Astm Bulletin 12 (1981) RECURSIVE EVALUATION OF A FAMILY OF COMPOUNI) DISTRIBUTIONS* HARRY H. PANJER of Waterloo, Ontario, Canada


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1 Astm Bulletin 12 (1981) RECURSIVE EVALUATION OF A FAMILY OF COMPOUNI) DISTRIBUTIONS* Umvelmty HARRY H. PANJER f Waterl, Ontari, Canada 1 INTRODUCTION Cmpund dlstributmns such as the cmpund Pmssn and the cmpund negative binmial are used extensively m the thery f risk t mdel the distributmn ff the ttal claims incurred m a fixed perid f time The usual methd f evaluating the dlqtributmn functmn requires the cmputatmn f many cnvlutins f the cnditinal d~atnbutmn f the amunt f a claim given that a clmm has ccurred When the expected number f claims is large, the cmputatmn can becme unwmldy even with mdern large scale electrnic cmputers In tlus paper, a recurs xe definitmn f the distributin f ttal clmms is develped fr a family f claml numbel distnbutmns and arbitrary claim amunt distributins When the clam1 amunt is discrete, the recursive dehnitmn can be used t cmpute the distributin f ttal claims withut the use f cnvlutins. This can reduce the number f required cmputatins by several rders f magnitude fr sufhcmntlv large prtflis Results fr sme spemfic dlatnbutmna have been prevmusly btained using generating functins and Laplace transfrms (see PANJER (1980) including dlscussmn). The simple algebraic prf f this paper yields all the previus results as special case~ 2. THE FAMILY OF CLAIM NUMBER DISTRIBUTIONS Cnsider the family f claim nunlber dlstribunns satisfying the recursmn (1) Pn = p,~t(a+ b/lz), n = 1, 2, 3... where p. dentes the prbabihty that exactly n clmms ccur in the fixed time interval. Members f flus family are a) Pissn chstrlbutmn' t" xkn, 1/. = O, 1 2, 1. Pn  i/i,.. * Tile authr is grateful t the leferec Ir pinting ut an errr m the riginal draft This research was supprted by the Natural Scmnces and Engineering Research Cunml f Canada
2 COMPOUND DISTI~.I BUTIONS p,/p,~_l = X/n, p = e ~' 3 a=, b=x b) BinlmaI distributin 1. p. = (.N) p" (l p)u",,~ =, 1, 2... N p,=, n=n+i,n P,dp,,1 = (1Vn+ I)p/(u(1p)), p = (lp)^' 3. a = p/(tp), b = (N+ l)p/op) c) Negative bmmml h~trlbtmn 1 p,, = (~'",~')p"(lp)%,~ =, l, 2, 2 p,dp,,1 = (~.+~lt)p/n, p = (tp)~, 3,7=p, b=(~~)p d) Gemetllc dlstnbutmn (Negative binmml with c~= l): t. p,, = (lp) p", n =, 1,2,.. 2. P,dP,,~ = P, p = I  p 3. a=p, b= SUNm and JEWELL (19Sl) shx~ that these are the nly members f this family. 3 THE RECURSIVE FORMULA CnsKler tile cmpund distributin with distributin functin (2) G(x) = ~Z p,~f*"(.a), x > ni = p0, V = O fr arbitrary claun amunt distributin F(~), x >. Fr ntatinal cnvenience, assume that F(x). x >, is f the cntlnuu~ frm Crrespnding results will be given fr tile discrete ca,it. Then thc density f ttal claims is (3) g(x) = ~ pr~f*"(x), x > ni = p0, 3.. = 0 when f(x) is the dcnslty assclated with F(x) In the thercm whmh fllws, the fllwing tw relatmns will be usecl' (I) ff(y)f,n(x_y)dy =f*(n+'l(x),,t = 1,2,3... 0
3 24 PANJ ER (II) f yf(y)f*'~(xy)[f*~n+~) (x) dy = x/q,+ I), n = I, 2, 3,.. Relatin (I) as the usual recurslve definitin f cnvlutirls The left side f relatin (II) is the cnchtlnal mean f any element f a sum cnsisting f n + 1 independeiit and identically distributed elements, given that the sum is exactly x The mean is x/(n + 11 as a result f the symmetry in the elements f the SUln. Relatin (II) is used lla Buhhnann and Gerber's discussmn t PAN JEll (1980) t develp an alternate prf f the result described m that paper. Therem Fr p. and g(x) defined by (l) and (3) iespcctwely, and f(x) any distributin f the cntinuus type fr x >, the fllwmg recursln hlds. (4) g(x) = p,f(x) + i (a + by~x)f(y) g(x y) dy, x >. Prf Substituting (3) int the right side f (4) results in x p~f(x) + f (a + by/x) f(y) g(x  y) dy = p,f(x) + J" (a + by~x) f(y) ni pnf*n(x  y) dy = p~f(x) + Z Pn f (a+by/x) f(y)f*n(xy)dy n i tl = p~f(x) + Y, ni p. {a + b/(n+ I)}f*(n+l) (x) (frm (I) aim (II)) = ptf(x) + Z p,, tf*(n+l)(x) (frm (1)) n i = p~f(x) + ~2 p.f*n(x) = ~ p,,f*~(x) nt (sincef*l(x) = f(x)) = g(x), the reqmred result. Q.E.D. If the claim ainunt distributin is discrete and defined n the psitwe 111 tegers, the crrespnding recursive deflmtln f ttal claims is (5) gt = E (a+t;j/7)fjg~_j, ~ = ~,2,3...
4 with g = p" whereas the usual frm is COMPOUND DISTRIBUTIONS 2 5 (6) g~ "" = = p.f,, i, 1,2,3... m The number f cmputatins required t btain gl is f rder,2 fr frmula (6) and f rder i fr frmula (5). Hence, fr large values f ~, the reductin in cmputatins is dramatic. 4' RESULTS FOR SPECIAL CASES The recursive definitins f the density f ttal claims fr the fur distributins cnsidered in Sectin 2 are given belw. a) P~ssn Cntinuus Mdel g(x) Discrete Mdel gt xev(x) + x yf(y) g(x  3') dy = jfig,_j ti b) Binmial P 1p [N('P)^~f(x) + f {(N + I)y/x I}f(y) g(xy) dy] c) Negative binmial z [ p ~(lp)~f(x) + {1 + ( 1)y/x}f(y)g(xy)dy] P ~ {(N + 1) Jl~  1 } fjg~_j 1 P II d) Gemetric P Z JI {1 + (~ 1) j/l}fjgtj p[(]p)f(x) + f f(y) g(x y) dy] p ~ fjg~_~ )I Tile recursin fr tile cmpund Pissn distributin with discrete claim amunt dzstributln waq riginally given by ADELSON (1966) in an inventry prblem. He used generating functins t btain his result,
5 26 PANJER SUNDT and JEWELL (1981) generahze the results f the present paper. They btain results fr a mre general recursin than (I) and btain results fr the case f pssibly negative claims. REFI~IIENCES ADELSON, R ~'[ (1966) Cmpund Plssn Dlstributlns, Operatzns Research Quarlerly, 17, PANJER, H H. (198) Tile Aggregate Clanns ])lstrlbutln and StpLss Reinsurance, Trans f tile Sctety f Actuarzes, XXX[I. SUNDT, B and JEWnLL, W S (~98~) Further Results n Recurslve Evaluatin f Cmpund Distributins Astm Bulletin 12
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