d v i c ev e r sa . C o n s i d e r t h e r a n d o m e f fe c t m o d e l

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MF ( t ) = Z O . r + b l / 2 X ( t ) , t E [0, 11,

v=lY ~ = F ( t i ) + e~ , i = 1 , . . . , n .

( 2 1 )

Le t {Ca," 9 , CM} span H0, the space o f po lynom ia ls o f to ta l deg ree l es s th an m , andH 1 be a R K H S w i t h t h e r e p r o d u c i n g ke r n e l d e f in e d by

E X ( s ) X ( t ) = K a(s , t) ,w h e r e K l ( s , t ) g i ve n by ( 19 ). T h e n , t h e m o d e l i n ( 21 ) w il l r e s u l t in t h e t h i n - p l a t esp l ine . To unde rs tand th i s r e su l t , l e t

Mv ~; = u , - Z o ~ r

u=land se t f ( t i ) = b l / 2 X ( t i ) . Then (21) becomes

= f ( t ~ ) + e~ , i = 1 , . . . , n ,

w i t h e = ( e l , . ' . , e , ) ' ,,~ N ( 0 , a 2 I ) a n d

E y ( s ) y ( t ) = E [ b l / 2 X ( s ) b l / 2 X ( t ) ]= b E [ X ( s ) X ( t ) ]= b K 1 ( s , t ).

Then , f rom Wahba (2000) ,

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E(f(t~)f(t~)f(t~)

0 .2l Y) = K I ( K 1 + - ~ - I ) - l Y

0 .2A ( A )y , w i t h A = y .

(22)

A(A) i s know n as t he in f luence m a t r ix and w e wi l l u se t he re su l t l a t e r .Now, cons ide r t he va r ia t iona l p robl em in H1 , we wan t t o f ind fx t o min imize

i = 1

w h e r e I I f I 1 ~ 1 i s t h e s qu a r e d n o r m i n H 1. I t c a n be s h o w n t h a t

[ (t~)E ( f ( t ~ )

:(t .)I Y ) = K I ( K ' + A I ) - l Y

--- A( A) ~.

I n su m m a r y , g i ve n t h e p r i o r f ~ N ( 0 , bK1) , a ze ro -mean Ganss ian s tochas t i cp rocess wi th e -~ (e l , " " , on ) ' "~ N (0 , a2 I ) , t he pos t e r io r m ean fo r f g iven y i s t heso lu t ion to a va r ia t iona l p robl em in an RKHS.

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4 B i v a r i a t e R e g r e s s i o n

L e t u s n ow r e t u r n t o c r e d ib i li t y p r ob l e m s . R e c a l l t h a t , i n t h e B t i h l m a n n - S t r a n bm o d e l , w e w i s h t o u s e t h e c o n d i t i o n a l d i s t r i b u t i o n f r r+ l l o (YT+ l IO ) o r t h e h y p o t h e t i c a lm e a n E ( Y T + I I @ = 8 ) - - , U T + i ( O ) f o r e s t i m a t i o n o f n e x t y e a r ' s c l a i m s . S i n c e w e h a v eo b s e r v e d y , o n e s u g g e s t i o n is t o a p p r o x i m a t e # T + i ( 0) b y a l i n e a r f u n c t i o n o f t h e p a s td a t a . I t t u r n s o u t t h a t t h e r e s u l t i n g c r e d i b i li t y p r e m i u m f o r m u l a Z ~ " + ( 1 - Z ) # i so f t h i s fo r m . T h e i d e a is t o r e s t r i c t e s t i m a t o r s o f t h e f o r m ao + ~T = i a t Y t , w h e r eaO , a l , . . . , a T n e e d t o b e c h o s e n . W e w i l l c h o o se t h e a ' s t o m i n i m i z e s q u a r e e r r o rlo ss , t ha t i s , { " }= E I , ~ + 1 ( 0 ) - ~ o - , _ _ ~ , r , ] ~ 9W e d eno te th e r e su l t b y ~o , ~ i , . . ., ~T fo r th e v a lues o f ao, a l , . .. , a T w h i c h m i n i m i z eQ . T h e n t h e c r e d i b i li ty p r e m i u m c a n b e w r i t t e n a s:

T&o + ~ ~tYt .t = t

M e a n w h i l e , t h e r e s u l t i n g a o , a l , ..., a r a l so m i n i m i z e

Q 1 = E [ E ( Y T + , I Y = y ) - a0 - atYt ] 2t = l

a n d

q ~ = E Y r + l - a o - a ~ B ] 2 .t = l )

Hence , th e c r e d ib i l i t y p r em ium ~0 + ~-~T= l~ t Y t is t h e b e s t l i n e a r e s t i m a t o r o f e a c ho f t h e h y p o t h e t i c a l m e a n E ( Y T + i I O = 0 ) , t h e B a y e s i a n p r e m i u m E ( Y T + I I Y = Y),

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a n d Y T + I i n th e sense o f sq ua r e e r ro r lo ss .N o w , w e w a n t t o e x t e n d t h e s t a n d a r d c r e d i b i l i t y t e c h n i q u e s i n t o n o n p a r a m e t r i c

r e g r e ss i o n m o d e l s . I n t h e c r e d i b i l it y c o n t e x t , s u p p o s e w e w i s h t o d e t e r m i n e h o w t h ecu r r e n t c l a im lo ss , Y~j, d epe nds on the pas t lo sses , say Y~,j -1 an d Y~, j-2. O u r app roa chi s t o e s t a b l i s h a m o d e l a s t h e n o n p a r a m e t r i c r e g r e s s i o n y ~ j = g ( Y i , j - l ,Y i , j - 2 ) + e ~ ,i = 1 , . . . , n , w h e r e g is a s m o o t h f u n c t i o n o f i t s a r g u m e n t s . W h a t w e w a n t t o a c -c o m p l i s h i s t o m o d e l t h e d e p e n d e n c y b e t w e e n t h e c u r r e n t o b s e r v a t i o n s y ~ j f o r a l lpo l i cyho lde r s i = 1 , . . . , n , and those pas t lo sses y i , j - 1 and y i ,3 -2 th rough a non -p a r a m e t r i c r e g r e s s i on a t o c c a s i o n j . O n c e t h e m o d e l i s e s t a b l i s h e d , w e c a n p e r f o r mone-s t ep ahe ad p r ed i c t i on on Y~d+ l b y u s ing y/~ and Yid-1 a s cov a r i a t e s . Fo r n o ta -t i o n a l c o n v e n ie n c e , w e l e t y i s t a n d s f o r t h e d e p e n d e n t v a r i a b l e ( c u r r e n t o b s e r v a t i o n )and u se s~ o r t i fo r cov a r i a t e s ( pas t l o sses ). W e dev e lop a me th odo lo gy to e s t im a tet h e f u n c t i o n g, g i v e n t h e d a t a , f r o m a n o n p a r a m e t r i c r e g r e s s io n p e r sp e c t i v e . W e w i llb e u s i n g a B a y e s i a n a p p r o a c h t o f i t t h e p r o p o s e d m o d e l u s i n g a G a u s s i a n p r i o r o nt h e u n k n o w n f u n c t i o n g , w h i c h u s e s t h e r e p r o d u c i n g k e r n e l o f a t h i n - p l a t e s p l i n e a sthe cov a r i ance o f th e p r i o r d i s t r i b u t i on (W ahb a , 19 90 , p .3 0 ) .

4 . 1 M o d e l a n d P r i o rW ith ou t lo ss o f gene ra l i ty , we a ssume v a r i ab le s s i, t i l ie i n the i n t e rv a l [ 0 ,1 ] . Cons ide rt h e m o d e l f r o m t h e b i v a r i a t e r e g r e ss i o n m o d e l

y~ = g ( s i , t ~ ) + e l , i = 1 , . . . , n , (23)w h e r e g i s a s m o o t h r e g r e s s io n o f t h e v a r i a b le s s a n d t , a n d t h e e r r o r s e, a r e i n d e-p e n d e n t N ( 0 , a 2 ) . I t i s c o n v e n i e n t t o w r i t e ( 23 ) a s

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y i = a o q - a l S i q - ~ 2 t i q - f ( s i , t i ) q - e i , i = 1 , . . . , n ,wi th f having the ze ro in i ti a l cond i tions :

(24)

w h i c h m e a n s t h a t

f(0, 0) = o,~ ( 0 , 0 ) = o ,~(0 ,o ) =o ,

(2s)

s 0 = g ( o , o ) ,~ 1 = ~ s ( 0 , 0 ) ,~ 2 = ~ - ( 0 , 0 ) .

Mode l (24) has t he same fo rm as (21 ) wi th

r t , ) = 1 , r t , ) = ~ , , 0 3 ( ~ , , t , ) = t , .Now we can spec i fy t he p r io r on (24) .

T h e p r i or f o r f ( s , t ) i s t he re p roduc ing ke rne l for t he t h in -p l a t e sp l ine in (19). Th ism e a n s t h a t f ( s , t ) i s o f ze ro -m ean Gauss ian random va r iab l e s wi th t h e cova r iancef u n c t io n be t w e e n f ( s , , t i ) a n d f ( s ~ , t j ) g i ve n by

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c~ v( f ( s i , t i) , f ( s t , t j ) ) = r~ K {( s l , t l ) , (Sj , t t ) } .

U sing resul ts f rom (19) , the kernel K is g iven by

(26)

where

K{ (s i , t , ) , (s t , t j ) } = E{(s , , t , ) , ( s t , t t ) }3

- ~__.2k(s~, t j ) E { u k , (si , t , )}k = l

3- ~ p k ( s , , t , ) E { ( s l , t t ) , u k }k = l

3 3+ E ~ v k ( s, , t , ) p , ( s t , t t ) E { u k , u , } ,

k=l 1=1

(2T)

E { ( s , , t , ) , ( s j , t ~ ) } = r 2 log( r ) , r = ~ / ( s , - ~ j ) 2 + ( t , - t t ) 2 . ( 2 8 )

This is because tha t , wi th d = 2 and m = 2 , we have j 2 ( f ) given by (14). Obviously ,d / 2 + 1 is even in (16), so E ( r ) i s propor t iona l to r21og( r ) . No te tha t 0 ~ can beabso rbed in to ci in (18) . Fur thermore , le t

P l (s~, t i) = - 1 + 2si + 2t~, p 2 ( s i , t i ) - - - - - 1 - 2si, p 3 ( 8 i , t i ) = 1 - 2t~.By choosing

1 1 1 ( ~ , 0 ) ,~ , = ( 5 , 5 ) , ~ 2 = ( 0 , ~ 1 , ~ =

(29)

we have Px ,P2 ,P3 be th e po lynomia ls o f to ta l deg ree l e ss than 2 sa t is fy ing p i ( u j ) =

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1 , i = j and pi(uj) = 0 , i ~ j . T h e n w e a r e n o w r e a d y t o a p p l y t h e r a n d o m e ff ec tm o d e l i n ( 2 1 ) .

T o c o m p l e t e t h e p r i o r s p e c i f i c a t i o n f o r m o d e l ( 2 4 ) , w e t a k e u n i n f o r m a t i v e p r i o r sf o r a l l u n k n o w n p a r a m e t e r s ( W o o d e t. a l. , 2 0 0 0 ) . W e t a k e u n i f o r m i n d e p e n d e n tp r i o r o n [ 0 , 1 0 l ~ f o r t h e s m o o t h i n g p a r a m e t e r T2 . Th e p r i o r f o r c~ = ( a 0 , a l , a 2 ) ' i s

~ N ( 0 , c I ) ,

wi th c -- -* oc . Th e p r i o r f o r a 2 i s

p(a 2) ~x ( a 2 ) - 1 - 1 ~ e x p ( - 1 0 - 1 ~T h e r e s u l t in g B a y e s e s t i m a t e w i ll b e t h e s o l u t i o n t o t h e v a r i a t i o n a l p r o b l e m i n ( 12 )wi th d = 2 a nd m = 2 .

4 . 2 M o d e l I m p l e m e n t a t i o nI n t h i s s u b s e c t i o n , w e w i ll d i s c u s s t h e i m p l e m e n t a t i o n o f t h e m o d e l i n ( 2 4 ) . T o m a k et h i s m o d e l c o m p u t a t i o n a l l y f e a si b l e, w e w i l l c o n s i d e r a t r a n s f o r m e d m o d e l . A s i nW o o d e t . a l. ( 2 0 0 0 ), t h e s a m p l i n g s c h e m e r e q u i r e s f a c t o r i n g t h e c o v a r i a n c e m a t r i xK a s Q D Q 1, w h e r e Q i s a n o r t h o n o r m a l m a t r i x a n d D i s t h e d i a g o n a l m a t r i x w i t hd i a g o n a l e l e m e n t s , d i, t h a t a r e t h e e i g e n v a lu e s o f K .

T o e a s e t h e n o t a t i o n , w e r e w r i t e m o d e l i n ( 2 4 ) a s

y~=c~o+c~ls~+c~t~+f~+e~, i = 1 , . . . , n , ( 30 )whe r e f ~ = f(si, ti) , a n d f = ( f l , " ' , f ~ ) l i s G a u s s i a n w i t h z e r o - m e a n a n d t h ec o v a r i a n c e T2K . L e t

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= ( , ~ o , , ~ a , ,~=)',Y = ( Y l , ' " , Y . ) ' ,f = ( e l , ' " " , en) ,

(31)

a n d1 s l t l1 s2 t2Z = :1 Sn tn

t h e n w e c a n w r i t e ( 3 0 ) i n t h e m a t r i x f o r m

(32)

t h a t i s ,

Y lY2

Y .

1 81 t 11 s2 t2! . . . i1 s . t .

~ 0GI +

G2

f lA +

A

E1E2

e n

(33)

w i t h t h e p r i o r s ,y = Z a + f + e , (34)

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~~N ( 0 , ~ I ) ,f ~ N ( 0 , r 2 K ),~ N ( 0 , a 2 I ) ,

"c 2 ~ u n i f [ O , 101~a 2 ~ IG (10 -s , 10-1~

(35)

where 72 is un i fo rmly d i s t r ibu t ed in t he in t e rva l [0, 10 ~ and a 2 has a inve rse Gam m ad i s tr i bu t i on w i t h p a r a m e t e r s 1 0 - s a n d 1 0 - l ~ L e t u s r e t u r n t o t h e c o va r i an c e m a t r i xK . S i n c e K i s p o s i ti v e d e fi n it e , w e c an f a c t o r K a s Q D Q ~ s u c h t h a t

Q Q ' = I . ( 3 6 )We can p re -mu l t ip ly Q ' t o (34), so we have y* = Q~y . An d the m ode l becomes

y* = Z*c~ + i f+ e* , (37)w h e r e

Z * = Q ' Z ,f * = Q ' f ,E* : Qte .

(38)

Th e p r io rs fo r ~ , 72 , a 2 wi l l r em a in t he same a s in (35) . Meanw hi l e , e* has t he sam ed is t r ibu t ion a s c ,- , N (0 , aZ I ) because o f (36) in N(0 , a2 Q ' IQ ) . How eve r , t he p r io r

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f o r f* b ecom es

b e c a u s e o ff* ,, , N(0 , T2D) , (39)

V a r ( Q ' f ) = Q ' V a r ( f )Q .= Q , r 2 K Q= r 2 Q ' Q D Q ' Q= 7-2D,

whe re r ~ and D as de f ined b e fo r e .

4 . 3 B i v a r i a t e R e g r e s s i o n f o r t h e B i i h l m a n n - S t r a u b M o d e lC o n s i d e r d a t a i n B t l h l m a i m - S t r a u b M o d e l, i t a ll ow s d i ff e r e n t n u m b e r o f e x p o s u r eu n i t s o r d i f fe r e n t d i s t r i b u t i o n o f c l a i m s iz e a c r os s p a s t p o l i c y ye a r s. T h i s c a n b eh a n d l e d i n m o d e l ( 3 0 ) b y a s s u m i n g

G2e i ~ g ( o , - ~ ) , i = l , . . . , n, (40)w h e r e w ~ i s t h e c o r r e s p o n d i n g w e i g h t f or d a t a v a l u e yi . W e c a n a l so h a v e t h e s a m em a t r i x f o r m a s ( 3 4 ) ,

y = Z a + f + e , (41)b u t w i t h t h e p r i or s

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~ , , ,N ( 0 , c I ) ,f ,~N(O, T2K) ,e ~ N ( O , 0 " 2 w - l ) ,

r 2 ~ u n i f [ O , 101~a 2 ~ I G( 1 0 - 8 , 1 0 - 1 ~

(42)

Here W i s the d iagona l ma t r ix w i th d iagona l e lemen ts , w~, the co r re spond ing we igh tfor da ta va lue Yi .

Th i s mode l can be eas i ly t r ans fo rmed to a s imi la r mode l a s in (34) and (35 ) .Th en we can im p lem en t the m od i f ied mo de l ana logous ly a s in Sec t ion 4 .2 . Now le t

An i n t e r im m o d e l i s g ive n by

wi th the p r io r s

y ' = v / - W yZ ' = v " W Z ,f ' = v ' - W f ,E~ = v / W e .

y~= Z I (~+ f l+e ~,

(43)

(44)

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~~N ( 0 , c I ) ,f 'N N ( 0 , T 2 ~ K ~ ) ,c ' ~ N ( 0 , a 2 I ) ,

T 2 ~ u n i f [O , 101~a 2 ,, , I G( 10 -s , 10-1~

(45)

W e c an t he n s et K ' = ~ K ~ . T h is m e an s t ha t f ' ( s , t ) is o f z er o- m ea n G au ss ia nr a n d o m va r i ab l e s w i t h t h e c o va r ia n c e f u n c t io n be t w e e n f ' ( s i , t i ) a n d f ' ( s j , tj ) g i ve nby

c o v { f ' ( s , , t~), f ' ( s j , t j ) } = K '{ (s i , t~ ) , ( s j , t j ) } .Th is i s j u s t a d i f fe ren t cho ice o f t he p r io r fo r t he fu l l b iva r ia t e su r face f ' ( s , t ) . W ecan then use resu l t s f rom (30) to (39) based on (44) to (45) .

We use t he G ibbs sam p l ing scheme w here t he b iva r ia t e reg re ssion su r face i s mod-e l ed by the t h in -p l a t e sp l ine p r io r a s desc r ibed ea r l i e r in t he pape r . A good in t ro-duc t ion to t he Gibb s samp le r i s g iven by Ge l fand and Sm i th (1990) . One o f t headvan tages o f Gibbs sam p l ing is t ha t i t c an take a dvan tage o f any add i t ive s t ruc tu rein t he m ode l a s exp la ined in Wong and K ohn (1996) . Th e sam p l ing scheme is s imi l a rto t he one used by Wood e t . a l . (2000) in a m ode l se l ec t ion con t ex t . In ou r ca se ,the es t im ates of c~, f ' , ~-2 , and ~2 ar e o bta in ed by gen era t in g t he i te ra t io ns c~lJ l, f'{31,T2[j] and r f rom the sam pl ing schem e descr ibe d in (45). T he constant c is chosento be a l a rge numb er (c = 102~ so as t o ensu re t ha t t he p r io r fo r ~ i s e s sen t ia l l y anon in fo rm a t ive f i a t p r ior .

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4 . 4 O n e - S t e p A h e a d P r e d i c t i o nM a n y p r o b l e m s i n a c t u a r i a l s c ie n c e i n v ol v e t h e b u i l d i n g o f a m a t h e m a t i c a l m o d e l t h a tc a n b e u s e d t o p r e d i c t i n s u r a n c e c o s t s o r t o f o r e c a s t l os s es i n t h e f u t u r e , p a r t i c u l a r l yt h e s h o r t - t e r m f u t u r e . O u r a p p r o a c h is t o e s t a b l i s h a n o n p a r a m e t r i c r e g r e ss i o n m o d e lYi3 = g(Y~j-1, Y~,j-2) + e~, i = 1 , . .. , n , w here g i s a sm oo th fun ct io n of i t s a rgu m ents .T h i s m o d e l a l l o w s u s t o d e s c r i b e t h e d e p e n d e n c y b e t w e e n t h e c u r r e n t o b s e r v a t i o n sy~j for a l l po l icy holde rs i = 1 , . . . ,n , and tho se pa st losses Y~,j -1 an d Y~,~-2 th ro ug h an o n p a r a m e t r i c r e g r e s s i o n f u n c t i o n a t o c c a s i o n j .

S u p p o s e t h a t w e a r e i n t e r e s t e d i n o n e s t e p a h e a d p r e d i c t i o n o f Y ~ , j + I . W e t a k ethe pos t e r i o r m ea n E (Y~,j+I I Y) a s th e b e s t p r e d i c to r o f Y~ ,j+I and u se the pos t e r i o rvar(Y~j+l I Y ) t o o b t a i n t h e p o s t e r i o r p o in t w i s e p r e d i c t i o n in t e r v a l . F o r c o n v e n i e nc e ,w e e s t i m a t e t h e p o s t e r i o r m e a n a n d v a r i a n c e o f Y ~,~+I u s i n g e m p i r ic a l e s t i m a t e s b a s e do n t h e v a l u e s o f yz,~ +~ g e n e r a t e d d u r i n g t h e s a m p l i n g p e r i o d b y u s i n g t h e m o d e l i n(3 0 ) , t ha t i s ,

y~j = a o + a l y ~ , j - l + a 2 y ~ ,j - 2 + f ~+ e ~, i = 1 , . . . , n .To gen e ra t e Y i,3+ l, we p lug i n y~j and Y~,j-1 as c o v a r ia t e s . F o r e a c h i t e r a t io n , w i t h

t h e g e n e r a t e d v a l u e s o f a a n d f f r o m t h e s a m p l i n g s c he m e , w e h a v e

y ~ , j + l = a o + a l y i j + c ~ 2 y i , j _ l + f i + e i , i = 1 , . . . , n .T h e r e f o r e , t h e p r e d i c t i o n i s

A A A AY ij+ l ~ 0 + + + f ~ ,c~lyi,j c~2yi,j-1A

w h e r e f i i s t h e e x p e c t e d n o i s e o n yi g i v e n o b s e r v e d d a t a f r o m ( 2 2) . A f t e r a b u r n -

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in pe riod , i t i s a s sumed the i t e ra t ions have conve rged to d raw s f rom t he pos t e r io rd i s t r ibu t ion . We es t im a t e t he pos t e r io r mean and the pos t e r io r va r iance o f Yi j+abased on the va lues o f Y~,j+ I gene ra t ed du r ing the samp l ing pe r iod .

5 N o n p a r a m e t r i c R e g r e s s io n w i t h H i g h e r D i m e n -s i o n sSuppose t he m ode l i s now ex t ended to hand l e t h ree va r iab l e s . S imi l a rly , we cant rea t s~ and t~ as the pas t losses and incorpora te o ther re levant informat ion as v, .F o r e xa m p l e , v i c a n be t h e n u m be r o f y e a rs a p o l i c y h ol d e r r e m a i n i n t h e s a m e p o l i c ywi th t he same in su re r , o r r ep re sen t s d i f fe ren t d r iv ing age g roup in au to in su rance .Then , t he reg re ss ion m ode l i s g iven by

y i = g ( s i , t i , v i ) + e i , i = 1 , . - . , n ,w i th e i independe n t N( 0 , a 2 ) and w i th g ( .) o f t he fo rm

y ~ = c ~ o + c q s ~ + c ~ 2 t i + c ~ a v ~ + f ( s i , t i , v ~ ) + q , i = 1 , . . . , n .Th e prior on f is specif ied s im ilar ly to (26) and (27) , tha t is

c o v { f ( s i , t i , vi) , f ( s j , t ~ , v j ) } = r 2 K { ( si , t i , v , ) , ( s j , t j , vj )}w h e r e

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K { ( s i , t~ , v i ) , ( s j , t j , v~)} = E { (s~, t~, v~), (sj, t3 , v3)}4

k = l4

k = l4 4

k = l / =1

E { ( s , , t, , , , ,) , ( s j , t j , . j ) } = ~ l o g ( ~ ) ,

pl( si , t i , v i) = - 1 + 2si + 2ti + 2v~,p2(s~, ti, v d = 1 - 2si,p3(s~, t~, v~) = 1 - 2t~ ,pa(s~, ti, vi) = 1 - 2vi

a n d

1 1 1 1 1 1 (1 ,~ ,0 ) .~ 1 = ( 8 9 2 ' ~ ) , ~ 2 = ( 0 , 2 ' ~ ) , u 3 = ( p o , ~ ) , ~ 4 =Thi s i s bec aus e of (13) , whe re m = 2 , d = 3 , and

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--

= 4 .

Wi thou t lo s s o f gene ra l i ty , we a s sume tha t the va r i ab le s s , t , and v a l l l i e in thein terv a l [0,1].

L e t a = ( a o, a l , a 2 , a 3 ) ' b e t h e ve c t o r o f l i n e a r r e g re s si o n p a r a me t e r s , a n d l e t

1 s l t l v l ]/

1 s,~ tn vn jTh e p r io r s fo r ~ , th e sm oo th in g pa rame te r r 2 , and a 2 a re the sam e as in (35 ) wh ichgives us

~~N(0,cI),f N N ( 0 , T 2 K ),e N N ( 0 , a 2 I) ,

r 2 " u n i f [ O , 101~o-2 ~ IG (IO -s , 10-1~

T h e mo d e l i mp l e m e n t a t i o n a n d t h e s a mp l i n g s c h eme w i l l b e e x a c t l y th e s a m e a s i n

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t h e b i v a r i at e m o d e l .

6 A p p l i c a t i o n t o m e d i c a l i n s u r a n c e d a t aI n t h i s s e c t io n , t h e r e s u l t s o f B a y e s i a n n o n p a r a m e t r i c r e g r e s s io n m o d e l f o r t h e B t i h l m a n n -S t r a n b t y p e d a t a w i t h u n e q u a l e x p o s ur e u n i t s w i ll b e i l l u s t r a t e d b y a n a p p l ic a t io nt o a c o l l e c ti v e m e d i c a l h e a l t h i n s u r a n c e d a t a f r o m a n i n s u r a n c e c o m p a n y i n T a i w a n.W e c o n s i d e r a p o r t f o l i o c o n s is t i n g o f th i r t y - f iv e g r o u p p o l i c y h o l d e r s t h a t h a s b e e no b s e r v e d f o r a p e r i o d o f t h r e e y e a rs . T h e c l a i m a ss o c i a t e d w i t h g r o u p j ( = 1 , . . . , 3 5 )i n y e a r o f o b s e r v a t i o n t ( = 1 , 2 , 3 ) i s r e p r e s e n t e d b y t h e r a n d o m v a r i a b l e Y jt , w h i c h isa n a v e r a g e t a k e n o v e r w it e m p l oy e e . W e c h o o s e g r o u p s w i t h m o d e r a t e g r o u p s iz e ( 23t o 8 0 in d i v i d u a ls ) , a n d a s s u m e t h a t t h e n u m b e r o f e m p l o y e e d o e s n o t c h a n g e ov e rthe pe r iods . The r e fo r e , w e hav e w3t = wj f o r a ll t , an d the c l a im Y~t w i th we igh t s w jf ul fi ll t h e B ~ l h l m a n n - S t r a n b a s s u m p t i o n s . T a b l e 2 s h ow s s o m e o b s e r v e d r e a l i z a ti o n so f t h e Y j t, a n d t h e n u m b e r s o f e m p l o y e e w j . W e w a n t t o d e t e r m i n e t h e e s t i m a t e dp r e m i u m t o b e c h a r g e d t o e a c h g ro u p i n y e a r 4. T h e d a t a i s i n n e w T a i w a n d o l la r( N T D ) . T h e e x c h a n g e r a t e i s a b o u t 1 U S d o l l ar t o 3 5 .1 6 n e w T a i w a n d o l l a r i n M a r c h ,2002.

W e c o n s i d e r t h e s e m i p a r a m e t r i c r e g r e s s i on m o d e l d i s c u s se d i n s e c t i o n 4 .3 ,

yj, t . - :aO+alyj, t -1+c~2yj,t -2+f(yj, t -x ,y j , t -2)+e~, j = l , - . . , n , (4 6)w h e r e e~ ~ N ( 0 , ~ ) , i = I , . . . , n . T h e f u n c t io n f(Y~,t-l,Y~,t-2) has ze ro i n i t i a lc o n d i t i o n s a n d i s e s t i m a t e d n o n p a r a m e t r i c a l l y u s in g t h e p r i o r s i n ( 42 ) . T h e s c a t t e rp l o t , f i t t e d s u r f a c e b y ( 4 6 ) , a n d f i t t e d s u r f a c e b y t h e B f i h l m a n n - S t r a u b ( B S ) m o d e la r e s h o w n i n F i g u r e 1 . T h e c o n t o u r p l o t s s h o w n i n F i g u r e 2 p r o v i d e a b e t t e r l o o k o fd i f feren t leve ls o f sur faces .

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To ex am ine the pe r fo rm ance o f the r eg re s s ion func t ion , the ave rage squa red e r ro r(ASE) was ca lcu la ted fo r the e s t ima te s o f the r eg re s s ion func t ions . T he A SE i sca lcula te d as fo llows

nA S E = , 1 E w j ( ~ ( s j , t j ) - g ( s j , ~ j ) ) 2 . (47)~w~5=1j = l

T h e ASE o f t h e b i va r i a t e s p l in e n o n p a r a m e t r i c re g r e ss io n m o d a l i s a b o u t 2 14 .2 6,wh i le tha t o f the B tL la lmann-Straub m ode l i s 2208 .07 . Clea r ly , the b iva r i a te sp l inenon pa ram e t r i c r eg re ss ion mo de l ou tpe r fo rm s the B t~.hlmarm-Straub mode l .

O u r g o a l i s t o d e t e r mi n e t h e e s t i ma t e d p r e m i u m t o b e c h a r g ed t o e a c h g ro u p i nyea r 4 . We pe r fo rm one - s tep ahead p red ic t ion d i scussed in s ec t ion 4 .5 . We es t im a tet l~e pos te r io r m ean and va r i ance o f Y~,4 us ing emp i r i ca l e s t im a te s based on the va lueso f Y j,4 gene ra ted d u r in g the sam phn g pe r iod . F igu re 3 shows some o f the pos te r io rd i s t r ibu t i o n s o f Y j, 4 . So me o f t h e e s t i ma t e d p r e m i u ms t o g e t h e r w i t h 95 p e r c e n tpos te r io r po in tw ise p red ic t ion in te rva l s ( in pa ren thesi s ) a re show n in Table 3 . Fo rexample , fo r g roup 3 , the e s t ima ted p remium i s 5965 .36 NTD fo r each employee inth i s g roup , and the to ta l e s t im a ted p rem ium i s 49 x (5965 .36) = 292302.64 N TD .Simi la r ca lcu la tions can be done fo r o the r g roups.

7 C o n c l u s i o n sM a n y p r o b l e ms i n a c t u a r ia l s c ie n c e i n vol ve t h e bu i l d i n g m a t h e m a t i c a l mo d e l s t h a tcan .be used to p red ic t in su ran ce cos t in the fu tu re , p a r t i cu la r ly the sh o r t - t e rm fu tu re .A Bayes ian nonpa rame t r i c app roach i s p roposed to the p roblem o f r i sk mode l ing .Th e m ode l inco rpo ra te s pas t and p re sen t obse rva t ions r e la ted to the r isk , a s wel l a sr e l eva n t c o var i at e i n f o rma t i o n , a n d u s es MC M C t e c h n i qu e t o c o m p u t e t h e p r e d ic t ive

2 4 5


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d i s t r i b u t i o n o f t h e f u t u r e c l a i m s b a s e d o n t h e a v a i l ab l e d a t a , w h e r e t h e c o v a r i a n c es t r u c t u r e i s b a s e d o n a t h i n - p l a t e s p l i n e ( W a h b a , 1 9 9 0 ).

W e h a v e i l l u s t r a t e d a p p l i c a t i o n s o f G i b b s s a m p l i n g w i t h i n t h e c o n t e x t o f n o n -p a r a m e t r i c r e g r e s si o n a n d s m o o t h i n g . G i b b s s a m p l i n g p r o v i de s fe a s ib l e a p p r o a c ht o t h e c o m p u t a t i o n o f po s t e r io r d i st r ib u t i o ns . C o m b i n e d w i t h a s s u m e d t h i n - p l a tes p l in e s t r u c t u r e o f t h e r e g r e s s io n s ur f a c e a n d t h e c o m p u t a t i o n a l a v a i l a b il i t y o f t h eb i v a r i a t e o r t r i v a r i a t e s u r f a c e e s t i m a t i o n , t h i s m e t h o d o l o g y o p e n s u p a n e w d i m e n -s i on t o c r e d i b i l it y l i t e r a t u r e . A l t h o u g h o u r d i s c u ss i o n c o n c e n t r a t e s p r i m a r i l y o n tw oa n d t h r e e - d i m e n s i o n a l a p p l i c a t i o n s , t h e t e c h n i q u e c a n b e e a s i l y e x t e n d e d t o h i g h e rd i m e n s i o n a l p r o b l e m s . O u r i n v e s t i g a t i o n s h o w s t h a t t h i s m e t h o d p e r f o r m s a t a s u-p e r i o r le v e l c o m p a r e d t o t h e e x i s t i n g t e c h n i q u e s i n t h e c r e d i b i l it y l it e r a t u r e .

I n t h i s p a p e r , w e h a v e o u t l i n e d a n e w a p p r o a c h t o m o d e l i n g a c t u a r i a l a n d f i n a n c i a ld a t a . T h e m o d e l u s e s a B a y e s i a n n o n p a r a m e t r i c p r o c e d u r e i n a no v e l m a n n e r b yi n c o r p o r a t i n g a G a u s s i a n p r i o r o n f u n c t i o n s p ac e . W e b e l i e v e t h a t t h i s p r o c e d u r ep r o v i d e s a f l e x i b l e a p p r o a c h t o f u n c t i o n e s t i m a t i o n a n d c a n b e u s e d s u c c e s s f u l l y i nt h e s t a t i s t i c a l a n a l y s e s o f a w i d e r a n g e o f i m p o r t a n t p r o b l e m s .

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Po l icyho lde rA ve r a g e c l a i m 1No . in g roup

A ve r a g e c l a i m 2No . in g roup:

A ve r a g e c l a im 35No . in g roup

Y e a r l Y e a r 2 Y e a r 3 Y e a r 45419.09 1691.38 5984.65 ?

74 74 74 745603.50 4150.12 5797.48 ?52 52 52 52

: : :

4554.38 4646.96 5059.80 ?80 80 80 80

Tabl e 2 : Av e rage c l a ims in g roup po l i cyholde rs du r ing th ree yea rs .

Po l i cyho lde r Yea r 4Av erag e c la im 3 5965.36 (3080.30, 8877.31)No. in group 49

Av erag e c la im 5 5485.61 (1429.58, 9398.640)No. in group 45

Av erage c l a im 9 5024 .05 ( -59 5 .66 , 10495 .44)No . in g roup 42Av erag e c la im 14 5000.68 (2173.55, 7648.52)No . in g roup 31Av erag e c la im 20 6437.74 (2558.72, 7116.71)No. in group 36Av erage c l a im 24 5217 .51 ( -36 1 .08 , 10462 ,63)No. in group 35Av erage c la im 26 4959.50 (308.02, 9345.16)N o i n g r o u p 42Av erag e c la im 35 5074.70 (4620.35, 5497.48)No . in g roup 80

Tabl e 3 : Es t im a t ed ave rage c l a im for yea r 4 wi th 95 pe rcen t pos t e r io r po in twisepredic t ion in terva l in parenthesis .

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Figu re 1: Surface Plo ts . (a ) Sca t te r p lo t . (b) Plo t of regress ion surface as a funct io nof y t-1 an d yt-2. (c ) Plo t o f BS m odel surface.

2 4 8


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5000

40OO.j,,

3000

20002000 3000 4000 5000 2( }00 3000 4000 5000y(t-2) y(t-2)

F i g u r e 2 : C o n t o u r P l o t s. ( a ) P l o t o f r e g r es s io n f u n c t i o n a s a f u n c t i o n o f yt-1 andyt-2. ( b ) P l o t o f B S m o d e l .

2 4 9


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( a )

3O0

1043.40 422 66 7 74C9,94 1O59321y , 3 . 4(c)

4 0 0 [: ~ 1 7 6 ~ I L0 .

-5897.74 708,57 7314.89 1392 120y . 9 . 4

( b )6~1760"" ' 9 '-5595 38 6159 9 6827.37 130387 4y . s . ~

(d)400 f3 ~ 1 7 6 ~ I I L~~176 III1[

4217650 4736.253 6254835 577345 7y.35.4

Figure 3: Posterior Plots. (a) Y3,,. (b) Y5,4. (c) Y9,4. (d) Y35,4.

250


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References1 . Aronszajn , N . The ory of Reproducing Kernels . Transac t ions of the Am erican

M athe m at ical Society, 68:337-404, 1950.2 . Bt thlmann, H. Experience Rat ing anf Credibi l i ty. ASTIN Bul let in, 4:199-207,

1967.3. Duchon, J . R m ct ion s Spl ines et Vecteurs Aleatoires. Tech nical Re port 213,

Sem inai re d 'Analyse Num eriaue , 1975.4. Duchon, J . Fun ct ions-spl ine et Espera nces Condit ionnel les de Cham ps Gaus-

s ians, A nnal Science Univers i ty C lermont F erran d I I M athem at ics , 14 :19-27 .1976.

5 . Duchon, J . S pl ine M inimizing Ro tat ion-Invax iant Sem i-Norm s in Sobolev Spaces.in Construct ive Theory of Funct ions of Several Variables, Springer-Verlag,Berl in, 1977.

6. Evgeniou, T . , Pont i l , M. , and Poggio, T. Reg ularizat ion Networks and Su ppo rtVector Machines. Advances in Com puta t ional Ma them at ics, 13:1-50, 2000.

7 . Gel fand, A .E . an d S mi th , A .F .M. Sampl ing-based Approach es to Calcula t ingM arginal Densi t ies. Journal of the A merican S ta t i s t ica l Associa t ion , 85 :398-4O9, 1990.

8 . Meinguet , J . Mul t ivar ia t e In t erpola t ion a t Ar bi t ra ry Point s Made Simple . Jour-nal of Appl ied M athem at ica l Physics (Z AM P), 30 :292-304, 1979.

9 . Wahba, G . Spl ine Model s for Observ at ional Da ta . SIAM , Phi l adephia , 69 ,1990.

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10. Wahba, G . An Int rodu ct ion to M odel Bui ld ing wi th R eproducing Kernel Hi lber tSpace . Technica l Re por t NO. 1020, De par tm ent of S ta t i s tics , U nivers i ty ofWisconsin, Madison, WI, 2000.

11. Wahba, G . an d We ndelberger , J . Some New Mathe m at ica l Methods for Var ia-t ional O bjec t ive Analys is us ing Splines and Cross-Val ida t ion. M onthly WeatherReview, 108:1122-1145, 1980.

12. Weiner t , H .L (Edi tor ) . Reproducing Kernel Hi lbe r t Spaces: appl ica t ions instat is t ical signal processing. Hutchinson Ross Pub . Co. , 1982.

13. Wendelberger , J . Sm othlng Noisy D ata w i th M ul t id imen sional Splines andGenera l ied Cross Val ida t ion. Ph.D. Thes is . D epar tme nt of S ta t i s t ics , Uni-ver si ty of Wisconsin, M adison, WI, 1982.

14. W hi t t aker , E .T . On A New M ethod of Gradu at ion. Proceedings of EdinburghM athe m at ical Society, 41:63-75, 1923.

15. Wang, C . and K ohn, R . A Bayes ian Approach to Ad di t ive Sem iparamet r icRegression. Jou rnal of Econom etrics, 7:209-223, 1996.

16. Wo od, S. , Koh n, R . , Shively, T. , a nd Jiang, W. M odel Select ion in Spl ineN onp aram etric Regression. Pre print , 2000.

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