Bayesian prediction with an asymmetric criterion in a nonparametric model of insurance risk

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1 Saisics, Vol. 4, No. 4, Augus 26, Bayesian predicion wih an asymmeric crierion in a nonparameric model of insurance risk WOJCIECH NIEMIRO* Faculy of Mahemaics and Compuer Science, Nicolaus Copernicus Universiy, Toruń, Poland (Received 2 March 23; in final form 19 January 26) We consider a nonparameric Bayesian insurance risk model. The claims are seen as a marked poin process (T i,y i ), where T i is he ime of occurrence of he ih claim and Y i is is size. We assume ha his is a nonhomogeneous Poisson process on R 2 + wih inensiy measure P. Here P describes he exposure o risk and i is known, whereas is regarded as an unknown risk characerisic. According o he Bayesian paradigm, we assume ha he measure is random. Processes wih independen incremens are used as prior disribuions. In paricular, Gamma processes are conjugae priors. The problem is o predic he sum of fuure claims in a given period, given he pas of he process. We consider he asymmeric crierion LINEX (linear-exponenial) ha penalizes underesimaion of claims more severely han overesimaion. For he conjugae Gamma prior, we consruc he bes predicor. Under a relaxed assumpion on he prior disribuion, we consruc he bes linear predicor. Keywords: Bayes; Nonparameric; Poisson process; Cox Process; Credibiliy heory; LINEX 1. Inroducion We consider insurance claims seen as a random collecion of pairs (T i,y i ), where T i is he ime of occurrence of ih claim and Y i is is size. Thus we model he risk process as a marked random poin process on R + or, equivalenly, a poin process on R 2 +. Our objecive, as in he classical heory of credibiliy, is o predic fuure claims in order o calculae an appropriae premium. General references on credibiliy are Goovaers e al. [1 and Klugman [2, see also a survey in Encyclopedia of Acuarial Science [3. This heory, generally speaking, deals wih nonhomogeneous porfolios of risks and adops he Bayesian or empirical Bayes perspecive. I shows how o combine he knowledge abou he average risk characerisics in he whole porfolio, wih less cerain knowledge abou an individual conrac. The former is usually modeled as a prior disribuion, and he laer is given in he form of a random sample. In his aricle, we carry over he ideas of credibiliy o our nonparameric model. We assume ha he poin process of claims is a Cox or doubly sochasic Poisson process. The unknown risk characerisic is described by a finie measure on R +. The oal mass (R + ) is he * wniem@ma.uni.orun.pl Saisics ISSN prin/issn online 26 Taylor & Francis hp:// DOI: 1.18/

2 354 W. Niemiro expeced number of claims per uni of exposure, whereas he normalized measure is he probabiliy disribuion of a single claim size, P (Y i y) = (,y/ (R + ). Condiional on, he poin process (T i,y i ) is a nonhomogeneous Poisson process wih inensiy P, where he measure P describes he exposure o risk, possibly varying in ime bu assumed o be known. According o he Bayesian paradigm, is considered as random. In our seing, a prior probabiliy disribuion is defined on he se of finie measures on R +. Naural conjugae priors are given by nonhomogeneous Gamma processes. Our model equipped wih he Gamma prior is closely relaed o nonparameric Bayesian models based on he Dirichle process [4; 5, chapers 8 and 9. We also consider more general priors given by some class of processes wih independen incremens. For Cox models, Grandell [6 considers he problems of opimal predicion and opimal linear predicion. In he spiri of classical credibiliy, he derives predicors ha minimize he mean square error. Our nonparameric Bayesian model is similar bu explicily includes also he size componen of he process. In conras wih he resuls of Grandell, we choose an asymmeric crierion funcion known as LINEX (linear-exponenial). LINEX seems o be an appealing and well-moivaed alernaive o he classical quadraic crierion. I is relaed o he exponenial premium principle and penalizes underesimaion of claims more severely han overesimaion. For he conjugae Gamma prior, he simpliciy of poserior disribuions allows us o derive he bes LINEX predicor. For more general prior disribuions, we only derive he bes linear predicor. For comparison, we also provide formulas for he bes predicors wih respec o he quadraic crierion. 2. Mixed marked process of claims Consider he following insurance risk model. Claims occur a random poins in ime and heir sizes are also random variables. Denoe by T i and Y i he ime of occurrence and he size, respecively, of he ih claim. The pairs (T i,y i ) are seen as a poin process on R 2 + = (, ) (, ). The associaed random measure N is generaed by N((, u,(x,y) = I( < T i u, x < Y i y), (1) i= <<u<, <x<y<. Thus, N((, u,(x,y) is he number of claims which occur during (, u and have size in (x, y. The oal number of claims in (, u is N((, u, R + ) = i= I( < T i u). Similarly, he oal amoun of claims in he same period is S(,u= Y i I( < T i u) = u i= yn(ds, dy). (2) As in he classical Bayesian heory of credibiliy, we inroduce a random quaniy inerpreed as a parameer ha characerizes he risk under consideraion. In our model, is a finie random measure on R +, which describes he inensiy of occurrence of claims per uni of exposure. The probabiliy disribuion of a single claim size Y i is ( )/ (R + ). Le us also inroduce a nonrandom measure P on R +, which describes he exposure o risk. Thus, P(,u is he oal amoun of risk exposed during (, u. Assume ha P is known and, condiional on, N is a nonhomogeneous Poisson process on R 2 + wih inensiy measure P. For a definiion, consrucion and properies of nonhomogeneous Poisson processes, we refer he reader o

3 Kingman [5. In paricular, we have Nonparameric model of insurance risk 355 N((, u,(x,y) Poisson(P (, u (x, y), (3) hence E N((, u,(x,y) = P(,u (x, y and consequenly, E S(,u=P(,u y (dy). (4) Here and in he sequel, E ( ) denoes he condiional expecaion E( ). Le us now make suiable assumpions abou he prior disribuion of he risk parameer. Noe ha we can idenify he random measure wih he increasing sochasic process (,. ASSUMPTION 1 Suppose ha α is a finie measure on a bounded inerval (,y and λ>. Assume ha is a Gamma process wih shape measure α and inverse scale parameer λ, i.e. has independen incremens and (x, y Gamma(α(x, y, λ). The Gamma process inroduced above is a conjugae prior for he Poisson process. We will also consider anoher weaker assumpion. ASSUMPTION 2 Suppose ha α is a finie measure on a bounded inerval (,y. Assume ha is a process wih independen incremens such ha he momen generaing funcion of is incremens is of he form E exp [r (x,y = exp [ψ(r)α(x,y, where ψ is some funcion. Le r = sup{r: ψ(r) < }. Assume ha r > and ψ(r) as r r. Clearly, if Assumpion 1 is fulfilled, hen Assumpion 2 holds wih and r = λ. ψ(r) = log λ λ r (5) 3. Bes predicion wih respec o he LINEX crierion Our objecive is o predic he sum of claims in some inerval of ime (, u in he fuure, given he pas hisory of he process, i.e. pairs (T i,y i ) wih T i. Thus, a predicor of S(,u is a F N -measurable random variable, where F N = σ (N((v, s,(x,y): <v<s, <x< y< ). To formalize he problem of predicion, we have o choose a suiable crierion (loss funcion) L. The bes predicor is a soluion o he following minimizaion problem EL(S(, u H) = min, (6) over all H ha are F N -measurable. The operaor E sands for he uncondiional expecaion wih respec o he join probabiliy disribuion of and N.

4 356 W. Niemiro The classical square crierion funcion L(y) = y 2 is symmeric. For insurance applicaions, symmery may no be desirable: he penaly for underesimaion should no be he same as for overesimaion. We will use an asymmeric crierion funcion ha seems o be especially suiable for he kind of problems we consider. I is called LINEX and is given by L(y) = e κy κy 1, for some fixed κ> [7. LINEX is a funcion wih he minimum a y =. I behaves roughly as an exponenial funcion for y and as a linear funcion for y (figure 1). I penalizes underesimaion much more severely han overesimaion. An appealing feaure of LINEX is he following fac. The bes predicor defined by equaion (6) is given by H = 1 κ log E ( e κs(,u F N ). (7) Derivaion of his formula is sraighforward. I is enough o condiion on F N and o minimize he poserior expecaion wih respec o H. Le us menion ha equaion (7) can be inerpreed as he poserior premium calculaed according o he well-known exponenial principle. Derivaion of his principle is usually based on uiliy heory. Discussion of he properies of he exponenial principle and is relaions wih oher principles of premium calculaion can be found in refs. [1; 8, chaper 5. Noe ha he minimizer of he square crierion is E ( ) S(,u F N, i.e. he ne poserior premium. Under he conjugae prior, i is possible o derive an explici expression for he bes predicor (7). Figure 1. The graph of he LINEX funcion L(y) for κ = 1.

5 Nonparameric model of insurance risk 357 THEOREM 1 If Assumpion 1 holds, hen he bes predicor of S(,u wih respec o he LINEX crierion is given by H = 1 κ ( ) log 1 P(,u(eκy 1) [α(dy) + N((,, dy), λ + P(, provided ha y <(1/κ) log [1 + (λ + P(,)/P (, u. Recall ha y bounds he suppor of α from above. Le us rewrie he conclusion of he heorem in a slighly more explici form H = 1 κ 1 κ i:t i ( log 1 P(,u(eκy 1) λ + P(, ( log 1 P(,u(eκY i 1) λ + P(, ) α(dy) The bes predicor is linear in N. We say a predicor H is a linear funcional of he couning process N if i is of he form H = b + c(s, y)n(ds, dy), for some number b and funcion c(s, y). The following resul describes he bes linear predicor under a weaker assumpion on he prior disribuion. THEOREM 2 If Assumpion 2 holds, hen he bes linear predicor of S(,u wih respec o he LINEX crierion is given by a funcion c ha depends only on y, i.e. c(s, y) = c(y). For every y, c(y) is he unique soluion o he following equaion ). ψ (P (, u(e κy 1) + P(,(e κc(y) 1)) = e κc(y) ψ (). The inercep erm of he bes predicor is b = 1 ψ(p(,u(e κy 1) + P(,(e κc(y) 1))α(dy). κ The above formulas hold under he condiion ha y < 1 [ κ log 1 + (r + P(,). P(,u We should noe ha he predicors in Theorems 1 and 2 are finie only when he measure α and consequenly also he claim size disribuion are concenraed on a bounded inerval (,y. Moreover, he upper bound for y depends on he exposure and ends o as P(,u. Therefore, he LINEX crierion combined wih he Poisson/Gamma model has applicabiliy limied o insurance producs for which he claims are bounded random variables. This condiion is saisfied by some producs wih a finie sum insured. The boundedness condiion is also saisfied under an excess-loss reinsurance agreemen, whereby he ceding company pays he amoun bounded above by he reenion level. I is perhaps of some ineres o compare he LINEX predicors o he bes predicors wih respec o he classical square crierion L(y) = y 2.

6 358 W. Niemiro PROPOSITION 1 If Assumpion 1 holds, hen he bes predicor of S(,u wih respec o he square crierion is given by H = P(,u y [α(dy) + N((,, dy) λ + P(, = λ λ + P(, ES(,u+ P(,u λ + P(, S(,. PROPOSITION 2 If Assumpion 2 holds, hen he bes linear predicor of S(,u wih respec o he square crierion is given by he following funcion c and he inercep erm b c(s, y) = c(y) = Pu differenly, he bes linear predicor is H = ψ ()P (, uy ψ () + ψ ()P (,, (ψ ()) 2 b = P(,u ψ () + P(,ψ () yα(dy). ψ () ψ () + ψ ()P (, ES(,u+ ψ ()P (, u ψ () + ψ ()P (, S(,. Le us menion ha he predicors in Proposiions 1 and 2 are no only linear funcionals of he couning process N bu also linear funcions of he random variable S(,, in conras o he LINEX predicors. Le us also noe ha he LINEX crierion for small κ becomes close o he square crierion. I is indeed no hard o verify ha, for κ, he formulas in Theorems 1 and 2 reduce o hose in Proposiions 1 and 2, respecively. 4. Proofs We will repeaedly use he following lemma. LEMMA Suppose ha Assumpion 2 holds and g is a Borel measurable real funcion such ha g(y) (dy) < a.s. Then, E g(y) (dy) = ψ () in he sense ha he LHS exiss iff he RHS exiss. If (8) is finie, hen Var g(y) (dy) = ψ () g(y)α(dy) (8) g(y) 2 α(dy). (9) Moreover, [ [ E exp g(y) (dy) = exp ψ (g(y)) α(dy), (1) where boh sides of his equaion may be finie or boh infinie.

7 Nonparameric model of insurance risk 359 This Lemma is well known and raher obvious. Analogous formulas hold also for he process N. Recall ha N is, condiional on, a Poisson process. We hus have E exp[rn((, u,(x,y) = exp[p(,u (x, y(e r 1). I follows ha E Var [ E exp u u u g(s, y)n(ds, dy) = g(s, y)n(ds, dy) = g(s, y)n(ds, dy) = exp u u g(s, y)p(ds) (dy), (11) g(s, y) 2 P(ds) (dy), (12) [ u (e g(s,y) 1)P (ds) (dy). (13) Proof of Theorem 1 The prior disribuion of is described by Assumpion 1. Applying equaions (1) and (13) combined wih equaion (5), we obain [ Ee κs(,u = EE exp κ [ = E exp u yn(ds, dy) P(,u(e κy 1) (dy) [ λ = exp log λ P(,u(e κy 1) α(dy). Hence, 1 κ log EeκS(,u = 1 ( ) log 1 P(,u(eκy 1) α(dy). κ λ Le us menion ha he above formula can be inerpreed as he bes prior LINEX predicor. I is easily seen ha he poserior disribuion of is also a Gamma process. The prior parameers (α, λ) are replaced wih he poserior ones, (α( ) + N((,, ), λ + P(,). Consequenly, in view of equaion (7), he poserior LINEX predicor H is 1 κ log E(eκS(,u F N ) = 1 κ ( ) log 1 P(,u(eκy 1) [α(dy) + N((,, dy). λ + P(, Proof of Theorem 2 We are o minimize he following funcional ha depends on a real number b and a funcion c = c(, ) where S = u Q(b, c) = EL(S Z(c) b), y dn(ds, dy), Z(c) = c(s, y)n(ds, dy) (14) and L(y) = e κy κy 1. In order o compue Q(b, c), le us firs condiion on.wehave E L(S Z(c) b) = E exp [κ (S Z(c) b) E [κ (S Z(c) b) 1 = E exp[κs E exp[ κz(c)e κb κe S + κe Z(c) + κb 1, (15)

8 36 W. Niemiro because S and Z(c) are condiionally independen, given. From equaion (13), i follows ha [ E exp[κs=e exp κ yn(ds, dy) [ = exp (16) (e κy 1)P (ds) (dy) and [ E exp[ κz(c) =E exp κ [ = exp c(s, y)n(ds, dy) (e κc(s,y) 1)P (ds) (dy). (17) Similarly, from equaion (11), i follows ha and E S = E E Z(c) = E yn(ds, dy) = c(s, y)n(ds, dy) = yp(ds) (dy) (18) c(s, y)p (ds) (dy). (19) We are now in a posiion o prove ha he opimum funcion c depends only on y and no on s. Indeed, le us define c(s, y)p (ds) c(y) =. (2) P(, In view of equaion (17), by Jensen s inequaliy, E exp[ κz(c) E exp[ κz( c). I is clear ha E Z(c) = E Z( c). Consequenly, E L(S Z(c) b) E L(S Z( c) b) and herefore Q(b, c) Q(b, c). Hence, i is sufficien o minimize Q(b, c) under he assumpion ha c(s, y) = c(y) = c(y). This will allow us o wrie equaion (15) in a simpler form. Le [ E (c) = exp (p 1 (e κy 1) + p (e κc(y) 1)) (dy) (21) and where R (c) = κ p = P(,, (p 1 y p c(y)) (dy), (22) p 1 = P(,u. Combining equaions (16) (19), we can rewrie equaion (15) as follows: E L(S Z(c) b) = E (c)e κb R (c) + κb 1. Our nex sep is o compue Q(b, c) = EE L(S Z(c) b). Le us wrie E(c) = EE (c) and R(c) = ER (c). We proceed in much he same way as before. Combining equaion (21)

9 wih equaion (1), we obain Nonparameric model of insurance risk 361 [ E(c) = exp ψ(p 1 (e κy 1) + p (e κc(y) 1) ) α(dy). (23) Similarly, combining equaion (22) wih equaion (8), we obain Clearly, R(c) = κψ () (p 1 y p c(y))α(dy). (24) Q(b, c) = EE L(S Z(c) b) = E(c)e κb R(c) + κb 1. We are now going o exploi he fac ha E(c) and R(c) are expressions ha depend only on c and no on b. For a given funcion c, i is easy o minimize Q(b, c) wih respec o b. Funcion Q(,c)is convex and is unique minimum can be found by equaing he derivaive o zero b Q(b, c) = κe(c)e κb + κ =, hence E(c) = e κb. Therefore, subsiuing he opimum value b = (1/κ) log E(c) ino Q(b, c), we obain Q(c) = min Q(b, c) = R(c) + log E(c). (25) b Subsiuing equaions (23) and (24) ino equaion (25), we obain [ψ(p1 Q(c) = (e κy 1) + p (e κc(y) 1)) κψ ()(p 1 y + p c(y)) α(dy). To minimize Q(c), i is sufficien o minimize he expression under he inegral wih respec o c(y) separaely for every y. Slighly modifying he noaion, le us fix y and consider he following funcion of a real variable c K(c) = ψ(p 1 (e κy 1) + p (e κc 1)) κψ ()(p 1 y + p c). From he Lèvy Khinchine represenaion formula, we know ha ψ is a convex funcion. As K (c) = d dc K(c) = κp [ ψ (p 1 (e κy 1) + p (e κc 1))e κc + ψ (), we infer ha he derivaive K is increasing, hus K is convex. I can easily be verified ha K (c) κp ψ () >for c.ifp 1 (e κy 1) <r, hen K () <, oherwise K (c) as c c for some c. Consequenly, he equaion K (c) = has a unique posiive soluion. Thus, we have derived he assered equaion for c, and we have shown ha i uniquely defines c. The formula for b follows immediaely from equaion (23) because b = (1/κ) log E(c). The proof is complee. Proof of Proposiion 1 The bes predicor wih respec o he square crierion is E(S(, u F N ). The compuaion is a sraighforward applicaion of equaions (8) and (11) o he poserior disribuion of. We omi easy deails.

10 362 W. Niemiro Proof of Proposiion 2 The proof is quie similar o ha of Theorem 2 bu wih equaions (9) and (12) aking over he roles played by equaions (1) and (13). We are o minimize Q(b, c) = E(S Z(c) b) 2, where S and Z(c) are defined in equaion (14). I is clear ha minimum is aained for and b = E(S Z(c)) Q(c) = min Q(b, c) = Var (S Z(c)) = Var E (S Z(c)) + E Var (S Z(c)). b By condiional independence and equaion (12), Var (S Z(c)) = Var S + Var Z(c) = u y 2 P(ds) (dy) + c(s, y) 2 P(ds) (dy). The condiional expecaions E S and E Z(c) are given by equaions (18) and (19). Le us define c(y) as in he proof of Theorem 2 by equaion (2). I is easily seen ha E Z(c) = E Z( c) and Var Z(c) Var Z( c). Consequenly, wihou loss of generaliy, we can assume ha c(s, y) = c(y) = c(y). Now we can wrie E (S Z(c)) = Var (S Z(c)) = [p 1 y p c(y) (dy), [ p1 y 2 + p c(y) 2 (dy), (26) where p = P(, and p 1 = P(,u. Applicaion of equaions (8) and (9) yields E Var (S Z(c)) = ψ () Var E (S Z(c)) = ψ () [ p1 y 2 + p c(y) 2 α(dy), [p 1 y p c(y) 2 α(dy). (27) Therefore, Q(c) = { ψ () [p 1 y p c(y) 2 + ψ () [ p 1 y 2 + p c(y) 2} α(dy). The minimum of Q(c) obains if, for every y, we minimize he funcion under he inegral wih respec o c(y). An easy compuaion gives he formula for c assered in he heorem. The formula for b follows rivially and he proof is complee. Noe ha in he proofs of Proposiions 1 and 2, we have no used he assumpion ha y < bounds he suppor of α.

11 Nonparameric model of insurance risk Concluding remarks The choice of he parameers of he prior disribuion is cerainly imporan for applicaions. This problem is no discussed in his aricle. A raher sraighforward par of modeling is o choose he prior inensiy measure E which is equal o α/λ under Assumpion 1. Is naural (empirical Bayes) esimae can be based on he empirical measure ha couns pas loss occurrences in he whole porfolio of risks. On he oher hand, choosing a suiable parameer λ is much more difficul. Acknowledgemen The auhor is graeful o he referee who suggesed ha he ime componen of he process can be inroduced in he model explicily. References [1 Goovaers, M.J., Kaas, R., van Heervaarden, A.E. and Bauwelinckx, T., 199, Effecive Acuarial Mehods (Norh Holland). [2 Klugman, S.A., 1992, Bayesian Saisics in Acuarial Science, Wih Emphasis on Credibiliy (Kluwer Academic Publishers). [3 Teugels, J.L. and Sund, B. (Eds), 24, Encyclopedia of Acuarial Science (Wiley). [4 Ferguson, T.S., 1973, A Bayesian analysis of some nonparameric problems. Annals of Saisics, 1, [5 Kingman, J.F.C., 1993, Poisson Processes (Clarendon Press). [6 Grandell, J., 1975, Doubly sochasic poisson processes. Insiue of Acuarial Mahemaics and Mahemaical Saisics, Universiy of Sockholm. [7 Zellner, A., 1986, Bayesian esimaion and predicion using asymmeric loss funcions. Journal of he American Saisical Associaion, 81(394), [8 Gerber, H.U., 1979, An Inroducion o Mahemaical Risk Theory (S.S. Huebner Foundaion for Insurance Educaion).

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