COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES * Yogo Purwono

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1 COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES * Yogo Purwono Department of Management Faculty of Economics, University of Inonesia Depok, INDONESIA Abstract. Copula moels are becoming increasingly popular tool for moeling epenencies between ranom variables, especially in such fiels as biostatistics, actuarial science, an finance. The purpose of the present paper is to provie preliminary iscussion about the project aime to investigate moels an inference methos for multiple lives base insurance ata by means of copula. In the life contingency moels for several iniviuals, the life time of insure iniviuals have been assume to be mutually inepenent. This means that for combination of lives, the life time probability of one live an life time of another lives o not irectly impact each other. Inepenence assumption in multiple lives contigency moels is often consiere becauce this is more mathematically tractable to compute straightforwarly actuarial present value or benefit premiums an reserves. However, this may be unrealistic because intutitively, life times of several associate iniviuals, such as marrie couples, can exhibit epenence because of such conitions as common isaster, common life style, or the broken heart synrome. Using actual ata on mortality of spouses that hol a last survivor annuity policies in Inonesia, this research project will apply the basic concepts of copula inference in empirically investigating the presence of epenence in multiple lives base insurance contract. There is a growing number of papers that explore the issue of epenencies on joint life times contracts, but no paper that has provie a etail lists of inference methos in copula moeling for multiple lives theory. Key-wors: copula function, multiple lives moel, joint survival function, conitional copula, IFM metho, Bayesian estimation. Introuction One important issue in actuarial practices is to moel, in a much efficient way, the epenence of ranom life time of several iniviuals. Developing an fining the goo moel for this problem become absolutely necessary in insurance. The theory an the markets urge the economic agents to combine iniviual components. Insurance or annuities proucts covering several lives, like last survivor annuities for a marrie couple for instance are increasingly emane an have to be fairly price. Similarly, finance instruments base on more than one asset, like multi name creit risk bearing erivatives, are wiely use an have become stanar. As part of a whole, ranom times to event behaviour are selom inepenent an can even have rather intricate epenence structures. Therefore, flexible multivariate moels an goo inference methos are calle for. * Submitte to the 3 th East Asian Actuarial Conference

2 Y PURWONO Formerly, in finance (mainly) an in insurance (to some extent) there were two main approaches to the epenence moelling question: inepenence or multivariate normality. Either the several risks were amitte to be inepenent like in a portfolio of insurance policies, or multivariate normality was assume as in the classical analysis of financial time series. In each case, risk aggregation is straightforwar but often too far from proviing realistic moels. Similarly, when we are ealing with insurance policies insuring a group of lives, the premiums or reserve that we calculate uner life time inepenencies assumption may be too far from the fair one. A multivariate moel has two components: the univariate (marginal) one, which characterizes each of the variables an the epenence structure between these marginal variables. This obvious separation can be specifie in mathematical terms an it is exploite for moelling in this propose stuy. The epenence structure of the ranom variables is known as the copula. Through the combination of copula with specific univariate istributions it is possible to construct an infinite number of multivariate istributions which we will refer to as copula base moels. The notion of a function characterizing the epenence structure between several ranom variables comes from the work of Hoeffing in the early forties. Inepenently, other authors introuce relate notions after-wars but it was in 959 that Sklar [36] efine an name as copula a functional that gives the multivariate istribution as a function of the univariate marginal istributions. Allowing for the stuy of the epenence structure apart from the univariate behaviour of each variable, copula became very useful. The literature kept growing as the interest in copula increase, for instance in environmental ata moelling. In the late nineties this notion was brought into finance an insurance problems. In 997, Wang [39] proposes copula for moelling aggregate loss istributions of correlate insurance policies. Frees an Valez [0] in 998 an Klugman an Parsa [22] in 999 use copula to moel bivariate insurance claim ata. Embrechts et al. [7] give a comprehensive overview of the application of copula to finance in a preprint alreay available on-line in 999. By now, the theory of copula is well establishe. On the other han, the literature on inference questions is still scarce. References like Genest et al. [2], Genest an Rivest [3], Joe [20] an Oakes [3] are important in this context. We can fin copula moels fitte to multiple lives base insurance in Anerson et al. [], Frees et al. [9], Youn an Shemyakin [40], an Youn et al. [4]. Our interest lies on parsimonious multivariate copula moels of joint survival analysis, able to capture the relevant facts of insurance ata an on the inference methos for those moels. Such fitte moels are essential to accurately estimate functionals of epenent life times, pricing insurance contracts base on more than one live an evaluate risk measures. The primary purpose of this research project is to investigate copula moels together with the proper inference methos for a broa context of multiple lives analysis an well on one particular problem of two associate lives. Straightforwar copula analysis base on mixing marginal istribution functions as suggeste in [] an [9] might work succesfully or not work at all epening on particular structure of association between two lives. Section 2 iscusses the basic efinitions, properties, an examples of several types of copula functions. Here we will iscuss in etail the types of copulas, which foun their applications, for instance, in the analysis of extremes in financial asset 2

3 COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES returns, an also the types of copulas, which seem to better fit the problems of life time analysis. Section 3 is eicate to the formulation of the range of multiple lives analysis problems, which we are going to consier. A special attention is pai to the effects of left truncation an right censoring on ata. Joint first life an joint last survivor functions, an their relationship to the bivariate survival function are iscusse. In section 4 we explain approaches we will consier in our project, which allows for the estimation of joint first life an joint last survivor functions with the help of joint bivariate survival function. We briefly iscuss the estimation methos we use to empirically estimate an infer copula moels for general multivariate survival functions. We also explain the properties of these estimators in brief. In section 5 we will iscuss the conitional bayesian copula moel suggeste first in [2]. Its construction is base on the application of copula mixing to conitional rather than to marginal survival functions. It allows for an explicit utilization of the prior information available on the conitional survival functions. The iscussion of why this approach shoul be sufficient for the estimation of the first life context, but may fail for the last survivor context, will be briefly given. In section 6 a numerical ata example for the application of copula analysis, which was previously iscusse in section 2, is explaine. It will be iscusse copula moels that will be consiere to analyse the ata. In the en part of this section, a potential application of join survival analysis to actuarial science is iscusse an some suggestions for that issue are given. Section 7 conclues. The results of this research can have far reaching implications to the practicing actuary who may be concerne about the financial consequences of assuming inepenent life length of a pair lives when in fact there may be forces riving epenencies. The actuary can be equippe with better methos to make more informe ecisions. Whilts the literature on copula is evote either to probabilistic theory, to inference methos or to applications, moreover, this research will combine of these three moeling aspects. Therefore, the results of this research will contribute consierably to the scientifics relate literatures an to making the new evelope tools known to a wier auience of practitioners an researchers alike. 2 Moelling Depenence with Copulas Suppose that a imensional ranom vector X = (X,..., X) has the istribution function F(x,..., x) = Pr(X x,..., X x). One can ecompose F into the univariate marginals of Xk for k =, 2,..., an another istribution function calle a copula. Refering to Joe in [20], a function F with support R an range [0, ] is a multivariate istribution function if it satisfies the following: i) It is right continuous; ii) iii) lim x k x k lim, k F(x,..., x) = 0, for k =, 2,..., ; F(x,..., x) = ; an 3

4 Y PURWONO iv) For all (a,..., a) an (b,..., b) with ak bk for k =, 2,...,, we have 2 2 i i... ( ) Fx ( i,..., x i ) i = i = 0, where xk = ak an xk2 = bk. A copula is a multivariate istribution function with stanar uniform (0, ) margins. An equivalent efinition that provies some copula properties follows. Supoose u = ( u,..., u) belong to the -cube [0, ]. A copula, C(u), is a function C: [0, ] [0, ] satisfying the conitions: i) For all (u,..., u) in [0, ], if at least one component ui is zero, C(u,..., u) = 0; ii) For ui [0, ], C(,...,, ui,,..., ) = ui for all i {, 2,..., }; iii) For all [u,u2] x... x [u, u2] imensioanl rectangles in [0, ], 2 2 i i... ( ) Cu ( i,..., ui ) i = i = 0. The ability of a copula to separate the epenence structure from the marginal behaviour in a multivariate istribution comes from Sklar s theorem in [36]. Accoring to Sklar s theorem, when we have a imensional istribution function F with univariate margins F, F2,..., F an the ranges of Fi are Ri, for i =, 2,...,, then there exists a unique function H, efine on R x R2 x... x R such that F(x,..., x) = H(F(x),..., F(x)). The extension of the function H to [0, ] is a copula C. For C such an extension of H, we have that F(x,..., x) = C(F(x),..., F(x)). The verifying of the theorem can be seen in [20, page 4]. If F,..., F in Sklar s theorem are continuous, the function H coincies with the copula C which is then unique. Besies the ability of isolating the epenence structure of a multivariate istributions, the copula provies a way of constructing istributions from given margins. Sklar then state that given univariate istribution functions F,..., F an a imensional copula C, the function efine by F(x,..., x) = C(F(x),..., F(x)) (2.) is a imensional istribution function with univariate margins F,..., F. For the proof, see [20, page 4]. So we are able to write the joint istribution as a function of the copula an the univariate marginal istributions. As the copula oes not epen on the univariate marginal istributions, we can say that it capture completely all the information about the epenence between the variables. For this exposition, from here on we will refer to a copula an a epenence structure inifferently. We will refer to a statistical moel given by a multivariate istribution written like (2.) as a copula base moel. 4

5 COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES Some important concepts in the construction of special types of copula base statistical moels are quantile functions, probability integral an quantile transformations. Suppose that ranom variables X, X,..., X have istribution functions F, F,..., F respectively. i) The quantile function of F is efine for all u in (0, ) by the generalise inverse of F: F (u) = inf {x R : F(x) u}. ii) The probability integral transformation is the mapping T : R [0, ], (x,..., x) (F(x),..., F(x)). iii) The quantile transformation is the operation T : [0, ] R, an (u,..., u) (F (u),..., F (u)). F enotes the usual inverse function of F an is a particular case of F when F is continuous an strictly increasing. It is well known that, if U is a uniform ranom variable on (0, ), then F (U) has istribution function F. On the other han, the istribution function F of a ranom variable X is continuous if an only if F(x) is uniformly istribute on (0, ). Suppose that Fi is continuous an write Fi(xi) = ui for i =,...,. Making these substitution in (2.), we obtain F ( u ),..., F ( u ) C(u,..., u) = F( ) (2.2) for (u,..., u) [0, ], which is an explicit expression for the copula as a function of the joint an the univariate marginal istribution functions. Consier one example as follows. Suppose that we have the bivariate logistic istribution function given by θ θ / θ F(x, y) = exp ( +, x > 0, y > 0, θ. ( x y ) ) We can rewrite this function in a more convenient form as follows: F(x, y) = exp / / θ x θ y θ (2.3). (( loge ) ( loge ) ) / + The margins of a bivariate logistic are stanar Fréchet. It means that they have istribution function F(z) = e /z for z > 0. Therefore, one can reaily obtain the copula function C of a pair of ranom variables with bivariate logistic istribution from (2.3) using (2.2), leaing to θ ( ( ) ) / θ 2 θ C(u, u2) = exp ( logu ) ( logu ) where θ an (u, u2) [0, ] 2. +, (2.4) The family of copula function, like in the previous example, is known as Gumbel, Gumbel Hougar or logistic copula. This copula family has the characteristics that are frequently suitable for moelling financial or insurance ata. Copula has a very interesting feature, that is invariance property. The formal explanation of invariance property is as follows. Suppose that C be the copula of the ranom vector X = (X,..., X) an Ri the range of Xi, for i =,...,. If the 5

6 Y PURWONO function gi : Ri R for i =,..., are continuous an strictly increasing a.s., C is still the copula of (g(x),..., g(x)). If the univariate marginal istributions are continuous then the functions gi only have to be increasing a.s. in orer to keep the invariance of C. The invariance property means that copula are the framework to stuy epenence properties which are invariant uner increasing transformations of the iniviual ranom variables. Consier a set of epenent insurance claims an we want to come up with a multivariate istribution as a moel for the losses. The joint istribution for the losses will have the same copula as the one for the logarithm of the losses or for the integral-transforms of the losses. So, in terms of copula moelling, we are inifferent with which of the three marginal scales we work (see 5). In many applications, the interesting moel is sometimes efine via the so calle survival family rather than the copula family itself. This moeling approach is usually use, in many cases, in moeling epenecies between life times of several objects. The survival copula appears as the function which relates the joint survival function of a multivariate istribution with the survival functions of the univariate margins. The formal proposition (for the proof see [27, page 28]) is as follows. Suppose we have the istribution function F of the ranom vector (X,..., X), with marginal istribution functions F,..., F respectively. Then, there exists a copula C such that Fx (,..., x ) CFx ( ( ),..., F( x )) =, where m m m in the bivariate case if C is the copula of (X, X2) then Fx (,..., x ) = Pr ( X > x,..., X > x ) for m =,...,. Furthermore, Cu (, u) = u + u + C( u, u) The proof of the above proposition we can see in [27, page 28]. In the final part of this section, we give some examples of copula important for life times epeencies moeling in insurance an finance. First we give an example of so calle explicit copula an then two examples of implicit copula. An implicit copula is obtaine from a multivariate istribution function thorough (2.2). The copula is sai to be explicit, when the expression which efines the copula is not written as a function of the joint an marginal istribution functions. One example of explicit copula is Clayton copula. The Clayton copula with parameter θ (0, + ) is given by: / θ θ Cu (,..., u) = ( ui ) +, (u,..., u) [0, ]. i = The first example of implicit copula is Gaussian copula. For a imensional ranom vector X that has a multivariate normal istribution with mean vector zero an correlation matrix Σ, the Gaussian copula is efine as Cu (,..., u ) = P ( Φ ( X) u,..., Φ ( X ) u ) 6

7 COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES = P ( X Φ ( X),..., X Φ ( X) ) = ( Φ Φ ( X),..., Φ ( X) ) t-copula is a secon example of implicit copula. Suppose that X has a zero mean imensional multivariate t istribution with ensity function f ( x,..., x ) = ν + Γ t 2 x Σ x ν + ν Γ ( πν ) Σ 2 ν + 2 where Σ is the correlation matrix an ν are the egrees of freeom. Analogously to the Gaussian copula, the t copula is given by ( ν ν ) Cu (,..., u ) = t t ( u ),..., t ( u ) (2.5) where (u,..., u) [0, ], t represents the istribution function of a - imensional ranom vector with ensity (2.5) an t ν enotes the istribution function of a stanar univariate stuent t ranom variable with ν egrees of freeom. 3 Multiple Lives Analysis The specific problem of epenence analysis consiere in the project eals with stuying associations between several lives. For lives Lj efine Xj as age at eath of life Lj ranom variables. Assume that associate pairs of imensional lives (L,..., L) are observe uring a certain limite perio of time T. An observation begins simultaneously at ranom entry age e for live L, e2 for live L2,..., an e for live L. The observation like this is applie to each observation. This conition represents the left truncation application to the ata. Life Lj is observe until Lj ies before the age ej + T or Lj is still alive at age ej + T, in which case the eath time will not be observe. This conition represents the right censoring. Therefore, in a sample of Y = (Y,..., Yn), each i th observation yi of an associate pair of lives (Li,..., Li) may be represente as a vector Yi = {(ei,..., ei), (Ti,..., Ti), (δi,..., δi)} (3.) where tij is the termination time length for life Lij, an δij is the right censoring inicator: δij = 0, when Tij = T an = when Tij < T. The range of application of such structure of observations is frequently encountere in health, epyemiologycal, or emographycal stuies. In these area of stuy, it is usually consiere two lives j =, 2 where T an T2 may represent times to failure of associate organs (eyes, kineys), see, e.g., [32], [35], or lives of twins [], [7] or lives of marrie couples. The latter example, playing important role in insurance pricing an first analyze by copula methos in [9], will be the numerical example in our project. 7

8 Y PURWONO One of the important tasks in many applications like in the multiple lives base insurance policies, is to estimate the future lifelength probabilities for given entry ages. As an example, consier an insurance covering benefits for a marrie couple, like last survivor life annuity. In this case, the interest are usually the joint first life survival function SFL(t; e, e2) = Pr{min(X e, X2 e2) > t min(x e, X2 e2) > 0} (3.2) an the joint last survivor function SLS(t; e, e2) = Pr{max(X e, X2 e2) > t min(x e, X2 e2) > 0} (3.3) A very natural approach to the estimation of SFL(t; e, e2) an SLS(t; e, e2) is to first estimate the bivariate survival function S(t, t2) as efine in section 2, an then use the formulas an SFL(t; e, e2) = SLS(t; e, e2) = Se ( + t, e2 + t) Se (, e) 2 Se (, e2 + t) + Se ( + t, e2) Se ( + t, e2 + t) Se (, e) 2 (3.4) We will consier this approach for explaining the numerical ata example in the final section. 4 Estimation for Copula Moels Estimation of S(t, t2,..., t) may be carrie out in two stages: at the first stage, the marginal survival functions Sj(tj) for j =,..., are estimate non parametrically (via prouct limit estimation metho or its moification) or with the help of a parametric techniques by proposing a number of parametric istributions (popular choices coul be Gompertz or Weibull istributions). At the secon stage, the estimate survival functions are mixe accoring to some copula moels (see section 2), an the association parameter is estimate via a statistical estimation metho we will consier in our stuy. An alternative approach consists of estimating simultaneously parameters of the marginal istributions an the association parameters as well. If there is a certain information available regaring the marginals, e.g., there exist aitional ata available on the iniviual lives, it is natural to consier Bayesian estimation methos with informative priors on the marginal parameters an non informative prior on the parameter of association. In the present project, we will restrict our shelves to only consier the secon approach as the main interest of our stuy. Specifically, we will pay attention to the case of estimating irectly the bivariate survival function as a copula S(t, t2; θ,θ2, α) = C(S(t; θ), S2(t2; θ2), α) where θj for j =, 2 are parameters vectors of the marginals an α is the parameter of association. In the presence of right censoring, the likelihoo function for the vector parameter θ = (θ,θ2, α) with ata y structure as escribe in (3.). 8

9 COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES In this section, we briefly escribe the proceure that may possibly be employe to estimate the moel parameters. Here we iscuss a general proceure for estimating parameters when a set of inepenent multivariate observations Xi = (Xi,..., Xi), i =,..., n are given with their corresponing marginal istribution functions Fk(.; θk) an ensity functions fk(.; θk), for k =,...,. Consier the ranom vector X = (X,..., X) t. Suppose we want to estimate the parametric copula base moel for X given by ( x ;,...,, ) = C F ( x ; ),..., F ( x ; ) F α α θ ( ) α α (4.) where Fi(xi; αi) is the istribution function of Xi with parameter vector αi with pi N for all univariate margins i =,..., an C is a copula family parameterise by the vector θ R q with q N. Assume that C has a ensity function c given by cu (,..., u; θ ) = Cu (,..., u ; θ) u... u (4.2) with ( u,..., u) [0,] an that Fi has a ensity fi for all i =,...,. For the case where the margins are iscrete, we enote by fi the probability mass function of Xi. As for our applications, margins are absolutely continuous, this will not be an issue. The ensity of the copula base moel (4.) for X is f ( x; α,..., α, θ ) = c F ( x ) F ( x ) ( ; α,..., ; α ; θ ) f i ( x i ; α i ) (4.3) i = Suppose that we have n ii imensional vectors of observations (x,..., xn). We assume that all the necessary regularity conitions (see [24]) on c an fi for i =,..., are met. The log likelihoo function for the univariate margins of moel (4.) takes the form n Li(αi; x) =, for i =,...,, (4.4) j = log f ( x ; α ) i ij i while the log likeihoo for the copula moel F is n L ( α,..., α, θ; x ) = l og f ( x j ; α,..., α, θ) (4.5) j = Assuming that the usual regularity conitions are fulfille, there is a vector solution α to each one of the systems of equations i L α i ( α ; x) L ( α ; x),..., i i i i α pi i = 0, for i =,..., (4.6) which is the maximum likelihoo estimator (MLE) for the marginal parameters. Note that these estimators are obtaine inepenently for each margin. p R i 9

10 Y PURWONO The process of fining estimators from the previous proceure will be complex when the function in (4.5) is to be complicate. Instea of using the full maximum likelihoo, we can use the so-calle Inference Function for Margins (IFM) Metho. This terminology comes from McLeish an Small in [26] an it has been followe by authors like Joe in [20] for copula-base moel statistical inference. The metho consists of estimating the moel parameters by fining the roots of a conveniently efine set of inference functions. In the case of maximum likelihoo estimation, the inference functions are the partial erivatives of the log likelihoo function. In the IFM metho, the score functions of the margins an of the copula constitute the set of estimating equations. The IFM metho consists first of obtaining the MLE vectors α i,..., α for the marginal parameters solving (4.6) then substitute these maginal estimates in (4.5) to maximise (in most cases numerically) L(θ; x, α i,..., n α ) = f xj j = l og (, α,..., α; θ ), (4.7) in orer to estimate θ, or to use the score function of (4.5) an estimate the epenence parameter vector θ, solving the following system of equations L ( θ; x, α,..., α ) L ( θ; x, α,..., α ) θ,..., θ q = 0. (4.8) This proceure is far more easy numerically an less computatioinally intensive than a irect optimisation of (4.5). However, in case it is feasible to obtain the MLE for the full vector of moel parameters from (4.5), we can use the IFM estimates as goo starting values for the optimiisation routine. Joe in [20] proves that the IFM estimator verifies, uner regular conitions (the research project will confirm these conitions), the property of asymptotic normality, an we have: T ( θifm θ0) N ( 0, ( θ 0 )) G (4.9) with G(θ0) is the Goambe information matrix. Thus, if we efine a score function s l l l = θ, (4.0) c ( θ),...,, θ θ 2 where lj enotes the log likelihoo of the jth marginal, an lc the log likelihoo for the copula itself, splitting the log-likelihoo in two parts l,..., l for each margin an lc for the copula, the Goambe information matrix takes the following form: G ( 0) = D V( D ) θ (4.) with () D E s θ = θ an V = E [ s () θs () θ ] 0

11 COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES The estimation of this covariance matrix requires us to compute many erivatives. Joe in [20] then suggests the use of the jacknife metho or other bootstrap methos to estimate it. Joe [20] points out that the IFM metho is highly efficient compare with the ML metho (we also confirm this in the research project). It is worth noting that the IFM metho may be viewe as a special case of the generalize metho of moments (GMM) with an ientity weight matrix (see [4]). In this project, we will use SAS to coe the maximum likelihoo proceure an in particular, use the Interactive Matrix Language (IML) proceure with optimization routines. These routines normally give estimates of the secon erivatives of the likelihoo function so that an estimate asymptotic covariance in (4.) is a stanar output. Using these asymptotic results, we can then construct confience intervals for our parameters of interest an evelop hypothesis tests. In particular, we will be intereste in constructing test for the presence of epenence an this will largely epen on the choice of the parametric copulas. 5 Conitional Copula for Joint Survival Shemyakin et al in [33] explaine that using (unconitional) copula moel to irectly estimate joint survival function S(t, t2) sometimes prove to be insufficient for some paire lives. The main reason for this is that the problem of imensionality will appears when the entry ages of the paire lives o not coincie, which is usually happene in the case of the spouses lives. Therefore, it seems that no single copula moel for multivariate survival function is going to provie a goo tool for estimation of joint first life an last survivor probabilities in (3.2) an in (3.3) respectively, for paire lives with ifferent ages an association tie to real time rather than to the age scale. For more etaile iscussion of this problem an its consequences, one can fin in [34]. A solution to this controversy, consists in creating separate copula moels with common structure an common value of the association parameter for each pair of entry ages. Instea of mixing two marginal survival functions into a joint bivariate function, one can mix two conitional survival functions conitione to the survival of paire lives to their entry ages. This approach appears to be efficient for estimation of joint first life, but faces serious problems when applie to estimation of last survivor probabilities. An alternative but, probably, inferior solution of introucing the age ifference as an aitional variable was iscusse in [33]. Consier the problem of estimating the joint first life survival function like in (3.2). such estimation requires observing of a pair of associate lives until first eath. A possible problem with the ata structure as escribe in (3.) in actuarial applications consists in the unerpinning of the time of the first eath in a pair, if only the secon eath brings about a change in the insurance payout. We will not iscuss this issue because it is not clear how serious this problem where it can be epeneing on the origin of ata. The iea of conitional copulas is to apply a copula function to conitional survival functions instea of the marginals so that PFL(t; e, e2) = C Se ( + t ) ( ), Se + t ;α 2 Se ( ) Se ( 2) (5.),

12 Y PURWONO where the type of copula use coul be the same as iscusse above. The imensionality problem, which was pointe out in the above, oes not arise in this situatio, because the joint life probability is clearly conitione to the entry ages. If the unerlying survival istributions of iniviual lives allow for close form conitional istributions, this approach oes not lea to a substantial increase of complexity of the likelihoo. For example see [42]. Bayesian moels base on the conitional copula approach with Weibull marginals an stable copulas were applie to martilities of marrie couples in [34]. The choice of priors an hyperparameters for the shape an scale istributions is base on the fitness to the existing male an female mortality tables. This choice reflecte the belief that a substantial amount of prior information on iniviual male an female mortality can be incorporate in a copula moel. However, this approach oes not promise an equally easy success while ealing with the joint last survivor probablities escribe in (3.3) for the same application to mortalities of the marrie couples. For more etail for this interesting phenomena, one can see [40]. For last survivor problem, shemyakin et al propose a reasonable approximation for the conitional copula of last survival probabilities (for etaile reasoning, see [34], page 0): PLS(t; e, e2) p(t; e : 0) + p2(t; e2 : 0) PFL(t; e, e2), (5.2) where p(t; e : 0) can be interprete as the probability of a woman alive an marrie at entry age e to survive for t more years, an p2 efine similarly. This woul ictate a ifferent choice of priors. A moel, which will make use of approximation (5.2), may look like this: where PLS(t; e, e2) p(t; e : 0) + p2(t; e2 : 0) PFL(t; e, e2) pj(t; ej : 0) = exp γ βj βj j j α j ej + t γ, (5.3) an the prior istributions for parameters βj an γj are set to reflect that they are relate not to entire male an female populations, but to the populations of marrie men an women. As it is evient from Table 0 in [8], marrie men an women ten to have lower mortality comparatively to men an women in general. When we introuce Weibull priors base on mortality tables, we have to take into account this effect, leaing to increase values in both scale an shape parameters. This moeling paraigm will be consiere as one of analysis tools we want to apply to the ata as an ilustration of the application of copula moels inference as escribe in the previous section. 6 Data an Analysis The most important piece of the project is ilustrating the application of copula moeling of multiple lives ata to the empirically estimation for the presence of epenence in the mortality of marrie couples. We will use the ata coming from a huge number of joint last survivor annuity contracts of a large Inonesian insurer in payoff status over 5 year observation perio beginning in 998 an ening in The ata structure follows situation in section 3, where we 2

13 COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES consiere right censoring an left truncation to the ata. In elicitation of the hyperparameters of the Bayesian moels we also use Table 0 of [8] proviing the group mortality rates by age, sex, an marital status. We will analyse the ata in five ways of moelling. For Moel I to IV, we assume that marginal survival functions for men an women escribe by two parameter Weibull istributions with scales βj an shapes γj, where j = correspons to female mortality, an j = 2 to male mortality. For the copula moel, we consier stable (Gumbel Hougaar) copula with the association parameter α. In Moel I, we assume that the inepenence of male an female mortalities may be treate as a egenerate case of stable copula with α =. In this moel the estimates for scale an shape parameters were obtaine by maximum likelihoo metho. In Moel II, the simultaneous IFM estimation was performe for all five parameters (scales, shapes, an association) follows the same likelihoo constructing paraigm one in [42]. Moel III was first introuce in [34] an requires Bayesian estimation of five parameters of the conitional copula function escribe in section 5 with the following priors istributions: βj N(φj,ηj), γj N(μj,σj), π(α) /α, where hyperparameters were etermine from US male an female mortality tables. The computation of last survivor probability can be carrie out with the help of approximate formula (7) in [42]. Moel IV uses the conitional copula function escribe in section 5 with a choice of priors motivate by the iscussion in section 6 of [34]. The general type of the prior istributions is similar to Moel III: βj N(φj,ηj), γj N(μj,σj), α U(, ), but the choice of hyperparameters is not relate to general male an female populations, but to marrie men an women. A possible way is to utilize the estimates of the marginal parameters obtaine in Moel I. This seems consistent with the results obtaine by fitting paramteric moels to group mortality rates in Table 0 of [8] an comparing scale an shape parameters of Weibull mortality moels for marrie men an women with those for the general population. Moel V is the one suggeste in the present research project. It uses the conitional copula functions escribe in section 5. The choice of priors istributions an the choice of hyperparamaters like the one use in Moel IV. In this Moel, we will not irectly assume that marginal survival functions are Weibull istribute, but here we infer the marginal survival functions irectly from the best fitte copula moel which we select base on an approach borrowing from a statistical inference metho. The choice of copula in the Moel V is one of the issues to resolve in the research project. Joe in [20] an Nelsen in [27] list several families of parametric copulas, an we escribe some of the more common ones in this proposal. Choosing the right copula is going to be a challenging aspect of the project because even in the theory of statistics, this area is not well evelope. Durrleman, et al. in [5] chooses a copula base on the construction of the empirical copula, but constructing this in themultivariate case is not starightforwar. Our suggeste strategy is to fit several families of copulas an choose among these families the best fitting moel. An approach similar to a gooness of fit arguments may then applie. 7 Concluing Remark This paper lays out the theoretical founations necessary to evelop the solution to the problems with which we wish to aress in the research project. In summary, our project aims to aress the following funamental issues: () How to empirically estimate an test the presence of epenencies in multiple life times of 3

14 Y PURWONO insure iniviuals?; (2) How to statistically select the appropriate copula moel that can be use to approximate the true behaviour of epenence structure of spouses life times?; an (3) If there is such presence, what are its implications in computing insurance an/or annuity premiums, insurance reserves an/or pension liabilities? Reference [] Anerson, J.E., Louis, T.A., Holm, N.V., an Harval, B. (992) Time epenent association measures for bivariate survival istributions, J. of the Amer. Statist. Assoc., 87, 49, [2] Bowers, N., Gerber, H., Hickman, J.C., Jones, D.A., an Nesbitt, C. (997) Actuarial Mathematics, Schaumburg, IL: Society of Actuaries. [3] Clayton, D.G. (978) A moel for association bivariate tables an its application in epiemiological stuies of family tenency in chronic isease incience, Biometrika, 65, 4-5. [4] Davison, R. an MascKinnon, J. (993) Estimation an inference in econometrics. Oxfor University Press, Oxfor. [5] Durrleman, V., Nikeghali, A. An Roncalli, T. (2000) Which Copula is the Right One?, working paper, Groupe e Recherche Operationelle, Creit Lyonnais, France. [6] Embrechts, P., McNeil, A. J., an Straumann, D. (2002) Correlation an epenence in risk management: Properties an pitfalls. In: Risk Management: Value at Risk an Beyon (E. M. Dempster), Cambrige University Press, Cambrige, [7] Embrechts, P., Linskog, F., McNeil, A. (2003) Moelling epenence with copulas an applications to risk management In: Hanbook of Heavy Taile Distributions in Finance, e. S. Rachev, Elsevier, Chapter 8, [8] Englan an Wales. Death rates. (2002) In: Mortality Statistics. Series DH no.35, Office of National Statistics, Lonon, [9] Frees, E., Carriere, J. an Valez, E. (996) Annuity valuation with epenent mortality, Journal of Risk an Insurance, Vol. 63, 229. [0] Frees, E. an Valez, E. (998) Unerstaning relationships using copulas, Actuarial Research Clearing House, Proceeings, 32n Actuarial Research Conference, August 6-8, 997, 5. [] Genest, C. (987) Frank's family of bivariate istributions. Biometrika 74, 3, [2] Genest, C., Ghoui, K., an Rivest, L.-P. (995) A semiparametric estimation proceure of epenence parameters in multivariate families of istributions. Biometrika 82, 3, [3] Genest, C., an Rivest, L.-P. (993) Statistical inference proceures for bivariate Archimeean copulas. J. Amer. Statist. Assoc. 88, 423, [4] Genest, C., an Werker, B. J. M. (2002) Conitions for the asymptotic semiparametric efficiency of an omnibus estimator of epenence 4

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16 Y PURWONO [32] Romeo, J.S., Tanaka, N.I., an Peroso e Lima, A.C. (2004) Bivariate survival moeling: a Bayesian approach base on copulas, Relatorio Tecnico, Universiae e Sao Paulo, Sao Paulo, Brasil., RT-MAE [33] Shemyakin, A. an Youn, H. (2000) Statistical aspects of joint-life insurance pricing, 999 Proceeings of Amer. Stat. Assoc., [34] Shemyakin, A. an Youn, H. (200) Bayesian estimation of joint survival functions in life insurance, In: Bayesian Methos with Applications to Science, Policy an Official Statistics, European Communities, [35] Shih, J.H. an Louis, T.A. (985) Inferences on the association parameter in copula moels for bivariate survival ata, Biometrika, 5, [36] Sklar, A. (959) Fonctions e répartition à n imensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, [37] Spiegelhalter, D.J., Thomas, A., an Best, N.G. (2003) WinBUGs.4. Computer Program, Imperial College an MRC Biostatistics Unit, Cambrige. [38] Valez E. A., an Chernih, A. (2003) Empirical estimation of epenence in a portfolio of insurance claims preliminaries, working paper, UNSW, Siney, AU. [39] Wang, S. (997) Aggregation of correlate risk portfolios: Moels an algorithms. Preprint, Casualty Actuarial Society (CAS). [40] Youn, H. an Shemyakin, A. (200) Pricing practices for joint last survivor insurance, Actuarial Research Clearing House, Vol [4] Youn, H., Shemyakin, A., an Herman, E. (2002) A re-examination of joint mortality functions, North Amer. Actuarial J., Vol. 6,, [42] Youn, H., an Shemyakin, A. (2004) Copula moels of joint survival analysis, working Paper, University of St. Thomas, MN, USA. YOGO PURWONO: Department of Management Faculty of Economics, University of Inonesia Depok, INDONESIA 6