A copula transformation in multivariate mixed discrete-continuous models
aa r X i v : . [ s t a t . M E ] A ug A copula transformation in multivariate mixeddiscrete-continuous models
Jae Youn Ahn † , Sebastian Fuchs ‡ , and Rosy Oh § Ewha Womans University, Republic of Korea Universit¨at Salzburg, Austria Ewha Womans University, Republic of Korea ** Corresponding authors
Abstract
Copulas allow a flexible and simultaneous modeling of complicated dependence struc-tures together with various marginal distributions. Especially if the density function canbe represented as the product of the marginal density functions and the copula densityfunction, this leads to both an intuitive interpretation of the conditional distribution andconvenient estimation procedures. However, this is no longer the case for copula modelswith mixed discrete and continuous marginal distributions, because the correspondingdensity function cannot be decomposed so nicely. In this paper, we introduce a copulatransformation method that allows to represent the density function of a distributionwith mixed discrete and continuous marginals as the product of the marginal probabil-ity mass/density functions and the copula density function. With the proposed method,conditional distributions can be described analytically and the computational complexityin the estimation procedure can be reduced depending on the type of copula used.
Along with random effect methods, copula methods are a widely used tool to model mul-tivariate distributions. In case of both, the univariate marginal distribution functions andthe copula associated with a ( d + 1)-dimensional distribution function H , are absolutelycontinuous, the density h of H fulfills h ( x, y , . . . , y d ) = c (cid:0) F ( x ) , G ( y ) , . . . , G d ( y d ) (cid:1) f ( x ) d Y i =1 g i ( y i ) , (1)where c denotes the copula density, F and G i denote the univariate marginal distributionfunctions and f and g i denote its corresponding density functions, i ∈ { , . . . , d } . The copularepresentation (1) enables the dependence structure to be separated from the marginal dis-tributions. Such a complete separation not only provides a meaningful interpretation of the † [email protected] ‡ [email protected] § [email protected] d discrete randomvariables, the evaluation of the likelihood function requires the calculation of 2 d terms whichprovoke computational difficulties in the estimation procedure and complicates an interpre-tation of the dependence structure [Smith and Khaled, 2012; Zilko and Kurowicka, 2016].Copula methods for mixed - discrete and continuous - marginals, mixed copula models forshort, suffer from similar difficulties.In this paper we investigate mixed copula models with a single discrete and several absolutelycontinuous variables and mainly focus on problems related to interpretation difficulties of the(conditional) dependence structure and computational difficulties in estimation.Often the dependence structure in the copula model is explained in terms of conditional dis-tributions. For instance, a wide range of copula families including Archimedean copulas andsome elliptical copula families are closed under the operation of conditioning [Mesfioui and Quessy,2008; Ding, 2016]. For such copula families, the conditional distribution has both an ana-lytical and an intuitive interpretation. However, this convenience is lost in copula modelswith discrete marginals. We note that the interpretability of conditional distributions whenconditioning with respect to a discrete random variable is important in several research ar-eas including case-control studies in medicine [He et al., 2012; de Leon and Wu, 2011] andfrequency-severity models in insurance [Czado et al., 2012; Kr¨amer et al., 2013].A second problem that occurs in mixed copula models is the computational complexityin the calculation of density functions [Kadhem and Nikoloulopoulos, 2019] since statisti-cal estimation procedures require the evaluation of the corresponding joint density func-tion for multiple terms. In the case of implicit copula models, for instance, the calcula-tion of the density function requires numerical integration [Nikoloulopoulos and Karlis, 2009;Kadhem and Nikoloulopoulos, 2019]. Such numerical difficulties may complicate the esti-mation procedure mainly due to the computational difficulties in calculating the likelihoodfunctions and subsequent derivatives [Song and Song, 2007].In the present paper we aim at providing a handy representation of the mixed copula model sothat both interpretation and calculation of the density function remain intuitive and simple.To this end, we start with a rather naive question: Can we reformulate a mixed copulamodel with some discrete distribution function F and some absolutely continuous distributionfunctions G , . . . , G n , and find some closely related distribution whose density satisfies (1)?An answer to that question requires a copula transformation that is presented in Section 2.With the rather appealing form of the density in (1), we expect that the proposed distributionmay provide a meaningful interpretation and excellence in computation in the estimation When pair copula construction is adapted, the computational burden of evaluating n -dimensional discreterandom variables only requires 2 n ( n −
1) terms [Panagiotelis et al., 2012]. I := [0 , N := N ∪ { } and let d ≥ y = ( y , . . . y d ) ∈ R d , or vectorsof functions, e.g., f = ( f , . . . , f d ). We denote by ζ d the d -dimensional Lebesgue measure; incase of d = 1 we simply write ζ . In this section, we consider a multivariate mixed model with a single discrete and severalabsolutely continuous variables and present a modification of this model that allows both ameaningful interpretation and a convenient estimation.First, fix some probability space (Ω , A , P ) and consider a random variable N (on this prob-ability space) with distribution function F such that N follows a discrete distribution on N . Additionally, we consider a d -dimensional random vector Y (on the same probabilityspace) whose margins Y i , i ∈ { , . . . , d } , follow an absolutely continuous distribution with ζ – densities g i and distribution functions G i , i ∈ { , . . . , d } . Then, by Sklar’s Theorem (see,e.g., ?? ), there exists some ( d + 1)-dimensional copula C such that the distribution function H of ( N, Y ) satisfies H ( n, y ) = C (cid:0) F ( n ) , G ( y ) (cid:1) (2)for every ( n, y ) ∈ N × R d . Note that the copula C fails to be unique, in general. In thefollowing, we assume that C is absolutely continuous, i.e. there exists some ζ d +1 - density c of C . Then, H has ( µ ⊗ ζ d )- density (where µ denotes the counting measure on the power setof N ) h satisfying h ( n, y ) = (cid:18) ∂ d ∂ y C (cid:0) F ( n ) , G ( y ) (cid:1) − ∂ d ∂ y C (cid:0) F ( n − , G ( y ) (cid:1)(cid:19) d Y i =1 g i ( y i ) (3)= Z ( F ( n − ,F ( n )] c (cid:0) u, G ( y ) (cid:1) d ζ ( u ) ! d Y i =1 g i ( y i )for ( µ ⊗ ζ d )– almost all ( n, y ) ∈ N × R d . While a model with a density function fulfilling (1)allows a meaningful interpretation of the underlying dependence structure and the univariate3arginal distributions, such a direct interpretation turns out to be difficult in a situation like(3). In addition to that, the estimation in (3) is usually cumbersome since the calculation ofthe likelihood function can be quite involved and may require numerical integration. There-fore, we are interested in the following naive question (Q1): Can we find a ( µ ⊗ ζ d )- density h ∗ of the following form(Q1) h ∗ ( n, y ) = c (cid:0) F ( n ) , G ( y ) (cid:1) P [ { N = n } ] d Y i =1 g i ( y i )for ( µ ⊗ ζ d )– almost all ( n, y ) ∈ N × R d , and, if yes, what are the distributional propertiesof h ∗ ?We additionally focus on the conditional version of the distribution function H of ( N, Y )in (2). Then, for P N – almost every n ∈ N , the conditional joint density h ( . | n ) of Y given N = n equals h ( y | n ) = 1 P [ { N = n } ] (cid:18) ∂ d ∂ y C (cid:0) F ( n ) , G ( y ) (cid:1) − ∂ d ∂ y C (cid:0) F ( n − , G ( y ) (cid:1)(cid:19) d Y i =1 g i ( y i ) (4)= 1 P [ { N = n } ] Z ( F ( n − ,F ( n )] c (cid:0) u, G ( y ) (cid:1) d ζ ( u ) ! d Y i =1 g i ( y i )for ζ d – almost all y ∈ R d . In contrast to the case ( N, Y ) would be absolutely continuous,equation (4) does not lead to a practicable interpretation. Therefore, we are also interestedin the following naive question (Q2) related to (Q1): For P N – almost every n ∈ N , can wefind a ζ d - density h ∗ ( . | n ) conditional on N = n of the following form(Q2) h ∗ ( y | n ) = c (cid:0) F ( n ) , G ( y ) (cid:1) d Y i =1 g i ( y i )for ζ d – almost all y ∈ R d , and, if yes, what are the distributional properties of h ∗ ( . | n )? Remark. 2.1.
If we assume c ( u, G ( y )) = c ( F ( n ) , G ( y )) on ( F ( n − , F ( n )] × R d for all n ∈ N , then the identities h ∗ ( n, y ) = h ( n, y ) and h ∗ ( y | n ) = h ( y | n ) hold for ( µ ⊗ ζ d )– almostall ( n, y ) ∈ N × R d . (cid:3) In the following, we use a generalization of the idea described in Remark 2.1 and constructa ‘copula’-density that is, with respect to the first coordinate, partially constant and hence astep function.
For α ∈ (0 , ⌈ . ⌉ α,F : (0 , → R by letting ⌈ u ⌉ α,F := X n ∈ N F α ( n ) ( F ( n − ,F ( n )] ( u ) (5)4here F α : N → I is given by F α ( n ) := (1 − α ) F ( n −
1) + α F ( n ), and canonically extend ⌈ . ⌉ α,F to I by putting ⌈ ⌉ α,F := 0. Then, 0 ≤ ⌈ u ⌉ α,F ≤ u ∈ I . Remark. 2.2.
We note that the identity ⌈ u ⌉ α,F = ( F α ◦ F ← )( u ) holds for all u ∈ (0 , F ← : (0 , → R denotes the pseudo inverse of F given by F ← ( u ) := inf { x ∈ R : F ( x ) ≥ u } . (cid:3) For the copula C with density function c , we further define the map c α,F,C : I × I d → R byletting c α,F,C ( u, v ) := c (cid:0) ⌈ u ⌉ α,F , v (cid:1) . (6)Then c α,F,C is positive, measurable and a ζ d +1 - density: Indeed, since ∂ C ( u, ) = 1 holds for ζ – almost all u ∈ I (where ∂ C denotes the partial derivative of C with respect to the firstcoordinate), we obtain Z I × I d c α,F,C ( u, v ) d ζ d +1 ( u, v ) = Z I × I d c (cid:0) ⌈ u ⌉ α,F , v (cid:1) d ζ d +1 ( u, v ) = Z I ∂ C ( ⌈ u ⌉ α,F , ) d ζ ( u ) = 1 . The following result is immediate from Equations (5) and (6):
Corollary. 2.3.
Consider v ∈ I d . • The identity c α,F,C ( u, v ) = c ( F α ( n ) , v ) holds for all n ∈ N and all u ∈ ( F ( n − , F ( n )] . • The map c α,F,C ( ., v ) is a positive step function. The next example, in which we consider a pertubation of the independence copula Π (see,e.g., ?? ), illustrates the construction principle and shows that c α,F,C fails to be a copuladensity, in general: Example. 2.4.
For θ ∈ [ − , C : I × I d → I given by C ( u, v ) := Π( u, v ) + θ u (1 − u ) v (1 − v ) d Y i =2 v i and the distribution function F : R → I given by F ( x ) := 13 [1 , ( x ) + 23 [2 , ( x ) + [3 , ∞ ) ( x ) . Then we have c ( u, v ) = 1 + θ (1 − u )(1 − v ) for all ( u, v ) ∈ I × I d and F α ( n ) = α/ n = 1;1 / α/ n = 2;2 / α/ n = 3;1 n ≥
4; and ⌈ u ⌉ α,F = u = 0; α/ u ∈ (cid:0) , (cid:3) ;1 / α/ u ∈ (cid:0) , (cid:3) ;2 / α/ u ∈ (cid:0) , (cid:3) . c α,F,C is a step function satisfying c α,F,C ( u, v ) = 1 + θ (1 − v ) u = 0; − α u ∈ (cid:0) , (cid:3) ; − α u ∈ (cid:0) , (cid:3) ; − − α u ∈ (cid:0) , (cid:3) , for all v ∈ I d . Since, for every α = 1 / θ = 0 and every v ∈ (0 , Z I × [0 ,v ] × I d − c α,F,C ( s, t ) d ζ d +1 ( s, t ) = v + θ v (1 − v ) (cid:18) − α (cid:19) = v , c α,F,C fails to be a copula density. (cid:3) Obviously, if c α,F,C fails to be a copula density, then c α,F,C = c . Note that the converseimplication is not true, in general. For instance, take α = 1 / θ = 0 in Example 2.4.Then, it is straightforward to show that c α,F,C is a copula density but fails to coincide with c . Nevertheless, there exist copulas satisfying c α,F,C = c . Example. 2.5.
Consider a d -dimensional copula A with ζ d - density a and define themap C : I × I d → I by letting C ( u, v ) := u A ( v ). Then, by [ ? , Theorem 6.6.3], C isa ( d + 1)-dimensional copula with ζ d +1 - density c satisfying c α,F,C ( u, v ) = c (cid:0) ⌈ u ⌉ α,F , v (cid:1) = a ( v ) = c ( u, v ) for all ( u, v ) ∈ I × I d . (cid:3) Since c α,F,C is a ζ d +1 - density, the map C α,F,C : I × I d → R given by C α,F,C ( u, v ) := Z [0 ,u ] × [ , v ] c α,F,C ( s, t ) d ζ d +1 ( s, t ) (7)is a distribution function on I d +1 following an absolutely continuous distribution. In the nextlemma we gather some properties of C α,F,C that turn out to be quite useful. Lemma. 2.6.
The identity C α,F,C ( u, v ) = n − X k =0 ∂ C ( F α ( k ) , v ) P [ { N = k } ] + ∂ C ( F α ( n ) , v ) (cid:0) u − F ( n − (cid:1) holds for all n ∈ N and all ( u, v ) ∈ ( F ( n − , F ( n )] × I d . In particular, we have C α,F,C ( u, ) = u and C α,F,C (1 , v ) = E (cid:2) ∂ C (cid:0) F α ( N ) , v (cid:1)(cid:3) for all u ∈ I and all v ∈ I d .Proof. For every n ∈ N and every u ∈ ( F ( n − , F ( n )], Corollary 2.3 yields Z [0 ,u ] c α,F,C ( s, v ) d ζ ( s ) = Z [0 ,u ] c (cid:0) ⌈ s ⌉ α,F , v (cid:1) d ζ ( s )6 n − X k =0 Z ( F ( k − ,F ( k )] c ( F α ( k ) , v ) d ζ ( s ) + Z ( F ( n − ,u ] c ( F α ( n ) , v ) d ζ ( s )= n − X k =0 c ( F α ( k ) , v ) (cid:0) F ( k ) − F ( k − (cid:1) + c ( F α ( n ) , v ) (cid:0) u − F ( n − (cid:1) = n − X k =0 c ( F α ( k ) , v ) P [ { N = k } ] + c ( F α ( n ) , v ) (cid:0) u − F ( n − (cid:1) for ζ d – almost all v ∈ I d . This proves the assertion. (cid:3) From a probabilistic viewpoint, if a random vector ( U, V ) is distributed according to C α,F,C ,then it follows from Lemma 2.6 that U is uniformly distributed, i.e. P [ { U ≤ u } ] = u for all u ∈ I , and that the identity P [ { V ≤ v } ] = E (cid:2) ∂ C (cid:0) F α ( N ) , v (cid:1)(cid:3) holds for all v ∈ I d .We illustrate the construction principle by completing Example 2.4. Example. 2.7.
For θ ∈ [ − , C and the distribution function F discussed in Example 2.4. Then C α,F,C ( u, v ) = Π( u, v ) + θ v (1 − v ) d Y i =2 v i u = 0 (cid:0) − α (cid:1) u u ∈ (cid:0) , (cid:3) + (cid:0) − α (cid:1) u u ∈ (cid:0) , (cid:3) + (cid:0) − − α (cid:1) u u ∈ (cid:0) , (cid:3) for all v ∈ I d and hence C α,F,C ( u, ) = u for every u ∈ I as well as C α,F,C (1 , v ) = E (cid:2) ∂ C (cid:0) F α ( N ) , v (cid:1)(cid:3) = d Y i =1 v i + θ v (1 − v ) d Y i =2 v i (cid:18) − α (cid:19) ∈ I for every v ∈ I d . Since, for every α = 1 / θ = 0 and every v ∈ (0 , C α,F,C (cid:0) , ( v , ) (cid:1) = v + θ v (1 − v ) (cid:18) − α (cid:19) = v , C α,F,C fails to be a copula. (cid:3) Choosing θ = 0 in the previous example yields C = Π and hence C α,F,C = Π = C . A moregeneral result is given by the following example which extends Example 2.5. Example. 2.8.
Consider a d -dimensional copula A with ζ d - density and the copula C : I × I d → I given by C ( u, v ) := u A ( v ). Then, C α,F,C = C . In particular, C α,F, Π = Π. (cid:3) .2 Transformation of the random vector Although the copula transformation C α,F,C of C fails to be a copula, in general, it is a dis-tribution function on I d +1 whose first coordinate is distributed uniformly. In this subsectionwe will use this copula transformation in combination with Sklar’s theorem to construct adistribution function that helps answering questions (Q1) and (Q2).For α ∈ (0 , C with density function c , the discrete distribution function F andabsolutely continuous distribution functions G , . . . , G d , we define the function H α,F, G ,C : R × R d → I by letting H α,F, G ,C ( x, y ) := C α,F,C (cid:0) F ( x ) , G ( y ) (cid:1) . Then it is straightforward to verify that H α,F, G ,C is a distribution function satisfyinglim t →∞ H α,F, G ,C ( x, t ) = F ( x ) and lim s →∞ H α,F, G ,C ( s, y ) = E (cid:2) ∂ C (cid:0) F α ( N ) , G ( y ) (cid:1)(cid:3) . The next result is immediate from Corollary 2.3 and solves question (Q1).
Theorem. 2.9.
The ( µ ⊗ ζ d ) - density h α,F, G ,C of H α,F, G ,C satisfies h α,F, G ,C ( n, y ) = c (cid:0) F α ( n ) , G ( y ) (cid:1) P [ { N = n } ] d Y i =1 g i ( y i ) for ( µ ⊗ ζ d ) - almost all ( n, y ) ∈ N × R d . From a probabilistic viewpoint, if a random vector ( M, T ) is distributed according to H α,F, G ,C ,then Lemma 2.6 shows that M is a random variable whose distribution function F M equals F M = F and that the distribution function F T of T satisfies F T = E (cid:2) ∂ C (cid:0) F α ( N ) , G ( . ) (cid:1)(cid:3) . Note that F T i = G i , i ∈ { , . . . , d } , in general; see Example 2.7. Now, we answer question(Q2). Theorem. 2.10.
Consider a random vector ( M, T ) distributed according to H α,F, G ,C .Then, for P M – almost every n ∈ N , the conditional joint density h α,F, G ,C ( . | n ) of T given M = n satisfies h α,F, G ,C ( t | n ) = c (cid:0) F α ( n ) , G ( t ) (cid:1) d Y i =1 g i ( t i ) (8) for ζ d – almost all t ∈ R d . Remark. 2.11.
In the special case α = 1, the results in Theorems 2.9 and 2.10 reduce to h α,F, G ,C ( n, y ) = c (cid:0) F ( n ) , G ( y ) (cid:1) P [ { N = n } ] d Y i =1 g i ( y i )8nd h α,F, G ,C ( y | n ) = c (cid:0) F ( n ) , G ( y ) (cid:1) d Y i =1 g i ( y i )for ( µ ⊗ ζ d )– almost all ( n, y ) ∈ N × R d . (cid:3) In this section, we perform a numerical study illustrating the impact of the copula transfor-mation method suggested in Section 2.1 by measuring the distance between the copula C andits transformed version C α,F,C .As distance measure we use the Kullback-Leibler divergence (KL divergence for short); see,e.g., Kullback and Leibler [1951]; Kullback [1997]. For two k -dimensional joint distributionfunctions P and Q having p and q as probability density functions, the KL divergence from P to Q is defined as D ( P, Q ) := Z R k p ( x ) log (cid:18) p ( x ) q ( x ) (cid:19) d ζ k ( x ) . We note that D ( P, Q ) ≥ P = Q . In addition, since D ( P, Q ) = D ( Q, P ), KL divergence fails to be symmetric, in general.In the first part of our numerical analysis, we consider bivariate copulas C θ from variousparametric copula families (Gaussian, Student t , Clayton and Gumbel) and put P = C θ and Q = C α,F,C θ where F denotes a Poisson distribution with mean ζ . The parameter θ is choosen in suchway that it corresponds to a certain value of bivariate Kendall’s tau and hence indicates thedegree of dependence represented by C θ . We measure the KL divergence from P to Q undervarious scenarios which are combinations of α = 0 . , . , . , . ζ = 0 . , . , . , . , . θ given in Table 1.Kendall’s tau -0.8 -0.3 -0.1 0 0.1 0.3 0.8 θ Gaussian / Student t -0.951 -0.454 -0.156 0.000 0.156 0.454 0.951Clayton - - - 0.000 0.222 0.857 8.000Gumbel - - - 1.000 1.111 1.429 5.000Table 1: Copula parameter θ corresponding to bivariate Kendall’s tauFor each scenario, we further calculate the values of Spearman’s rho ρ ( P ) and ρ ( Q )to illustrate how the dependence structure of the copula changes with the proposed transfor-mation; recall that Spearman’s rho of a bivariate distribution function H with marginals F and G is defined as ρ ( H ) := 12 Z R F ( x ) G ( y ) d H ( x, y ) −
39n the second part of the numerical analysis, we leave the bivariate setting and consider 3-dimensional copulas. Here, we restrict ourselves to positive dependence (symmetric Gaussianand Clayton copulas) whereas the parameter θ is chosen in such a way that it correspondsto a certain value of the bivariate Kendall’s tau as in Table 1.The results of the numerical analysis are summarized in Table 2 to 11: • Table 2 to 5: KL divergence for Gaussian, Student t , Clayton and Gumbel copula indimension 2. • Table 6 to 9: Spearmans’s rho for Gaussian, Student t , Clayton and Gumbel copula indimension 2. • Table 10 and 11: KL divergence for the 3-dimensional Gaussian and Clayton copula.From Table 2 and 3, one may observe that, for each α and ζ , the values of KL divergence aresymmetric about 0 with respect to θ which is due to the fact that the density functions oftwo Gaussian copulas (t-copulas) having parameters of opposite signs, ± ρ , are reflection ofeach other over the horizon line x = 0 .
5. Similarly, from Table 6 and 7, one may also observethat, for each α and ζ , the values ρ ( P ) and ρ ( Q ) are symmetric about 0.As expected from the definition of the transformation C α,F,C θ , we observe that copulas withweaker dependence tend to have smaller KL divergence and smaller discrepancy between ρ ( P ) and ρ ( Q ). We also observe that for each θ , as ζ increases, KL divergence decreasesand the discrepancy is diminished, which is also an expected result from the definition of thetransformation C α,F,C θ . However, one interesting phenomenon discovered in this numericalanalysis is that, around α = 0 .
5, we observe the smallest KL divergence for each combinationof ζ and θ . Finally, as can be shown in Table 10 and 11, we note that 3-dimensional copulasshow similar patterns as in 2-dimensional copulas. In this section, we apply the proposed copula transformation method to the collective riskmodel (CRM, for short). In the CRM, the aggregate severity in insurance portfolio is modelledas the random sum of individual severities. Specifically, for the nonnegative integer valuedrandom variable N and the positive random variables Y j , the aggregate severity S is definedby S := N P j =1 Y j , if N = n ∈ N ;0 , if N = 0 . Note that the aggregate severity can be expressed as S = M N where M is the averageseverity given by M := N N P j =1 Y j , if N = n ∈ N ;0 , if N = 0 . We first review two CRMs in the insurance literature. The first model that we will revisit isso called the two part
CRM where dependence between frequency and severity is induced byusing the frequency as an explanatory variable of the severities; see, e.g, Frees et al. [2014];10hi et al. [2015]; Garrido et al. [2016]; Park et al. [2018]. As a result, in this model thedistribution of the aggregate severity can be easily determined. However, it is known that thedependence structure in the two part model is quite limited; see, e.g., Liu and Wang [2017];Shi et al. [2020]. The second model that we will revisit is the copula-based
CRM, which cancover the full spectrum of dependencies by describing the dependence for the frequency andthe individual severities based on copula function [Cossette et al., 2019; Oh et al., 2020]. Notethat the description of the aggregate severity, which is the main concern of insurance industry,under the copula-based CRM can be inconvenient as will be explained below. Finally, weapply the proposed copula transformation to the copula based CRM which enables convenienthandling of the distribution of the aggregate severity, and we provide an example where thelinkage between the two CRMs is demonstrated.
Model. 4.1.
The two part CRM for frequency and aggregate severity ( N, M )is defined within the framework of the exponential dispersion family (EDF) as follows (seeFrees et al. [2014]; Garrido et al. [2016]):(i) We specify the frequency component N as N ∼ F (9)where F can be any discrete distribution function on N .(ii) We specify the conditional distribution of the average severity conditional on the numberof claims N = n ∈ N as M (cid:12)(cid:12) N = n i . i . d . ∼ ED (cid:0) µ n , σ n (cid:1) (10)where ED (cid:0) µ n , σ n (cid:1) is the reproductive exponential dispersion model with mean µ n anddispersion parameter σ n ; see, e.g., Jørgensen [1987]; Jorgensen [1997]. Here, the meanparameter µ n is implicitly given by η ( µ n ) = β + ψ ( n ) for properly chosen function ψ and link function η , and the dispersion parameter is given by σ n := σ n . (11)The choice of the dispersion parameter in (11) can be justified by the following distributionalassumption on the individual severities Y , . . . , Y n | N = n i . i . d . ∼ ED (cid:0) µ n , σ (cid:1) n ∈ N (12)which implies (10) by the convolutionary property of EDF. As a result, one may replacethe description of the average severity in Model 4.1 with the description of the individualseverities in (12). We call such model as the two part CRM for frequency and the individualseverities. However, while convenient in many ways, the conditional independence assumptionin (12) is a rather restrictive dependence assumption of frequency and individual severities aspointed out in Liu and Wang [2017] and Shi et al. [2020]. Therefore, the two part CRM with11he functional form of the dispersion parameter in (11) can accommodate only restrictivedependence structures of frequency and individual severities.Alternatively, one may choose a more complicated functional form of σ n as mentioned inLee et al. [2019] to consider more general dependence structures in the CRM. Depending onthe purpose of the data analysis, one may use, for instance, advanced regression modelingstrategies such as non-parametric regression or additive modeling; see, e.g., Hastie and Tibshirani[1990]; Faraway [2005]. However, in such a case the important linkage between the averageseverity in (10) and the individual severities in (12) is violated, in general. We now consider the copula-based CRM for the frequency and the individual severities dis-cussed in Cossette et al. [2019] and Oh et al. [2020] where a wider variety of dependencestructures is possible depending on the particular choice of the used copula family.
Model. 4.2.
The copula-based CRM for frequency and individual severities(
N, Y , Y , . . . , Y N )is defined as follows:(i) We specify the frequency component N as N ∼ F (13)where F can be any discrete distribution function on N ; compare (9).(ii) We specify the conditional distribution of the vector of individual severities ( Y , · · · , Y n )conditional on the number of claims N = n ∈ N as( Y , . . . , Y n ) (cid:12)(cid:12) N = n ∼ W ( n ) (14)for some distribution function W ( n ) given by W ( n ) ( y , . . . , y n ) := C ( n ) (cid:0) F ( n ) , G ( y ) , . . . , G ( y n ) (cid:1) − C ( n ) (cid:0) F ( n − , G ( y ) , . . . , G ( y n ) (cid:1) P [ { N = n } ]with C ( n ) being an absolutely continuous ( n + 1)-dimensional copula.In this model, the density function of ( Y , Y , · · · , Y n ) at point ( y , y , · · · , y n ) conditional on N = n ∈ N satisfies ∂ n ∂y . . . ∂y n W ( n ) ( y , . . . , y n ) = Z ( F ( n − ,F ( n )] c ( n ) (cid:0) u, G ( y ) , . . . , G ( y n ) (cid:1) d ζ ( u ) ! Q ni =1 g ( y i ) P [ { N = n } ] . (15)Hence, loosely speaking, the copula-based CRM can be understood as( N, Y , . . . , Y N ) ∼ C ( N ) ( F, G, . . . , G ) . (16)We refer to Cossette et al. [2019] and Oh et al. [2020] for the natural linkage between (15)and the density function in (16). 12he copula-based CRM allows modeling both, the dependence among the individual severitiesand the dependence between frequency and severities. Hence, in terms of the dependencestructure, the copula-based CRM provides a wider range of dependence than the two partCRM. But while the main interest of insurance industry lies in the aggregate severity S ,the copula-based CRM does not allow an analytical interpretation of S , in general. For ananalysis of S one may consider to use the conditional distribution in (14). However, mainlydue to the non-continuous nature of the frequency N , an analytical interpretation of theconditional distribution in (14) or equivalently in (15) is difficult, in general. In the following, we provide an example where the proposed copula transformation methodallows an analytical interpretation of the conditional dependence among severities in (14)as well as the dependence between frequency and individual severities. The following modelis a modification of the previous copula-based CRM where the copula C is replaced by itstransformed version C α,F,C ( n ) . Model. 4.3.
For α ∈ (0 , transformed copula-based CRM for( N, Y , Y , . . . , Y N )is defined as follows:(i) We specify the frequency component N as N ∼ F (17)where F can be any discrete distribution function on N ; compare (9) and (13).(ii) We specify the conditional distribution of the vector of individual severities ( Y , . . . , Y n )conditional on the number of claims N = n ∈ N as( Y , . . . , Y n ) (cid:12)(cid:12) N = n ∼ W ∗ ( n ) (18)for some distribution function W ∗ ( n ) given by W ∗ ( n ) ( y , . . . , y n ) := C α,F,C ( n ) (cid:0) F ( n ) , G ( y ) , . . . , G ( y n ) (cid:1) − C α,F,C ( n ) (cid:0) F ( n − , G ( y ) , . . . , G ( y n ) (cid:1) P [ { N = n } ] (19)with C ( n ) being an absolutely continuous ( n + 1)-dimensional copula.Note that, due to Theorem 2.10, the conditional distribution function in (18) can be repre-sented as W ∗ ( n ) ( y , . . . , y n ) = ∂ C ( n ) (cid:0) F α ( n ) , G ( y ) , . . . , G ( y n ) (cid:1) and the corresponding density function at ( y , y , · · · , y n ) conditional on N = n ∈ N satisfies ∂ n ∂y . . . ∂y n W ∗ ( n ) ( y , . . . , y n ) = c ( n ) (cid:0) F α ( n ) , G ( y ) , · · · , G ( y n ) (cid:1) n Y i =1 g ( y i ) (20)13imilar to the case of copula-based CRM, the transformed copula-based CRM can be under-stood as ( N, Y , . . . , Y N ) ∼ C α,F,C ( N ) ( F, G, . . . , G ) (21)Now we are ready to provide an example which shows a link between two part CRM andcopula-based CRM. First, we define the following matrices for ρ , ρ ∈ [ − , k ∈ N and l ∈ { , } , define the k × k matrix Σ [ k,l ] ρ by letting h Σ [ k, ρ i i,j := ( , if i = j ; ρ , if i = j ;and h Σ [ k, ρ i i,j := ( , if i = j ; ρ | i − j | , if i = j. (ii) For k ∈ N and l ∈ { , } , define the ( k + 1) × ( k + 1) matrix Σ [ k,l ] ρ ,ρ by letting Σ [ k,l ] ρ ,ρ := ρ ( k ) T ρ k Σ [ k,l ] ρ ! , k ∈ N ;1 , k = 0;where k is a column vector with entries 1 of length k .As shown in Oh et al. [2020], the condition( ρ , ρ ) ∈ (cid:8) ( ρ , ρ ) ∈ ( − , (cid:12)(cid:12) ρ < ρ < (cid:9) (22)implies positive definiteness of the matrix Σ [ k, ρ ,ρ for every k ∈ N . Similarly, using the wellknown result on the Schur complement of a block matrix [Haynsworth, 1968], the matrix Σ [ k, ρ ,ρ is positive definite for any ρ , ρ ∈ ( − ,
1) satisfying1 − ρ ( k (1 − ρ ) + 2 ρ ) (1 − ρ ) > . (23)We denote by C (cid:0) · ; Σ [ n,l ] ρ ,ρ (cid:1) the ( n + 1)-dimensional Gaussian copula with correlation matrix Σ [ n,l ] ρ ,ρ and by c (cid:0) · ; Σ [ n,l ] ρ ,ρ (cid:1) its density.In the sequel, we consider Model 4.3 for ( N, Y , Y , · · · , Y N ) assuming a symmetric dependencestructure for the individual severities: Assumption. 4.4. • ρ and ρ satisfy condition (22) . • C ( n ) is an ( n + 1) -dimensional Gaussian copula with correlation matrix Σ [ n, ρ ,ρ . • G is a normal distribution with mean ξ and variance σ . In this case, the conditional density function of ( Y , Y , · · · , Y n ) conditional on N = n ∈ N satisfies ∂ n ∂y . . . ∂y n W ∗ ( n ) ( y , · · · , y n ) = c ( n ) (cid:16) F α ( n ) , Φ ξ,σ ( y ) , . . . , Φ ξ,σ ( y n ); Σ [ n, ρ ,ρ (cid:17) n Y i =1 φ ξ,σ ( y i )(24)14nd part i of Lemma A.1 shows that, for n ∈ N , ( Y , · · · , Y n ) conditional on N = n follows amultivariate normal distribution with mean (cid:0) ξ + σρ Φ − , ( F α ( n )) (cid:1) n and covariance matrix σ (cid:16) Σ [ n, ρ − ρ J n × n (cid:17) where J n × n denotes an n × n matrix with entries 1. By the convolutionary property of themultivariate normal distribution, we then obtain Y + · · · + Y n n (cid:12)(cid:12)(cid:12) N = n ∼ N (cid:0) µ n , σ n (cid:1) where µ n = ξ + σρ Φ − , ( F α ( n )) and σ n = 1 n σ (cid:0) ( n − ρ − nρ + 1 (cid:1) As a result, the distribution of S can be expressed in closed form which in turn allows aclosed form expression for subsequent statistics of S . Specifically, for s ≥
0, we have P [ { S ≤ s } ] = ∞ X n =0 P [ { S ≤ s }|{ N = n } ] P [ { N = n } ]= F (0) + ∞ X n =1 Φ (cid:18) s/n − µ n σ n (cid:19) P [ { N = n } ]Additionally, we obtain E [ S ] = E [ N E [ M | N ]] = ∞ X n =1 nµ n P [ { N = n } ]and Var [ S ] = E (cid:2) N Var [ M | N ] (cid:3) + Var [ N E [ M | N ]]= ∞ X n =1 (cid:0) n σ n + n µ n (cid:1) P [ { N = n } ] − ∞ X n =1 nµ n P [ { N = n } ] ! . Note that, while the dependence structure of the conditional severities in (12) under two partCRM is restricted to conditional independence, the dependence structure of the conditionalseverities in (19) under transformed copula-based CRM allows more general dependence struc-tures. In Lemma A.1, we also provide the condition where two part CRM and transformedcopula-based CRM are equivalent.Finally, we consider Model 4.3 for (
N, Y , Y , · · · , Y N ) assuming an autoregressive dependencestructure for the individual severities. Assumption. 4.5. • F is a discrete distribution function with finite support on N having κ ∈ N as theessential supremum of F . • k = κ satisfies (23) . • C ( n ) is an ( n +1) -dimensional Gaussian copula with correlation matrix Σ [ n, ρ ,ρ for n ≤ κ . G is a normal distribution with mean ξ and variance σ . Following the same procedure as above for n ∈ N yields Y + · · · + Y N n (cid:12)(cid:12)(cid:12) N = n ∼ N (cid:0) µ n , σ n (cid:1) (25)where µ n = ξ + σρ Φ − ( F α ( n )) and σ n = (cid:0) − nρ (cid:1) n + 2 n ρ − ρ (cid:0) ρ n − − (cid:1) . Again, the distribution of S can be expressed in closed form which in turn allows a closedform expression for subsequent statistics of S . Acknowledgements
Rosy Oh was supported by the Basic Science Research Program through the National Re-search Foundation of Korea (NRF) funded by the Ministry of Education (2019R1A6A1A11051177and 2020R1I1A1A01067376). Jae Youn Ahn was supported by an NRF grant funded by theKorean Government (2020R1F1A1A01061202).
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A Appendix
Lemma. A.1.
Consider α ∈ (0 , , a random vector ( N, Y , . . . , Y N ) from a transformedcopula-based CRM (Model 4.3) satisfying Assumption 4.4. Then, we have following proper-ties:i. For n ∈ N , consider a random vector ( Z , Z , · · · , Z n ) following a multivariate normaldistribution with mean (cid:0) , ξ, · · · , ξ (cid:1) T and covariance matrix diag (1 , σ, . . . , σ ) Σ [ n, ρ ,ρ diag (1 , σ, . . . , σ ) where J n × n denotes an n × n matrix with entries . Then, the conditional distributionof ( Y , · · · , Y n ) conditional on N = n equals the conditional distribution of ( Z , · · · , Z n ) conditional on Z = Φ − , ( F α ( n )) which satisfies ( Z , · · · , Z n ) | Z = Φ − , ( F α ( n )) ∼ MVN (cid:16)(cid:16) ξ + σρ Φ − , ( F α ( n )) (cid:17) n , σ (cid:16) Σ [ n, ρ − ρ J n × n (cid:17)(cid:17) (26) ii. Consider a random vector ( X, Y , · · · , Y X ) from a two part CRM (Model 4.1) and as-sume that X d = N shares the same distribution function F with N ∼ F . If we furtherassume ρ = ρ and that µ n and σ in (12) satisfying µ n = σρ Φ − ( F α ( n )) and σ = σ (cid:0) − ρ (cid:1) for X = n ∈ N , then the two random vectors ( N, Y , · · · , Y N ) and ( X, Y , · · · , Y X ) havethe same distribution. roof. We first prove part i. By Oh et al. [2020], the matrix Σ [ n, ρ ,ρ is positive definite.Furthermore, since the corresponding marginals and copula function of a multivariate nor-mal distribution are normal distributions and Gaussian copula, respectively, the conditionaldensity function of ( Z , . . . , Z n ) at point ( z , . . . , z n ) conditional on Z = z equals c ( n ) (cid:16) Φ , ( z ) , Φ ξ,σ ( z ) , . . . , Φ ξ,σ ( z n ) ; Σ [ n, ρ ,ρ (cid:17) n Y i =1 φ ξ,σ ( z i ) . (27)On the other hand, the conditional distribution of ( Z , · · · , Z n ) conditional on Z = z satisfies ( Z , · · · , Z n ) | Z = z ∼ MVN (cid:16) ( ξ + ρ σz ) n , σ (cid:16) Σ [ n, ρ − ρ J n × n (cid:17)(cid:17) (28)Since (27) and (28) describe the same distribution, the conditional distribution of ( Y , . . . , Y n )conditional on N = n and the conditional distribution of ( Z , . . . , Z n ) conditional on Z =Φ − , ( F α ( n )) coincide which follows from (24). This proves part i. The proof of part ii isimmediate from part i. (cid:3) D ( P, Q ) from P to Q under various parameter settings for Gaussiancopula (a) α = 0 . ζ . . . . . θ -0.951 4.532 1.932 0.998 0.149 0.072-0.454 0.124 0.053 0.028 0.004 0.002-0.156 0.012 0.005 0.003 0 00 0 0 0 0 00.156 0.012 0.005 0.003 0 00.454 0.124 0.054 0.028 0.004 0.0020.951 4.526 1.932 1.004 0.149 0.071(b) α = 0 . ζ . . . . . θ -0.951 3.142 1.422 0.731 0.089 0.042-0.454 0.087 0.039 0.020 0.002 0.001-0.156 0.008 0.004 0.002 0 00 0 0 0 0 00.156 0.008 0.004 0.002 0 00.454 0.086 0.039 0.020 0.002 0.0010.951 3.156 1.417 0.726 0.088 0.041(c) α = 0 . ζ . . . . . θ -0.951 4.962 2.246 1.153 0.153 0.072-0.454 0.135 0.062 0.031 0.004 0.002-0.156 0.013 0.006 0.003 0 00 0 0 0 0 00.156 0.013 0.006 0.003 0 00.454 0.136 0.062 0.032 0.004 0.0020.951 4.955 2.244 1.150 0.153 0.072(d) α = 1 . ζ . . . . . θ -0.951 13.068 4.369 2.174 0.334 0.161-0.454 0.359 0.119 0.060 0.009 0.004-0.156 0.034 0.012 0.006 0.001 00 0 0 0 0 00.156 0.035 0.012 0.006 0.001 00.454 0.358 0.120 0.060 0.009 0.0040.951 13.074 4.375 2.166 0.335 0.16220able 3: KL divergence D ( P, Q ) from P to Q under various parameter settings for Studentt copula (a) α = 0 . ζ . . . . . θ -0.951 2.559 1.450 0.878 0.165 0.081-0.454 0.139 0.069 0.040 0.006 0.003-0.156 0.026 0.017 0.012 0.002 0.0010 0 0 0 0 00.156 0.025 0.017 0.012 0.002 0.0010.454 0.138 0.069 0.040 0.006 0.0030.951 2.562 1.447 0.879 0.166 0.081(b) α = 0 . ζ . . . . . α = 0 . ζ . . . . . θ -0.951 2.472 1.419 0.893 0.174 0.082-0.454 0.151 0.086 0.051 0.007 0.003-0.156 0.028 0.023 0.016 0.002 0.0010 0 0 0 0 00.156 0.028 0.022 0.016 0.002 0.0010.454 0.151 0.085 0.050 0.006 0.0030.951 2.477 1.423 0.892 0.171 0.082(d) α = 1 . ζ . . . . . θ -0.951 4.576 2.454 1.568 0.361 0.181-0.454 0.328 0.146 0.085 0.013 0.006-0.156 0.057 0.033 0.024 0.004 0.0020 0 0 0 0 00.156 0.057 0.032 0.023 0.004 0.0020.454 0.327 0.146 0.085 0.013 0.0060.951 4.572 2.447 1.565 0.360 0.18121able 4: KL divergence D ( P, Q ) from P to Q under various parameter settings for Claytoncopula (a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θθ
Consider α ∈ (0 , , a random vector ( N, Y , . . . , Y N ) from a transformedcopula-based CRM (Model 4.3) satisfying Assumption 4.4. Then, we have following proper-ties:i. For n ∈ N , consider a random vector ( Z , Z , · · · , Z n ) following a multivariate normaldistribution with mean (cid:0) , ξ, · · · , ξ (cid:1) T and covariance matrix diag (1 , σ, . . . , σ ) Σ [ n, ρ ,ρ diag (1 , σ, . . . , σ ) where J n × n denotes an n × n matrix with entries . Then, the conditional distributionof ( Y , · · · , Y n ) conditional on N = n equals the conditional distribution of ( Z , · · · , Z n ) conditional on Z = Φ − , ( F α ( n )) which satisfies ( Z , · · · , Z n ) | Z = Φ − , ( F α ( n )) ∼ MVN (cid:16)(cid:16) ξ + σρ Φ − , ( F α ( n )) (cid:17) n , σ (cid:16) Σ [ n, ρ − ρ J n × n (cid:17)(cid:17) (26) ii. Consider a random vector ( X, Y , · · · , Y X ) from a two part CRM (Model 4.1) and as-sume that X d = N shares the same distribution function F with N ∼ F . If we furtherassume ρ = ρ and that µ n and σ in (12) satisfying µ n = σρ Φ − ( F α ( n )) and σ = σ (cid:0) − ρ (cid:1) for X = n ∈ N , then the two random vectors ( N, Y , · · · , Y N ) and ( X, Y , · · · , Y X ) havethe same distribution. roof. We first prove part i. By Oh et al. [2020], the matrix Σ [ n, ρ ,ρ is positive definite.Furthermore, since the corresponding marginals and copula function of a multivariate nor-mal distribution are normal distributions and Gaussian copula, respectively, the conditionaldensity function of ( Z , . . . , Z n ) at point ( z , . . . , z n ) conditional on Z = z equals c ( n ) (cid:16) Φ , ( z ) , Φ ξ,σ ( z ) , . . . , Φ ξ,σ ( z n ) ; Σ [ n, ρ ,ρ (cid:17) n Y i =1 φ ξ,σ ( z i ) . (27)On the other hand, the conditional distribution of ( Z , · · · , Z n ) conditional on Z = z satisfies ( Z , · · · , Z n ) | Z = z ∼ MVN (cid:16) ( ξ + ρ σz ) n , σ (cid:16) Σ [ n, ρ − ρ J n × n (cid:17)(cid:17) (28)Since (27) and (28) describe the same distribution, the conditional distribution of ( Y , . . . , Y n )conditional on N = n and the conditional distribution of ( Z , . . . , Z n ) conditional on Z =Φ − , ( F α ( n )) coincide which follows from (24). This proves part i. The proof of part ii isimmediate from part i. (cid:3) D ( P, Q ) from P to Q under various parameter settings for Gaussiancopula (a) α = 0 . ζ . . . . . θ -0.951 4.532 1.932 0.998 0.149 0.072-0.454 0.124 0.053 0.028 0.004 0.002-0.156 0.012 0.005 0.003 0 00 0 0 0 0 00.156 0.012 0.005 0.003 0 00.454 0.124 0.054 0.028 0.004 0.0020.951 4.526 1.932 1.004 0.149 0.071(b) α = 0 . ζ . . . . . θ -0.951 3.142 1.422 0.731 0.089 0.042-0.454 0.087 0.039 0.020 0.002 0.001-0.156 0.008 0.004 0.002 0 00 0 0 0 0 00.156 0.008 0.004 0.002 0 00.454 0.086 0.039 0.020 0.002 0.0010.951 3.156 1.417 0.726 0.088 0.041(c) α = 0 . ζ . . . . . θ -0.951 4.962 2.246 1.153 0.153 0.072-0.454 0.135 0.062 0.031 0.004 0.002-0.156 0.013 0.006 0.003 0 00 0 0 0 0 00.156 0.013 0.006 0.003 0 00.454 0.136 0.062 0.032 0.004 0.0020.951 4.955 2.244 1.150 0.153 0.072(d) α = 1 . ζ . . . . . θ -0.951 13.068 4.369 2.174 0.334 0.161-0.454 0.359 0.119 0.060 0.009 0.004-0.156 0.034 0.012 0.006 0.001 00 0 0 0 0 00.156 0.035 0.012 0.006 0.001 00.454 0.358 0.120 0.060 0.009 0.0040.951 13.074 4.375 2.166 0.335 0.16220able 3: KL divergence D ( P, Q ) from P to Q under various parameter settings for Studentt copula (a) α = 0 . ζ . . . . . θ -0.951 2.559 1.450 0.878 0.165 0.081-0.454 0.139 0.069 0.040 0.006 0.003-0.156 0.026 0.017 0.012 0.002 0.0010 0 0 0 0 00.156 0.025 0.017 0.012 0.002 0.0010.454 0.138 0.069 0.040 0.006 0.0030.951 2.562 1.447 0.879 0.166 0.081(b) α = 0 . ζ . . . . . α = 0 . ζ . . . . . θ -0.951 2.472 1.419 0.893 0.174 0.082-0.454 0.151 0.086 0.051 0.007 0.003-0.156 0.028 0.023 0.016 0.002 0.0010 0 0 0 0 00.156 0.028 0.022 0.016 0.002 0.0010.454 0.151 0.085 0.050 0.006 0.0030.951 2.477 1.423 0.892 0.171 0.082(d) α = 1 . ζ . . . . . θ -0.951 4.576 2.454 1.568 0.361 0.181-0.454 0.328 0.146 0.085 0.013 0.006-0.156 0.057 0.033 0.024 0.004 0.0020 0 0 0 0 00.156 0.057 0.032 0.023 0.004 0.0020.454 0.327 0.146 0.085 0.013 0.0060.951 4.572 2.447 1.565 0.360 0.18121able 4: KL divergence D ( P, Q ) from P to Q under various parameter settings for Claytoncopula (a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θθ D ( P, Q ) from P to Q under various parameter settings for Gumbelcopula (a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θθ
Consider α ∈ (0 , , a random vector ( N, Y , . . . , Y N ) from a transformedcopula-based CRM (Model 4.3) satisfying Assumption 4.4. Then, we have following proper-ties:i. For n ∈ N , consider a random vector ( Z , Z , · · · , Z n ) following a multivariate normaldistribution with mean (cid:0) , ξ, · · · , ξ (cid:1) T and covariance matrix diag (1 , σ, . . . , σ ) Σ [ n, ρ ,ρ diag (1 , σ, . . . , σ ) where J n × n denotes an n × n matrix with entries . Then, the conditional distributionof ( Y , · · · , Y n ) conditional on N = n equals the conditional distribution of ( Z , · · · , Z n ) conditional on Z = Φ − , ( F α ( n )) which satisfies ( Z , · · · , Z n ) | Z = Φ − , ( F α ( n )) ∼ MVN (cid:16)(cid:16) ξ + σρ Φ − , ( F α ( n )) (cid:17) n , σ (cid:16) Σ [ n, ρ − ρ J n × n (cid:17)(cid:17) (26) ii. Consider a random vector ( X, Y , · · · , Y X ) from a two part CRM (Model 4.1) and as-sume that X d = N shares the same distribution function F with N ∼ F . If we furtherassume ρ = ρ and that µ n and σ in (12) satisfying µ n = σρ Φ − ( F α ( n )) and σ = σ (cid:0) − ρ (cid:1) for X = n ∈ N , then the two random vectors ( N, Y , · · · , Y N ) and ( X, Y , · · · , Y X ) havethe same distribution. roof. We first prove part i. By Oh et al. [2020], the matrix Σ [ n, ρ ,ρ is positive definite.Furthermore, since the corresponding marginals and copula function of a multivariate nor-mal distribution are normal distributions and Gaussian copula, respectively, the conditionaldensity function of ( Z , . . . , Z n ) at point ( z , . . . , z n ) conditional on Z = z equals c ( n ) (cid:16) Φ , ( z ) , Φ ξ,σ ( z ) , . . . , Φ ξ,σ ( z n ) ; Σ [ n, ρ ,ρ (cid:17) n Y i =1 φ ξ,σ ( z i ) . (27)On the other hand, the conditional distribution of ( Z , · · · , Z n ) conditional on Z = z satisfies ( Z , · · · , Z n ) | Z = z ∼ MVN (cid:16) ( ξ + ρ σz ) n , σ (cid:16) Σ [ n, ρ − ρ J n × n (cid:17)(cid:17) (28)Since (27) and (28) describe the same distribution, the conditional distribution of ( Y , . . . , Y n )conditional on N = n and the conditional distribution of ( Z , . . . , Z n ) conditional on Z =Φ − , ( F α ( n )) coincide which follows from (24). This proves part i. The proof of part ii isimmediate from part i. (cid:3) D ( P, Q ) from P to Q under various parameter settings for Gaussiancopula (a) α = 0 . ζ . . . . . θ -0.951 4.532 1.932 0.998 0.149 0.072-0.454 0.124 0.053 0.028 0.004 0.002-0.156 0.012 0.005 0.003 0 00 0 0 0 0 00.156 0.012 0.005 0.003 0 00.454 0.124 0.054 0.028 0.004 0.0020.951 4.526 1.932 1.004 0.149 0.071(b) α = 0 . ζ . . . . . θ -0.951 3.142 1.422 0.731 0.089 0.042-0.454 0.087 0.039 0.020 0.002 0.001-0.156 0.008 0.004 0.002 0 00 0 0 0 0 00.156 0.008 0.004 0.002 0 00.454 0.086 0.039 0.020 0.002 0.0010.951 3.156 1.417 0.726 0.088 0.041(c) α = 0 . ζ . . . . . θ -0.951 4.962 2.246 1.153 0.153 0.072-0.454 0.135 0.062 0.031 0.004 0.002-0.156 0.013 0.006 0.003 0 00 0 0 0 0 00.156 0.013 0.006 0.003 0 00.454 0.136 0.062 0.032 0.004 0.0020.951 4.955 2.244 1.150 0.153 0.072(d) α = 1 . ζ . . . . . θ -0.951 13.068 4.369 2.174 0.334 0.161-0.454 0.359 0.119 0.060 0.009 0.004-0.156 0.034 0.012 0.006 0.001 00 0 0 0 0 00.156 0.035 0.012 0.006 0.001 00.454 0.358 0.120 0.060 0.009 0.0040.951 13.074 4.375 2.166 0.335 0.16220able 3: KL divergence D ( P, Q ) from P to Q under various parameter settings for Studentt copula (a) α = 0 . ζ . . . . . θ -0.951 2.559 1.450 0.878 0.165 0.081-0.454 0.139 0.069 0.040 0.006 0.003-0.156 0.026 0.017 0.012 0.002 0.0010 0 0 0 0 00.156 0.025 0.017 0.012 0.002 0.0010.454 0.138 0.069 0.040 0.006 0.0030.951 2.562 1.447 0.879 0.166 0.081(b) α = 0 . ζ . . . . . α = 0 . ζ . . . . . θ -0.951 2.472 1.419 0.893 0.174 0.082-0.454 0.151 0.086 0.051 0.007 0.003-0.156 0.028 0.023 0.016 0.002 0.0010 0 0 0 0 00.156 0.028 0.022 0.016 0.002 0.0010.454 0.151 0.085 0.050 0.006 0.0030.951 2.477 1.423 0.892 0.171 0.082(d) α = 1 . ζ . . . . . θ -0.951 4.576 2.454 1.568 0.361 0.181-0.454 0.328 0.146 0.085 0.013 0.006-0.156 0.057 0.033 0.024 0.004 0.0020 0 0 0 0 00.156 0.057 0.032 0.023 0.004 0.0020.454 0.327 0.146 0.085 0.013 0.0060.951 4.572 2.447 1.565 0.360 0.18121able 4: KL divergence D ( P, Q ) from P to Q under various parameter settings for Claytoncopula (a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θθ D ( P, Q ) from P to Q under various parameter settings for Gumbelcopula (a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θθ ρ ( P ) / ρ ( Q ) under various parameter settings for Gaussiancopula (a) α = 0 . ζ . . . . . θ -0.951 -0.946 / -0.259 -0.946 / -0.746 -0.946 / -0.884 -0.946 / -0.939 -0.946 / -0.943-0.454 -0.438 / -0.148 -0.437 / -0.358 -0.437 / -0.411 -0.439 / -0.438 -0.439 / -0.437-0.156 -0.150 / -0.053 -0.149 / -0.123 -0.150/ -0.147 -0.149 / -0.149 -0.151 / -0.1490 0 / -0.001 -0.001 / 0 -0.001 / 0 0 / 0.001 0.001 / 0.0060.156 0.150 / 0.053 0.148 / 0.118 0.147 / 0.143 0.149 / 0.145 0.149 / 0.1480.454 0.436 / 0.156 0.438 / 0.366 0.437 / 0.413 0.438 / 0.437 0.437 / 0.4370.951 0.946 / 0.258 0.947 / 0.747 0.947 / 0.881 0.946 / 0.934 0.946 / 0.945(b) α = 0 . ζ . . . . . θ -0.951 -0.946 / -0.256 -0.946 / -0.745 -0.946 / -0.876 -0.946 / -0.934 -0.947 / -0.942-0.454 -0.437 / -0.128 -0.438 / -0.304 -0.439 / -0.369 -0.437 / -0.428 -0.438 / -0.433-0.156 -0.150 / -0.041 -0.150 / -0.103 -0.148 / -0.121 -0.149 / -0.149 -0.149 / -0.1490 0 / 0 0.001 / 0 0.001 / -0.001 0 / -0.003 0 / -0.0040.156 0.149 / 0.038 0.149 / 0.104 0.149 / 0.121 0.148 / 0.145 0.149 / 0.1470.454 0.439 / 0.125 0.437 / 0.308 0.438 / 0.364 0.436 / 0.426 0.437 / 0.4350.951 0.946 / 0.256 0.946 / 0.749 0.946 / 0.881 0.946 / 0.938 0.946 / 0.944(c) α = 0 . ζ . . . . . θ -0.951 -0.946 / -0.26 -0.946 / -0.739 -0.946 / -0.872 -0.947 / -0.936 -0.946 / -0.939-0.454 -0.437 / -0.104 -0.437 / -0.280 -0.436 / -0.353 -0.436 / -0.425 -0.438 / -0.430-0.156 -0.150 / -0.035 -0.148 / -0.097 -0.150 / -0.116 -0.150/ -0.144 -0.149 / -0.1490 0 / 0.004 -0.002 / 0.003 -0.001 / 0.001 0.002 / -0.001 0.001 / 0.0030.156 0.148 / 0.034 0.150 / 0.087 0.149 / 0.116 0.148 / 0.144 0.150 / 0.1370.454 0.437 / 0.105 0.438 / 0.278 0.436 / 0.356 0.437 / 0.420 0.437 / 0.4340.951 0.947 / 0.263 0.946 / 0.737 0.946 / 0.871 0.946 / 0.933 0.947 / 0.943(d) α = 1 . ζ . . . . . θ -0.951 -0.946 / -0.259 -0.946 / -0.737 -0.947 / -0.873 -0.946 / -0.934 -0.946 / -0.942-0.454 -0.439 / -0.099 -0.438 / -0.271 -0.438 / -0.355 -0.438 / -0.425 -0.438 / -0.430-0.156 -0.152 / -0.027 -0.149 / -0.089 -0.151 / -0.114 -0.150 / -0.141 -0.148 / -0.1440 0.001 / 0.002 -0.001 / 0.003 -0.001 / -0.006 -0.001 / 0 0 / -0.0020.156 0.149 / 0.033 0.150 / 0.092 0.149 / 0.122 0.150 / 0.144 0.150 / 0.1480.454 0.437 / 0.099 0.437 / 0.276 0.437 / 0.348 0.437 / 0.423 0.438 / 0.4310.951 0.947 / 0.249 0.946 / 0.737 0.946 / 0.877 0.946 / 0.937 0.947 / 0.941 ρ ( P ) / ρ ( Q ) under various parameter settings for Student tcopula (a) α = 0 . ζ . . . . . θ -0.951 -0.942 / -0.258 -0.942 / -0.743 -0.942 / -0.879 -0.942 / -0.935 -0.942 / -0.938-0.454 -0.427 / -0.144 -0.427 / -0.353 -0.427 / -0.407 -0.427 / -0.425 -0.426 / -0.428-0.156 -0.144 / -0.050 -0.145 / -0.115 -0.144 / -0.136 -0.144 / -0.143 -0.148 / -0.1430 -0.001 / 0.001 -0.002 / -0.002 -0.002 / -0.005 0 / 0.004 0 / -0.0040.156 0.147 / 0.053 0.146 / 0.121 0.146 / 0.141 0.147 / 0.144 0.145 / 0.1460.454 0.428 / 0.145 0.428 / 0.352 0.426 / 0.407 0.429 / 0.429 0.427 / 0.4300.951 0.942 / 0.256 0.942 / 0.742 0.942 / 0.871 0.942 / 0.936 0.943 / 0.936(b) α = 0 . ζ . . . . . θ -0.951 -0.942 / -0.258 -0.942 / -0.739 -0.942 / -0.869 -0.942 / -0.930 -0.942 / -0.937-0.454 -0.426 / -0.122 -0.427 / -0.302 -0.426 / -0.364 -0.427 / -0.419 -0.425 / -0.425-0.156 -0.146 / -0.043 -0.146 / -0.106 -0.146 / -0.120 -0.145 / -0.144 -0.145 / -0.1410 0 / -0.002 0.001 / -0.003 -0.001 / 0 -0.001 / -0.004 0.001 / 0.0020.156 0.145 / 0.048 0.145 / 0.101 0.147 / 0.123 0.146 / 0.141 0.145 / 0.1410.454 0.426 / 0.125 0.426 / 0.304 0.427 / 0.366 0.427 / 0.418 0.427 / 0.4200.951 0.942 / 0.249 0.942 / 0.737 0.942 / 0.871 0.942 / 0.935 0.942 / 0.933(c) α = 0 . ζ . . . . . θ -0.951 -0.942 / -0.256 -0.942 / -0.731 -0.942 / -0.865 -0.942 / -0.932 -0.942 / -0.935-0.454 -0.428 / -0.117 -0.428 / -0.286 -0.426 / -0.351 -0.428 / -0.415 -0.428 / -0.421-0.156 -0.145 / -0.039 -0.147 / -0.096 -0.145 / -0.121 -0.145 / -0.144 -0.145 / -0.1450 0 / 0.001 0 / 0 0.001 / 0 0 / 0.004 0.001 / 0.0030.156 0.146 / 0.035 0.146 / 0.089 0.146 / 0.121 0.146 / 0.140 0.144 / 0.1460.454 0.429 / 0.111 0.429 / 0.277 0.428 / 0.352 0.429 / 0.411 0.427 / 0.4240.951 0.942 / 0.257 0.942 / 0.725 0.942 / 0.866 0.942 / 0.932 0.942 / 0.934(d) α = 1 . ζ . . . . . θ -0.951 -0.942 / -0.255 -0.942 / -0.730 -0.942 / -0.867 -0.942 / -0.930 -0.942 / -0.935-0.454 -0.428 / -0.108 -0.428 / -0.283 -0.428 / -0.355 -0.427 / -0.415 -0.428 / -0.417-0.156 -0.147 / -0.039 -0.146 / -0.095 -0.146 / -0.118 -0.146 / -0.141 -0.145 / -0.1360 0 / 0.001 0.001 / 0.001 -0.001 / 0.001 0 / 0.003 -0.002 / -0.0010.156 0.146 / 0.033 0.147 / 0.096 0.147 / 0.119 0.147 / 0.143 0.146 / 0.1430.454 0.428 / 0.115 0.428 / 0.282 0.427 / 0.354 0.426 / 0.416 0.427 / 0.4210.951 0.942 / 0.252 0.942 / 0.729 0.942 / 0.866 0.942 / 0.933 0.942 / 0.939 ρ ( P ) / ρ ( Q ) under various parameter settings for Claytoncopula (a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θ ρ ( P ) / ρ ( Q ) under various parameter settings for Gumbelcopula (a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θ D ( P, Q ) from P to Q under various parameter settings for Gaussiancopula of dimension 3(a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θ D ( P, Q ) from P to Q under various parameter settings for Claytoncopula of dimension 3 (a) α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 0 . ζ . . . . . θ α = 1 . ζ . . . . . θθ