Copula 1 - Modelling dependence through copulas Oana...

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Unformatted text preview: Modelling dependence through copulas Oana Purcaru∗ and Yuri Goegebeur∗∗ ∗ Institut de statistique & Institut des sciences actuarielles, ´ Universite catholique de Louvain ∗∗ Department of Applied Economics & University Center for Statistics, Katholieke Universiteit Leuven BELGIUM O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.1/138 Summary • Motivation • Normal copula • Bivariate copulas • Properties • Statistical inference • Archimedean copulas • Applications O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.2/138 Motivation Consider the Loss-ALAE data studied by F REES and VALDEZ (1998) and K LUGMAN and PARSA (1999). 1500 general liability claims : • Loss : indemnity payment ; • ALAE (allocated loss adjustment expense) : types of insurance company expenses that are specifically attributable to the settlement of individual claims (e.g. lawyers’ fees, claims investigation expenses). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.3/138 Motivation (ctd) 13 logALAE 9 5 1 2 6 10 14 logLoss → Estimate the joint distribution of ( Loss, ALAE ). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.4/138 Motivation (ctd) How to estimate joint distribution of (Loss, ALAE) ? • Marginal distributions • Correlation coefficient Except for some very specific classes of joint distributions, knowledge of marginal distributions and correlation coefficient is insufficient to construct the joint distribution. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.5/138 Motivation (ctd) x2 6 0 0 2 2 4 4 6 x2 8 8 10 10 12 12 14 Simulated data : 1000 samples from bivariate distributions with G(3, 1) marginals and τ = 0.25, but different dependence structures . 0 2 4 6 8 10 x1 Clayton copula 12 14 0 2 4 6 8 10 12 x1 Gumbel-Hougaard copula O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.6/138 Motivation (ctd) x2 0 0 2 2 4 4 6 x2 6 8 8 10 12 10 Simulated data : 1000 samples from bivariate distributions with G(3, 1) marginals and τ = 0.5, but different dependence structures . 0 2 4 6 8 x1 Clayton copula 10 0 2 4 6 8 10 12 x1 Gumbel-Hougaard copula O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.7/138 Motivation (ctd) x2 4 6 2 4 0 2 0 x2 6 8 8 10 10 12 Simulated data : 1000 samples from bivariate distributions with G(3, 1) marginals and τ = 0.75, but different dependence structures . 0 2 4 6 8 x1 Clayton copula 10 12 0 2 4 6 8 10 x1 Gumbel-Hougaard copula O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.8/138 Motivation (ctd) • Correlation and independence → independence implies zero correlation → zero correlation does not always imply independence → true for the multivariate normal distribution ⇒ Correlation coefficients do not contain enough information about the dependence structure to construct the joint distribution. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.9/138 Normal copula Cα (u1 , u2 ) = Hα Φ−1 (u1 ), Φ−1 (u2 ) , i.e. Φ= Hα = Cα (u1 , u2 ) = 1 α ∈ [−1, 1] univariate N (0, 1) cdf bivariate N2 (0, 1) cdf with correlation coefficient α √ 2π 1 − α2 Φ−1 (u1 ) Φ−1 (u2 ) exp ξ1 =−∞ ξ2 =−∞ 2 2 −(ξ1 − 2αξ1 ξ2 + ξ2 ) 2(1 − α2 ) ¡ where Normal copula : dξ1 dξ2 . ⇒ Variables with N (0, 1) marginal distributions and with joint cdf Cα (Φ(x1 ), Φ(x2 )) are standard bivariate normal variables with correlation coefficient α. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.10/138 Normal copula Increasing α increases the strength of dependence, in the sense that Cα (u) ≤ Cα (u) for all u ∈ [0, 1]2 whenever α ≤ α . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.11/138 Normal copula: derivation of the pdf cα (u1 , u2 ) = 1 √ exp 2 2π 1 − α 2 2 −(ζ1 − 2αζ1 ζ2 + ζ2 ) 2(1 − α2 ) ¡ The corresponding pdf is d −1 d −1 Φ (u1 ) Φ (u2 ) du1 du2 where ζi = Φ−1 (ui ), i = 1, 2. Denoting as ϕ = Φ the pdf of the N (0, 1) law, ¡ √ d −1 1 2 Φ (ui ) = = 2π exp(ζi /2) dui ϕ Φ−1 (ui ) cα (u1 , u2 ) = √ exp 2 1−α exp 2 2 ζ1 + ζ 2 2 ¡ 2 2 −(ζ1 − 2αζ1 ζ2 + ζ2 ) 2(1 − α2 ) ¡ 1 so that . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.12/138 Normal copula (Ctd) 6 (u_ c_a 4 2 a lph 0 c_a 0.5 1 a lph (u_ 1.5 1,u _2 ) 2 1,u _2 ) 8 2.5 10 • The pdf of the Normal copula, cα (u), for different values of Kendall’s τ (0.1 and 0.4): 0 1 0.8 1 0.8 1 0.6 0.8 u_ 0.4 2 0.6 0.2 0.4 u_1 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 u_1 0.2 0.2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.13/138 Normal copula (Ctd) _1 (u ha lp c_a 00 0 0 60 80 1 -20 0 2 4 ,u_ 2) h alp c_ 0 5 30 0 5 10 15 2 2 _ a(u 1,u _2 ) • The pdf of the Normal copula, cα (u), for different values of Kendall’s τ ( 0.7 and 0.9) : 1 1 0.8 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 0.2 u_1 0.8 1 0.6 0.8 u_ 0.4 2 0.6 0.2 0.4 u_1 0.2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.14/138 Normal copula with N (0, 1) margins 0 0 0.05 0.05 pdf 0.1 pdf 0.1 0.15 0.15 0.2 0.2 • The pdf of a random couple with N (0, 1) marginals and Normal copula, for τ = 0.1 and 0.4. 2 2 1 2 0 x_ 2 1 2 1 -1 -1 -2 -2 0 x_1 0 x_ 2 1 -1 -1 -2 0 x_1 -2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.15/138 Normal copula with N (0, 1) margins pdf 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 pdf 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 • The pdf of a random couple with N (0, 1) marginals and Normal copula, for τ = 0.7 and 0.9. 2 2 1 2 0 x_ 2 1 0 -1 -1 -2 -2 x_1 1 2 0 x_ 2 1 -1 -1 -2 0 x_1 -2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.16/138 Normal copula with N (0, 1) margins (Ctd) 2 1 0 x_2 -2 -1 0 -2 -1 x_2 1 2 • The image plots for the pdf of a random couple with N (0, 1) marginals and Normal copula, for τ = 0.1, 0.4, 0.7 and 0.9. -2 -1 0 1 2 -2 -1 1 2 1 2 2 1 -2 -1 0 x_2 1 0 -1 -2 x_2 0 x_1 2 x_1 -2 -1 0 x_1 1 2 -2 -1 0 x_1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.17/138 Normal copula with G(3, 1) margins 0 0.0 2 f pd 4 0.0 0.0 6 f pd 2 2 4 06 .08 .1 0.1 0 0.0 0.0 0. 0 0 0.0 8 • The pdf of a random couple with G(3, 1) marginals and Normal copula for τ = 0.1, 0.4, 0.7 and 0.9. 8 8 6 8 6 8 6 x_ 4 2 4 2 6 x_ 4 2 x_1 4 2 2 x_1 0 0.0 5 f pd 0.1 0.1 5 0.2 f pd 3 .4 .5 0.6 0 0.1 0.2 0. 0 0 2 8 8 6 8 6 x_ 4 2 4 2 2 x_1 6 8 6 x_ 4 2 4 2 x_1 2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.18/138 Normal copula with G(3, 1) margins (Ctd) 8 6 x_2 4 2 2 4 x_2 6 8 • The image plots for the pdf of a random couple with G(3, 1) marginals and Normal copula for τ = 0.1, 0.4, 0.7 and 0.9. 2 4 6 8 2 4 6 8 6 8 6 x_2 4 2 2 4 x_2 6 8 x_1 8 x_1 2 4 6 x_1 8 2 4 x_1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.19/138 Normal copula with G(3, 1) margins (Ctd) f pd 2 4 2 4 06 .08 .1 .1 0.1 0 0.0 0.0 0. 0 0 0 f pd 2 04 .06 .08 0.1 0 0.0 0. 0 0 • The pdf of a random couple with G(3, 1) marginals and Normal copula for τ = −0.4 and τ = −0.7. 8 8 6 8 6 6 x_2 4 4 2 8 6 x_2 4 x_1 4 2 2 x_1 8 6 x_2 4 2 2 4 x_2 6 8 2 2 4 6 x_1 8 2 4 6 8 x_1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.20/138 Bivariate distributions A function FX : R2 → [0, 1] is a bivariate distribution function if FX is a non-decreasing and right-continuous function satisfying : (i) limxi →−∞ FX (x) = 0, i = 1, 2, (ii) limxi →∞, i=1,2 FX (x) = 1, (iii) FX (v1 , v2 ) − FX (v1 , u2 ) − FX (u1 , v2 ) + FX (u1 , u2 ) ≥ 0 for any u1 ≤ v1 and u2 ≤ v2 . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.21/138 Bivariate copulas A copula is the restriction to the unit square [0, 1]2 of a joint cdf for a bivariate random vector with unit uniform marginals. or, equivalently, C : [0, 1] × [0, 1] → [0, 1] , non-decreasing and right-continuous satisfying: (i) limui →0 C(u1 , u2 ) = 0 for i = 1, 2; (ii) limu1 →1 C(u1 , u2 ) = u2 and limu2 →1 C(u1 , u2 ) = u1 ; (iii) C(v1 , v2 ) − C(u1 , v2 ) − C(v1 , u2 ) + C(u1 , u2 ) ≥ 0 for any u1 ≤ v1 , u 2 ≤ v2 . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.22/138 Sklar’s theorem Let us consider X = (X1 , X2 ) a random vector with marginals F1 (·), F2 (·) and bivariate cdf FX (·, ·). Then, there exists a copula C : [0, 1] × [0, 1] → [0, 1] such that FX (x1 , x2 ) = C F1 (x1 ), F2 (x2 ) , x = (x1 , x2 ) ∈ R2 . ( Sklar’s theorem separates FX (·, ·) into a part which describes the dependence structure (C(·, ·)) and parts which describe the marginal behaviors only (F1 (·) and F2 (·)) ) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.23/138 Sklar’s theorem (ctd) If the Fi (·)’s are continuous then C(·, ·) is unique and is given by −1 −1 C(u) = FX F1 (u1 ), F2 (u2 ) , u ∈ [0, 1]2 ; ( in this case, C(·, ·) is the joint cdf for F1 (X1 ), F2 (X2 ) .) Otherwise, C(·, ·) is uniquely determined on Range(F1 )×Range(F2 ). Henceforth, the marginal cdf’s are assumed to be continuous. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.24/138 Functional invariance • Let X1 , X2 be two continuous r.v. with copula C and h1 (·), h2 (·) : R → R be two strictly ↑ functions. ⇒ (h1 (X1 ), h2 (X2 )) has the same copula C. (The dependence structure is invariant under increasing transformations of the marginals.) The cdf of hi (Xi ), Gi (·) is given by: ¡ ¡ x ∈ , i = 1, 2. Gi (x) = Pr[hi (Xi ) ≤ x] = Fi h−1 (x) , i ¡ ⇒ Pr[X1 ≤ x1 , X2 ≤ x2 ] = F (x1 , x2 ) = C F1 (x1 ), F2 (x2 ) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.25/138 Functional invariance (Ctd) Pr[X1 ≤ h−1 (x1 ), X2 ≤ h−1 (x2 )] 1 2 C F1 h−1 (x1 ) , F2 h−1 (x2 ) 1 2 = C G1 (x1 ), G2 (x2 ) ¡ ¡ ¡ ¢ ¡ = = Pr h1 (X1 ) ≤ x1 , h2 (X2 ) ≤ x2 ] =⇒ (X1 , X2 ) and (Y1 , Y2 ) have the same copula C(·, ·) if, and only if, there exist 2 increasing functions h1 (·) and h2 (·) such that (Y1 , Y2 ) =d h1 (X1 ), h2 (X2 ) . =⇒ The dependence structure is entirely described by C(·, ·) and dissociated from the marginal cdf’s F1 (·) and F2 (·). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.26/138 Exercises Ex1 : Let X ∼ N2 (µ, Σ); derive the copula of X. Ex2 : Let X ∼ LN 2 (µ, Σ); derive the copula of X. Ex3 : Let X conform to the cdf FX (x) = F1 (x1 )F2 (x2 ) 1+ F 1 (x1 )F 2 (x2 ) , x ∈ R2 , ∈ [−1, 1]; derive the copula of X. It is referred to as the Morgenstern copula. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.27/138 Exercises - solutions Ex1 : Let X ∼ N2 (µ, Σ); derive the copula of X. Solution : Let Φ be the cdf associated to N2 (µ, Σ) Φ1 be the cdf associated to N1 (µ1 , Σ1 ) Φ2 be the cdf associated to N2 (µ2 , Σ2 ) By Sklar’s theorem we have that: C(u1 , u2 ) = Φ(Φ−1 (u1 ), Φ−1 (u2 )) = C ∗ (u1 , u2 ) 1 2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.28/138 = (X1 , X2 ) ∼ LN 2 ( , Σ) ⇒ (Y1 , Y2 ) = (lnX1 , lnX2 ) ∼ N2 ( , Σ). ¡ Let . ¡ Solution : ∼ LN 2 ( , Σ); derive the copula of ¡ Let Ex2 : Exercises - solutions (ctd) The marginals of (X1 , X2 ) are given by: Fi (xi ) = Pr[lnXi ≤ lnxi ] = Φi (lnxi ), Pr[X1 ≤ x1 , X2 ≤ x2 ] i = 1, 2 = Pr[Y1 ≤ lnx1 , Y2 ≤ lnx2 ] = Φ(lnx1 , lnx2 ) = Φ Φ−1 (Φ1 (lnx1 )), Φ−1 (Φ2 (lnx2 )) 1 2 = Φ Φ−1 (F1 (x1 )), Φ−1 (F2 (x2 )) 1 2 = C ∗ (F1 (x1 ), F2 (x2 )) ¢ ¡ ¢ C(F1 (x1 ), F2 (x2 )) ¡ = where C ∗ is the copula of (Y1 , Y2 ). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.29/138 Exercises - solutions(ctd) Ex3: Let X conform to the cdf FX (x) = F1 (x1 )F2 (x2 ) 1 + F 1 (x1 )F 2 (x2 ) , x ∈ R2 , ∈ [−1, 1]; derive the copula of X. It is referred to as the Morgenstern copula. Solution : C(u1 , u2 ) = u1 u2 {1 + (1 − u1 )(1 − u2 )} O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.30/138 Copulas CI , CU and CL • Independence copula CI : CI (u1 , u2 ) = u1 u2 , (u1 , u2 ) ∈ [0, 1]2 X1 and X2 are independent ⇔ C = CI . • Fréchet upper bound CU : CU (u1 , u2 ) = min{u1 , u2 }, (u1 , u2 ) ∈ [0, 1]2 X1 and X2 are perfectly positively dependent, i.e. X2 =a.s. h(X1 ) with h(·) ↑ ⇔ C = CU O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.31/138 Copulas CI , CU and CL (Ctd) • Fréchet lower bound CL : CL (u1 , u2 ) = max{u1 + u2 − 1, 0} X1 and X2 are perfectly negatively dependent, i.e. X2 =a.s. t(X1 ) with t(·) ↓ ⇔ C = CL =⇒ For any copula C(·, ·), CL (u1 , u2 ) ≤ C(u1 , u2 ) ≤ CU (u1 , u2 ) for all u ∈ [0, 1]2 . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.32/138 Copulas CI , CU and CL (Ctd) Example ( Normal copula ) : C1 = CU = min{u1 , u2 } (perfect positive dependence) C0 = CI = u 1 u 2 (independence) C−1 = CL = max{0, u1 + u2 − 1} (perfect negative dependence) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.33/138 Copulas CI , CU and CL (Ctd) 0.8 0.6 0 0.4 v 0.6 ula Independence cop 0.2 0.4 0.6 0.8 0.8 1 1.0 • The Independence copula CI 0.4 0.6 0.8 2 0.2 0.6 0.4 0.2 0.4 u_1 0.2 0.0 u_ 0.2 0.8 0.0 0.2 0.4 0.6 0.8 1.0 v O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.34/138 Copulas CI , CU and CL (Ctd) 0.8 0.6 0.4 0.2 0 v 0.6 0.4 1 nd Frechet upper bou 0.2 0.4 0.6 0.8 0.8 0.2 1.0 • The Fréchet upper bound CU 0.8 0.6 0.8 2 0.4 0.6 0.2 0.4 0.2 u_1 0.0 u_ 0.0 0.2 0.4 0.6 0.8 1.0 v O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.35/138 Copulas CI , CU and CL (Ctd) 0.8 0.6 v 0.6 nd Frechet lower bou 0.2 0.4 0.6 0.8 0.8 1 1.0 • The Fréchet lower bound CL 0 0.4 0.4 0.8 0.8 2 0.6 0.4 0.2 0.4 0.2 u_1 0.0 u_ 0.2 0.2 0.6 0.0 0.2 0.4 0.6 0.8 1.0 v O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.36/138 Copula and dependence structures Let X = (X1 , X2 ) be a random vector. Pr[X1 ≤ x1 , X2 ≤ x2 ] ≥ Pr[X1 ≤ x1 ] Pr[X2 ≤ x2 ], ∀x1 , x2 ∈ • X is Positive Quadrant Dependent (PQD) if Pr[X1 > x1 , X2 > x2 ] ≥ Pr[X1 > x1 ] Pr[X2 > x2 ], ∀x1 , x2 ∈ or, equivalently O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.37/138 Copula and dependence structures (ctd) ⇒ PQD is a property of the copula, that is, if C is a copula for X, X PQD ⇒ C PQD ⇔ C ≥ CI ; (the reciprocal implication holds true for continuous random vectors.) Example ( Normal copula ) : Cα is PQD for α ≥ 0 and NQD for α ≤ 0. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.38/138 Copula and dependence structures (ctd) • X2 is Stochastically Increasing in X1 (SI(X2 |X1 )) if Pr[X2 > x2 |X1 = x1 ] is a nondecreasing function of x1 , for all x2 . ⇒ If X has continuous marginals and copula C, then SI(X2 |X1 ) ∀u2 ∈ [0, 1], for almost all u1 , ⇔ ∂C(u1 , u2 ) ∂u1 is nonincreasing in u1 . ( for all u2 ∈ [0, 1], C(u1 , u2 ) is a concave function of u1 ) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.39/138 Copula and dependence structures (ctd) • X is Associated (A) if, for all ↑ g, h : R2 → R, E[g(X)h(X)] ≥ E[g(X)]E[h(X)] (⇔ XA ⇒ (F1 (X1 ), F2 (X2 ))A ⇒ Cov[g(X), h(X)] ≥ 0) (U1 , U2 ) with copula C A ( the reciprocal implication is true for continuous marginals F1 and F2 .) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.40/138 Copula and dependence structures (ctd) • X is Comonotonic if there exists a random variable Z and non-decreasing functions g1 and g2 such that X =d (g1 (Z), g2 (Z)). • X is Countermonotonic if there exists a random variable Z and non-decreasing and a decreasing function, g1 and g2 respectively, such that X =d (g1 (Z), g2 (Z)). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.41/138 Copulas and Kendall’s τ The Kendall’s correlation coefficient τ for a random vector X with cdf FX is defined by τ (X1 , X2 ) = Pr[(X1 − X1 )(X2 − X2 ) > 0] − Pr[(X1 − X1 )(X2 − X2 ) < 0] where (X1 , X2 ) and (X1 , X2 ) are independent realizations of FX . • Pr[(X1 − X1 )(X2 − X2 ) > 0] → probability of concordance • Pr[(X1 − X1 )(X2 − X2 ) < 0] → probability of discordance • −1 ≤ τ (X1 , X2 ) ≤ 1. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.42/138 Copulas and Kendall’s τ (ctd) Let X = (X1 , X2 ) be a r.v. with continuous marginals Fi and copula C. Kendall’s τ can be expressed as 1 1 τ (X1 , X2 ) = 4 u1 =0 u2 =0 C(u1 , u2 )dC(u1 , u2 ) − 1 = 4E[C(U1 , U2 )] − 1, where (U1 , U2 ) stands for a couple of rv’s uniformly distributed over [0, 1] with joint cdf C. Example ( Normal copula ) : τ = 2 π arcsin(α). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.43/138 Copulas and Spearman’s ρ The Spearman’s rank correlation coefficient ρ for a random vector X with joint cdf FX and marginals F1 and F2 is defined by ρ(X1 , X2 ) = E F1 (X1 ) − E[F1 (X1 )] F2 (X2 ) − E[F2 (X2 )] V ar[F1 (X1 )]V ar[F2 (X2 )] ⇒ ‘classical’ linear correlation between F1 (X1 ) and F2 (X2 ). • −1 ≤ ρ(X1 , X2 ) ≤ 1. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.44/138 Copulas and Spearman’s ρ (ctd) Let X = (X1 , X2 ) be a r.v. with continuous marginals Fi and copula C. 12 ¢ F1 (X1 )F2 (X2 ) − 3 1 = 12 u1 =0 1 = u2 =0 1 u2 =0 1 12 u1 =0 u1 u2 dC(u1 , u2 ) − 3 1 12 u1 =0 = 1 u2 =0 C(u1 , u2 )du1 du2 − 3 ¢ = ¡ ρ(X1 , X2 ) ¡ Spearman’s ρ can be expressed as C(u1 , u2 ) − u1 u2 du1 du2 . (ρ is a measure of "average distance" between the distribution of X 1 , X2 (represented by C) and independence (represented by the independence copula)) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.45/138 Copulas and Spearman’s ρ (ctd) Example ( Normal copula ) : ρ= α 6 arcsin( ). π 2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.46/138 Copulas and Pearson’s r The Pearson’s linear correlation coefficient r for a random vector X is defined by r(X1 , X2 ) = E X1 − E[X1 ] X2 − E[X2 ] V ar[X1 ]V ar[X2 ] • −1 ≤ r(X1 , X2 ) ≤ 1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.47/138 Copulas and Pearson’s r (ctd) Let X = (X1 , X2 ) be a r.v. with continuous marginals Fi and copula C. Pearson’s linear correlation can be expressed as: r(X1 , X2 ) ¡ ar[X1 ] ar[X2 ] (u1 ,u2 )∈[0,1]2 −1 −1 C(u1 , u2 ) − u1 u2 dF1 (u1 )dF2 (u1 ). £ ¢ 1 ¡ = • Pearson’s r does not only depend on the copula C but also on the marginals F1 and F2 ⇒ Pearson’s r is not an appropriate dependence measure. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.48/138 Conditional distribution and copulas Let (U1 , U2 ) be a random couple with joint cdf C, where U1 , U2 are uniformly distributed on the unit interval [0, 1]. • The conditional cdf of U2 given U1 = u1 is given by C2|1 (u2 |u1 ) C(u1 + ∆u1 , u2 ) − C(u1 , u2 ) ∆u1 = Pr[U2 ≤ u2 |U1 = u1 ] = = ∂ C(u1 , u2 ) ≡ C (1,0) (u1 , u2 ). ∂u1 lim ∆u1 →0 • Let C be a copula for X = (X1 , X2 ); then the conditional cdf of X2 given X1 = x1 is Pr[X2 ≤ x2 |X1 = x1 ] = F2|1 (x2 |x1 ) = C (1,0) (F1 (x1 ), F2 (x2 )). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.49/138 Pdf’s and copulas Assume that the marginal cdf’s F1 and F2 are continuous with respective pdf’s f1 and f2 . The joint pdf of X is then given by fX (x) = f1 (x1 )f2 (x2 )C (1,1) (F1 (x1 ), F2 (x2 )), where C (1,1) ∂2 (u1 , u2 ) = C(u1 , u2 ). ∂u1 ∂u2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.50/138 Statistical inference • We have at our disposal a bivariate random sample {xi = (xi1 , xi2 ), i = 1, . . . , n} of size n from a parent cdf FX . • Consider the model FX (x|α1 , α2 , θ) = Cθ (F1 (x1 |α1 ), F2 (x2 |α2 )) where F1 and F2 are univariate cdf’s with respective (vector) parameters α1 and α2 , and Cθ belongs to some family C(Θ) = {Cθ , θ ∈ Θ} of copulas parameterized by a (vector) parameter θ. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.51/138 Statistical inference (Ctd) Procedure : Step 1 : fit the 2 univariate marginal cdf’s F1 and F2 with the help of the observations {x11 , x21 , . . . , xn1 } and {x12 , x22 , . . . , xn2 } respectively. Let α1 and α2 be the corresponding MLE’s of α1 and α2 . Step 2 : estimate θ with the parameters α1 and α2 fixed at the estimated values from Step 1; let the result be θ. Step 3 : using α1 , α2 and θ as starting values, determine the global MLE’s α1 , α2 and θ of the parameters α1 , α2 and θ. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.52/138 Statistical inference (Ctd) • Several authors suggest to compare the values (α1 , α2 , θ) to (α1 , α2 , θ) as an estimation consistency check to evaluate the adequacy of the copula. • Note that for multivariate normal distribution with parameters 2 θ = R, the correlation matrix, and αj = (µj , σj ), j = 1, . . . , n, the marginal mean and variance, we have αj = αj and θ = θ. However, the equivalence of the estimators does not hold in general. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.53/138 Maximum likelihood estimation • Univariate case Consider a random variable X with cdf F (.|α) and let f (.|α) denote the probability / density function. For independent realizations X 1 , . . . , Xn from F (.|α) the likelihood function is given by n L(α) = i=1 f (Xi |α). ˆ The maximum likelihood estimator (MLE) α of α is defined as the value of α that maximizes the likelihood function. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.54/138 Maximum likelihood estimation - univariate case (ctd) • Likelihood contribution in case of deductible d: f (x|α) . 1 − F (d|α) • Likelihood contribution of grouped observation l ≤ X ≤ u: F (u|α) − F (l|α). e.g. right censoring: u = ∞ O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.55/138 Maximum likelihood estimation - univariate case (ctd) • Inference about α: limiting distribution of MLE. Under certain regularity conditions (Lehmann, 1998), for n → ∞, √ L n(ˆ − α) → N (0, I −1 (α)), α where I −1 (α) is the inverse Fisher information matrix whose i, j-th element is given by ∂ 2 log f (X|α) . Ii,j (α) = −E ∂αi ∂αj O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.56/138 Maximum likelihood estimation • Bivariate case Consider a random couple X = (X1 , X2 ) with bivariate cdf FX : FX (x|α1 , α2 , θ) = Cθ (F1 (x1 |α1 ), F2 (x2 |α2 )). Assume continuous marginal distributions, fX : joint density function, f1 , f2 : marginal density functions. For independent realizations (x11 , x12 ), . . . , (xn1 , xn2 ) from FX the likelihood function is given by f ( ¡ = i=1 n L(α1 , α2 , θ) i=1 i |α1 , α2 , θ). (1,1) Cθ = ¡ n (F1 (x1 |α1 ), F2 (x2 |α2 ))f1 (x1 |α1 )f2 (x2 |α2 ). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.57/138 Maximum likelihood estimation - bivariate case (ctd) • Likelihood contribution in case of deductibles d1 , d2 : fX (x|α1 , α2 , θ) = 1 − F1 (d1 |α1 ) − F2 (d2 |α2 ) + FX (d1 , d2 |α1 , α2 , θ) (1,1) Cθ (F1 (x1 |α1 ), F2 (x2 |α2 ))f1 (x1 |α1 )f2 (x2 |α2 ) 1 − F1 (d1 |α1 ) − F2 (d2 |α2 ) + Cθ (F1 (d1 |α1 ), F2 (d2 |α2)) . • Likelihood contribution in case of grouped X1 observation l1 ≤ X 1 ≤ u 1 : ∂FX (l1 , x2 |α1 , α2 , θ) ∂FX (u1 , x2 |α1 , α2 , θ) − = ∂x2 ∂x2 (0,1) Cθ (0,1) Cθ (F1 (u1 |α1 ), F2 (x2 |α2 ))f2 (x2 |α2 ) − (F1 (l1 |α1 ), F2 (x2 |α2 ))f2 (x2 |α2 ). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.58/138 Maximum likelihood estimation - bivariate case (ctd) • Likelihood contribution in case of grouped X1 and X2 observations l1 ≤ X1 ≤ u1 , l2 ≤ X2 ≤ u2 : FX (u1 , u2 |α1 , α2 , θ) − FX (u1 , l2 |α1 , α2 , θ)− FX (l1 , u2 |α1 , α2 , θ) + FX (l1 , l2 |α1 , α2 , θ) = Cθ (F1 (u1 |α1 ), F2 (u2 |α2)) − Cθ (F1 (u1 |α1 ), F2 (l2 |α2)) − Cθ (F1 (l1 |α1 ), F2 (u2 |α2)) + Cθ (F1 (l1 |α1 ), F2 (l2 |α2)). • Inference about model parameters: similar to univariate case. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.59/138 Omnibus procedure : to estimate the dependence parameter of a copula model from a random sample {xi , i = 1, . . . , n}, where xi = (xi1 , xi2 ) is a realization from the random couple X i with cdf FX , copula C and marginal cdf’s Fj , j = 1, 2. Goal : select specific forms for each of the Fi ’s (like Lognormal or Pareto, for instance) and maximize the joint likelihood in two steps. The simple approach ⇒ inappropriate choices for the marginals can affect appreciably the estimation of θ. ⇒ seek semi-parametric procedures for selecting the dependence parameter. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.60/138 Omnibus procedure (Ctd) The procedure : Let 1 (n) Fj (x) = n+1 n i=1 I[xij ≤ x] be the (rescaled) empirical distribution function corresponding to the jth component of the vector observations. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.61/138 Omnibus procedure (Ctd) The estimation θ of θ is the value that maximizes the pseudo-log-likelihood : n (n) (n) ln cθ F1 (xi1 ), F2 (xi2 ) i=1 where cθ is the pdf associated with Cθ (assuming it is strictly positive on [0, 1]2 ). The resulting estimator is consistent and asymptotically normally distributed. (for more details see e.g. OAKES (1994), S HIH AND L OUIS (1995), G ENEST AND W ERKER (2001).) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.62/138 Regression analysis ⇒ Copula : understand the joint distribution of several random variables. ⇒ Regression : identify one rv as measure of primary interest (dependent or response variable) and other variables as supporting or explaining (independent variables or covariates). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.63/138 Regression analysis (ctd) Consider a random vector X with joint cdf FX and continuous marginal distributions F1 , F2 . • the joint cdf: FX (x) = C(F1 (x1 ), F2 (x2 )) • the conditional cdf: F2|1 (x2 |x1 ) = C (1,0) (F1 (x1 ), F2 (x2 )). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.64/138 Regression analysis (ctd) • Classical regression - conditional mean : E[X2 |X1 = x1 ] = ∞ −∞ x2 dF2|1 (x2 |x1 ). • Quantile regression - conditional quantile function : given p, solve p = F2|1 (x2,p |x1 ), for x2,p . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.65/138 Simulation Given marginal distributions F1 , F2 and a copula C(u1 , u2 ), a unique joint distribution can be found, FX (x) = C(F1 (x1 ), F2 (x2 )). Simulation algorithm : 1. simulate u1 from U([0, 1]), 2. given u1 , simulate u2 from C2|1 (u2 |u1 ), −1 −1 3. transform (u1 , u2 ) into (x1 , x2 ) using (F1 (u1 ), F2 (u2 )) where F −1 (p) = inf{x|F (x) ≥ p}. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.66/138 Simulation (ctd) Example (Simulation from normal copula) : Cα (u1 , u2 ) = Hα (Φ−1 (u1 ), Φ−1 (u2 )). Simulation algorithm : 1. compute the Choleski decomposition of the covariance matrix V, i.e. construct a lower triangular matrix B such that V = BB , 2. generate independent standard normal random variables Z = (Z1 , Z2 ), 3. set Y = BZ, 4. set Ui = Φ(Yi ), i = 1, 2. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.67/138 Archimedean copulas: the idea • For independent U1 and U2 , Pr[U1 ≤ u1 , U2 ≤ u2 ] = CI ( ) = u1 u2 . ¡ • Let us “distort" the cdf’s involved in the latter equation: let ϕ : [0, 1] → [0, 1] be ↑ and continuous, such that ϕ(0) = 0 and ϕ(1) = 1, and assume that ϕ Pr[U1 ≤ u1 , U2 ≤ u2 ] = ϕ(u1 )ϕ(u2 ) ∈ [0, 1]2 . • If ϕ(t) = t (no distortion) then U1 and U2 are independent. ¡ • If we define φ(t) = − ln ϕ(t) for 0 < t ≤ 1 then φ Pr[U1 ≤ u1 , U2 ≤ u2 ] = φ(u1 ) + φ(u2 ). O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.68/138 Archimedean copulas: definition + • Consider a function φ : [0, 1] → R having continuous first and second derivative on (0,1) and satisfying φ(1) = 0, (1) (τ ) < 0 and φ(2) (τ ) > 0 for all τ ∈ (0, 1). ¡ ¢ Every such function φ generates a copula Cφ given by £ Cφ (u1 , u2 ) = φ−1 {φ(u1 ) + φ(u2 )} , 0, if φ(u1 ) + φ(u2 ) ≤ φ(0), otherwise; The copula Cφ is called an archimedean copula and the function φ is called the generator of the copula. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.69/138 Archimedean copulas (ctd) Basic properties : • symmetry : Cφ (u, v) = Cφ (v, u), for all u, v ∈ [0, 1]; • if c > 0 is any constant, then φ and cφ generate the same copula. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.70/138 Archimedean copulas (ctd) • A bivariate function FX with continuous margins is said to be generated by an archimedean copula x ∈ R2 ⇔ FX (x) = Cφ (F1 (x1 ), F2 (x2 )), for Cφ satisfying the conditions of the definition. • Examples of generators : Independence : Clayton : Frank : Gumbel-Hougaard : φ(t) = − ln(t) φα (t) = t−α − 1, α > 1 exp(−αt) − 1 , φα (t) = − ln exp(−α) − 1 φα (t) = [− ln(t)]α , α ≥ 1. α∈R O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.71/138 The frailty construction • Let Θ be a (frailty) non-negative random variable with pdf fΘ , cdf FΘ and Laplace transform LΘ . • Suppose that X1 and X2 are conditionally independent given the value of Θ. The joint distribution of X1 and X2 can be derived as follows: ∞ FX (x) = 0 ∞ = 0 ∞ = 0 Pr[X1 ≤ x1 , X2 ≤ x2 |Θ = θ]fΘ (θ)dθ Pr[X1 ≤ x1 |Θ = θ] Pr[X2 ≤ x2 |Θ = θ]fΘ (θ)dθ F1 (x1 |θ)F2 (x2 |θ)fΘ (θ)dθ = EΘ [F1 (x1 |Θ)F2 (x2 |Θ)]. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.72/138 The frailty construction (ctd) • Assume that X1 and X2 each obey to Pr[X1 ≤ x1 |Θ = θ] = B1 (x1 ) θ and Pr[X2 ≤ x2 |Θ = θ] = B2 (x2 ) θ for some baseline cdf’s B1 and B2 . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.73/138 The frailty construction (Ctd) In this case the marginal distributions are Fi (xi ) = Θ EΘ [Bi (xi )] = EΘ [exp(Θ ln Bi (xi ))] = LΘ (− ln Bi (xi )), and hence L−1 (Fi (xi )) = − ln Bi (xi ). Θ O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.74/138 The frailty construction (Ctd) Then, the joint cdf of X is FX (x) = E Pr[X1 ≤ x1 , X2 ≤ x2 |Θ] = E B1 (x1 ) Θ B2 (x2 ) Θ = LΘ − ln B1 (x1 ) − ln B2 (x2 ) = LΘ L−1 (F1 (x1 )) + L−1 (F2 (x2 )) . Θ Θ where LΘ is the Laplace transform of Θ. ⇒ Such FX is generated by an archimedean copula with φ−1 = LΘ and Fi (xi ) = LΘ (− ln Bi (xi )), i = 1, 2. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.75/138 The frailty construction (Ctd) Example ( Clayton copula ) : ⇒ The Clayton family is obtained when Θ is Gamma distributed with LΘ (t) = (1 + t)−1/α , α > 0 t > −1. φ−1 φ(u1 ) + φ(u2 ) −1+ u−α 2 = ¢ 1+ u−α 1 ¡ = ¢ = ¡ Cφ ( ) ¢ ¡ ⇒ In this case, φ(t) = t−α − 1, and u−α + u−α − 1 1 2 −1 −1/α −1/α where we recognize Clayton’s copula. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.76/138 Pdf’s of archimedean copulas ∂ Cφ ( ) = φ(1) (u1 ). ∂u1 φ(1) Cφ ( ) ¡ • Let us first differentiate with respect to u1 : • Substituting the expression obtained for yields ¡ ¡ ∂ ∂2 ∂ (1) Cφ ( ) Cφ ( ) Cφ ( ) + φ Cφ ( ) Cφ ( ) = 0. ∂u2 ∂u1 ∂u1 ∂u2 ∂ ∂u1 Cφ (u) and ∂ ∂u2 Cφ (u) ¡ ¡ ¡ φ(2) Cφ ( ) φ(1) (u1 )φ(1) (u2 ) ∂2 cφ ( ) = Cφ ( ) = − . 3 ∂u1 ∂u2 φ(1) Cφ ( ) φ (2) • Let us now differentiate the latter expression with respect to u2 to get O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.77/138 • Let Selecting φ: preliminary results be a random couple with uniform marginals and joint cdf C φ . • Let us define the rv’s V = φ(U1 ) φ(U1 ) + φ(U2 ) and Z = Cφ (U1 , U2 ), as well as the function λ(ξ) = φ(ξ) , 0 < ξ ≤ 1. φ(1) (ξ) • Then, (i) V is uniformly distributed on [0, 1]; (ii) the cdf FZ of Z is given by FZ (z) = z − λ(z); (iii) V and Z are independent rv’s. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.78/138 Nonparametric estimation of φ • Note that Kendall’s τ can be cast into 1 λ(ξ)dξ + 1. τ = 4 [Z] − 1 = 4 ξ=0 φ(z) = z − FZ (z) ⇒ φ(z) = exp φ(1) (z) z ξ=z0 ¡ • Given FZ , it is possible to recover φ by solving the differential equation 1 dξ λ(ξ) where 0 < z0 < 1 is an arbitrary choosen constant; φ given above is a bona fide generator if, and only if, FZ (z−) = lim FZ (t) > z t→z− for all 0 < z < 1. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.79/138 Nonparametric estimation of φ (Ctd) • Under the assumption that the dependence function associated with FX is archimedean, a natural estimator λn of λ can be derived from Fn through the relation λn (z) = z − Fn (z), 0 < z < 1. ¢ ¡ • Provided Fn (z−) > z for all z, z 1 φ(z) = exp ξ=z0 λn (ξ) dξ provides an estimator of Cφ within the class of archimedean copulas. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.80/138 Selecting φ (i) (i) Let {(x1 , x2 ), i = 1, 2, . . . , n} denote a sample of size n from a continuous bivariate distribution. Assume that the underlying distribution function FX has an associated archimedean copula Cφ . • The aim : identify φ. (i) (i) • The idea : work with the intermediate rv Zi = FX (X1 , X2 ) that has cdf FZ . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.81/138 Selecting φ • Step 1 : Estimate Kendall’s τ using the nonparametric estimator: τ = n 2 −1 (i) i<j (j) (i) (j) sign (x1 − x1 )(x2 − x2 ) . Fn (z) = where zi = O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.82/138 ¡ 1 (j) (j) (j) (i) (j) (i) # (x1 , x2 ) x1 < x1 , x2 < x2 ; n−1 1 #{i|zi ≤ z} n ¢¡ • Step 2 : Construct a nonparametric estimate of FZ as Selecting φ (Ctd) • Step 3 : Construct a parametric estimate of FZ : For various choices of φ use τ to estimate α and α to estimate φα ; then ¡ φα (z) Fφα (z) = z − . ¡ (1) φα (z) • Step 4 : Compare the parametric estimates to the nonparametric estimate from Step 2 and select φ so that the parametric estimate resembles the nonparametric one. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.83/138 Simulations • Simulating couples (u1 , u2 ) with joint cdf the archimedean copula Cφ : 1. generate t1 , t2 from U([0, 1]), independent; −1 2. determine z = FZ (t2 ); 3. take (u1 , u2 ), where : u1 = φ−1 (t1 φ(z)), u2 = φ−1 ((1 − t1 )φ(z)) (for more details see e.g. N ELSEN (1999)) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.84/138 Clayton copula • For α > 0, consider the copula Cα (u1 , u2 ) = u−α + u−α − 1 1 2 −1/α , u ∈ [0, 1]2 . • The pdf associated to Cα is given by ∂2 1+α cα (u) = Cα (u) = u−α + u−α − 1 2 ∂u1 ∂u2 (u1 u2 )α+1 1 1 −2− α . α • Kendall’s τ = α+2 ; it increases monotonically from 0 to 1 as α goes from 0 to ∞. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.85/138 Clayton copula (Ctd) • It can be shown that lim Cα (u1 , u2 ) = min{u1 , u2 } = CU (u) α→+∞ so that the dependency structure approaches its maximum (i.e. comonotonicty) when α increases to +∞. • On the other hand, lim Cα (u1 , u2 ) = u1 u2 = CI (u); α→0 hence the independence is obtained when α tends to 0. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.86/138 Clayton copula (Ctd) • The parameter α controls the amount of dependence since Cα (u) ≤ Cα (u) for all u ∈ [0, 1]2 whenever α ≤ α . • Consequently, for any α CI (u) ≤ Cα (u) ≤ CU (u) for all u ∈ [0, 1]2 whence it follows that Cα is PQD for all α. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.87/138 Clayton copula (Ctd) • The conditional cdf of U2 given U1 = u1 is given by ∂ Cα (u) = 1 + uα (u−α − 1) C2|1 (u2 |u1 ) = 1 2 ∂u1 1 −1− α . • If Cα is a copula for X with continuous margins, Pr[X2 ≤ x2 |X1 = x1 ] = C2|1 F2 (x2 )|F1 (x1 ) . ⇒ C2|1 (·|u1 ) can be inverted as −1 C2|1 (q|u1 ) = (q α − 1+α − 1)u−α 1 +1 1 −α . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.88/138 Clayton copula (Ctd) 10 0 f c_ pd 5 alp (u ha _1 15 2) ,u_ 20 • The pdf cα (u) for τ = 0.25. 1 0.8 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 u_1 0.2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.89/138 Clayton copula (Ctd) alp f c_ pd 0 0 0 0 10 20 3 4 5 (u ha _1 ,u_ 2) 60 • The pdf cα (u) for τ = 0.50. 1 0.8 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 u_1 0.2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.90/138 Clayton copula (Ctd) 10 0 pd alp f c_ 50 (u ha _1 0 2) ,u_ 15 0 • The pdf cα (u) for τ = 0.75. 1 0.8 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 u_1 0.2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.91/138 Clayton copula with N (0, 1) margins 0 0 0.0 0.0 5 f pd 0.1 0.1 5 f pd 5 5 0.1 0.15 0.2 0.2 0.2 • The pdf of a random couple with N (0, 1) marginals and Clayton copula for τ = 0.25, 0.5 and 0.75. 2 2 1 1 2 2 1 -1 -1 -2 0 x_2 0 x_1 1 0 -1 -1 -2 -2 x_1 -2 f pd 3 .4 .5 0.6 0.7 0 0.1 0.2 0. 0 0 0 x_2 2 1 2 0 x_2 1 -1 -1 -2 0 x_1 -2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.92/138 Clayton copula with N (0, 1) margins (Ctd) 2 1 1 2 • The contour plots for the pdf of a random couple with N (0, 1) marginals and Clayton copula for τ = 0.25, 0.5 and 0.75. 0.1 0.05 0 -1 0.2 -2 -1 0 x_2 0.2 -2 0 1 2 -2 -1 x_1 0 1 2 2 x_1 1 0.1 0.2 0.3 0.4 0 x_2 -1 0.5 0.6 0.6 0.6 0.6 0.6 -1 -2 0.5 0.4 0.3 -2 x_2 0.05 0.15 0.1 0.15 0.2 0.1 -2 -1 0 1 2 x_1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.93/138 Clayton copula with N (0, 1) margins (Ctd) 2 1 0 -1 -2 -2 -1 0 1 2 • The image plots for the pdf of a random couple with N (0, 1) marginals and Clayton copula for τ = 0.25, 0.5 and 0.75. 1 2 -2 -1 0 1 2 0 1 2 0 -1 -1 -2 -2 -2 -1 0 1 2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.94/138 Clayton copula with G(3, 1) margins 0 0.0 f pd 2 04 .06 .08 0.1 0 0.0 0. 0 0 f pd 5 5 0.1 0.15 0.2 0.2 • The pdf of a random couple with G(3, 1) marginals and Clayton copula for τ = 0.25, 0.5 and 0.75. 8 8 6 6 8 8 x_2 4 4 2 6 x_2 4 6 4 2 x_1 x_1 2 f pd 3 .4 .5 .6 0.7 0 0.1 0.2 0. 0 0 0 2 8 6 8 6 x_2 4 4 2 x_1 2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.95/138 Clayton copula with G(3, 1) margins (Ctd) 8 6 x_2 4 2 2 4 x_2 6 8 • The image plots for the pdf of a random couple with G(3, 1) marginals and Clayton copula for τ = 0.25, 0.5 and 0.75. 4 6 8 2 4 6 8 x_1 4 x_2 6 8 x_1 2 2 2 4 6 8 x_1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.96/138 Frank copula 1 (exp(−αu1 ) − 1)(exp(−αu2 ) − 1) ln 1 + α exp(−α) − 1 Cα (u1 , u2 ) = − ¡ • For α = 0, Frank’s copula is given by . ¡ exp − α(u1 + u2 ) − exp(−αu1 ) − exp(−αu2 ) + exp(−α) ¡ ¡ α exp − α(u1 + u2 ) 1 − exp(−α) cα ( ) = ¡ • The pdf is 2 . • There is no closed form for Kendall’s τ . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.97/138 Frank copula (Ctd) • We have that lim Cα = CL , α→−∞ lim Cα = CU α→+∞ and lim Cα = CI . α→0 • On the other hand, provided that α ≤ α Cα (u) ≤ Cα (u) for all u ∈ [0, 1]2 . ⇒ Hence, Cα is PQD when α ≥ 0 and NQD for α ≤ 0. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.98/138 Frank copula (Ctd) exp − α(u1 + u2 ) − exp(−αu1 ) ¡ ( )= C (1,0) ¡ • The conditional cdf of U2 given U1 = u1 is exp − α(u1 + u2 ) − exp(−αu1 ) − exp(−αu2 ) + exp(−α) . 1 1 − exp(−α) ln 1 − α 1 + (q −1 − 1) exp(−αu1 ) −1 C2|1 (q|u1 ) = − ¡ • C2|1 (·|u1 ) can be inverted as . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.99/138 Frank copula (Ctd) exp(−αu1 ) − 1 ¡ ¡ ¡ 1 − exp(−α) u1 exp(−αu1 ) + exp(−α) exp(−αu1 ) − 1 = (u1 , u2 ) du2 ¡ u2 =0 (1,0) 1 − Cα = [U2 |U1 = u1 ] ¡ 1 • Now, having a random couple U with joint cdf Cα , exp(−α) − exp(−αu1 ) . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.100/138 Frank copula (Ctd) 3 _1 lp c_a 2 1 (u ha 0 lp c_a .6 0.8 1 0 (u ha _1 ,u_ 2) 4 1.2 1. ,u_ 2) 4 1.6 5 • The pdf cα (u) for τ = 0.1 (α = 0.91) and τ = 0.4 (α = 4.16) 1 0.8 1 0.8 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 u_1 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 u_1 0.2 0.2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.101/138 Frank copula (Ctd) 20 lph c_a 5 10 15 1,u _ a(u 0 2 c_a 4 a lph 6 (u_ 1,u 8 _2 ) _2 ) 10 12 25 • The pdf cα (u) for τ = 0.7 (α = 11.4) and τ = 0.9 (α = 20.9). 0 1 1 0.8 0.8 1 0.6 0.8 u_ 0.4 2 0.6 0.2 0.4 u_1 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 u_1 0.2 0.2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.102/138 Frank copula (Ctd) 1,u 0 0 2 c_ 4 alp alp c_ 1 2 ha 6 _ (u (u ha _1 3 ,u_ _2 ) 2) 4 5 2 8 10 1 • The pdf cα (u) for τ = −0.4 and τ = −0.7: 1 1 0.8 0.8 1 0.6 u_ 2 0.4 0.6 0.4 0.2 1 0.6 0.8 0.8 u_ 2 0.4 u_1 0.6 0.4 0.2 u_1 0.6 0.8 1.0 0.2 0.0 0.2 0.4 u_2 0.0 0.2 0.4 u_2 0.6 0.8 1.0 0.2 0.0 0.2 0.4 0.6 u_1 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u_1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.103/138 Frank copula with N (0, 1) margins 0 0 0.0 0.0 5 f pd 0.1 0.1 5 f pd 5 5 0.1 0.15 0.2 0.2 0.2 • The pdf of a random couple with N (0, 1) marginals and Frank copula for τ = 0.1, 0.4, 0.7 and 0.9 . 2 2 1 1 2 0 x_2 1 -1 -1 1 0 -1 -1 x_1 -1 -2 -2 0 x_1 -2 f pd 4 .6 .8 0 0.2 0. 0 0 f pd 2 .3 .4 0.5 0 0.1 0. 0 0 1 -2 2 0 x_2 0 x_1 2 2 1 1 2 2 0 x_2 1 -1 -1 -2 -2 0 x_1 0 x_2 1 -1 -2 -2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.104/138 Frank copula with N (0, 1) margins (Ctd) 2 1 0 x_2 -2 -1 0 -2 -1 x_2 1 2 • The image plots for the pdf of a random couple with N (0, 1) marginals and Frank copula for τ = 0.1, 0.4, 0.7 and 0.9. -2 -1 0 1 2 -2 -1 1 2 1 2 2 1 -2 -1 0 x_2 1 0 -1 -2 x_2 0 x_1 2 x_1 -2 -1 0 x_1 1 2 -2 -1 0 x_1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.105/138 Frank copula (Ctd) 0 0.0 f pd 2 .3 .4 0.5 0 0.1 0. 0 0 f pd 5 5 0.1 0.15 0.2 0.2 • The pdf of Frank copula with N (0, 1) marginals for τ = −0.4 and τ = −0.7. 2 2 1 1 2 0 x_2 0 x_2 0 x_1 -1 -1 1 0 x_1 -1 -1 -2 -2 1 0 -1 -2 -2 -1 0 x_2 1 2 -2 2 -2 x_2 2 1 -2 -1 0 x_1 1 2 -2 -1 0 1 2 x_1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.106/138 Gumbel-Hougaard copula ¢ ¡ • For α ≥ 1, Gumbel-Hougaard copula is given by £ − (− ln(u1 ))α + (− ln(u2 ))α ¢ Cα (u1 , u2 ) = exp 1/α . • The pdf is ¡ ¢ (ln(u1 ) ln(u2 ))α−1 Cα (u1 , u2 ) cα ( ) = × u1 u2 × (− ln(u1 ))α + (− ln(u2 ))α ¡ ¡ (− ln(u1 ))α + (− ln(u2 ))α 1 2− α 1/α +α−1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.107/138 Gumbel-Hougaard copula (Ctd) • Kendall’s τ = 1 − α−1 , which increases monotonically from 0 to 1, as α goes from 1 to ∞. • It can be shown that lim Cα (u1 , u2 ) = min{u1 , u2 } = CU (u) α→+∞ so that the dependency structure approaches its maximum (i.e. comonotonicty) when α increases to +∞. • On the other hand, C1 (u1 , u2 ) = u1 u2 = CI (u); O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.108/138 Gumbel-Hougaard copula (Ctd) • The parameter α controls the amount of dependence since Cα (u) ≤ Cα (u) for all u ∈ [0, 1]2 whenever α ≤ α . • Consequently, for any α CI (u) ≤ Cα (u) ≤ CU (u) for all u ∈ [0, 1]2 whence it follows that Cα is PQD for all α. In fact Cα is only used to model positive dependence. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.109/138 Gumbel-Hougaard copula (Ctd) • the generator : φα (t) = [− ln(t)]α , • the FZ function: FZ (t) = t 1 − α ≥ 1. ln(t) α O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.110/138 Gumbel-Hougaard copula (Ctd) c_alpha(u_1,u_2) 05000 6000 0 100020003000 400 c_alpha(u_1,u_2) 8000 10000 0 2000 4000 6000 • The pdf of Gumbel-Hougaard copula for τ = 0.1, 0.4, 0.7 and 0.9. 1 1 0.8 0.8 1 0.6 u_ 2 0.4 0.4 0.8 u_ 0.6 2 0.4 u_1 0.6 0.2 0.2 0.4 u_1 0.2 c_alpha(u_1,u_2) 4000 5000 0 1000 2000 3000 c_alpha(u_1,u_2) 060007000 200 300 0 1000 0 04000500 0.2 1 0.6 0.8 1 1 0.8 0.8 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 0.2 u_1 1 0.6 0.8 u_ 2 0.4 0.6 0.2 0.4 u_1 0.2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.111/138 Gumbel-Hougaard copula (Ctd) 0 0 C_alpha(u_1,u_2) 5 10 15 20 C_alpha(u_1,u_2) 30 5 10 15 20 25 25 • The pdf of Gumbel-Hougaard copula with G(3, 1) margins, for τ = 0.1, 0.4, 0.7 and 0.9. 8 8 6 6 8 6 u_ 4 2 4 2 8 6 u_ 4 2 u_1 4 2 u_1 2 0 0 C_alpha(u_1,u_2) 15 10 5 20 C_alpha(u_1,u_2) 30 5 10 15 20 25 2 8 8 6 8 6 u_ 4 2 4 2 2 u_1 6 8 6 u_ 4 2 4 2 u_1 2 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.112/138 Application 1 : Reinsurance premiums • We consider the Loss-ALAE data studied by F REES and VALDEZ (1998) and K LUGMAN and PARSA (1999). (Provided by Insurance Services Office, Inc.) • The data : 1500 observed values of the pair (loss, ALAE), as well as a Policy Limit: loss = indemnity payment ALAE = allocated loss adjustment expense (expenses that are specifically attributable to the settlement of individual claims such as lawyers’ fees and claims investigation expenses.) Policy Limit = the maximal claim amount O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.113/138 Application 1 : Reinsurance premiums • Because of policy limits, some Loss variables were censored (since the amount of a claim cannot exceed the stated policy limit); claims that equaled the policy limit are then considered as censored. (Ti , ALAEi ), i = 1, . . . , n where Ti = min(lossi , Li ), 1, if loss > L → censored claim amount i i δ = I[T = L ] = i i i 0, if lossi ≤ Li → uncensored claim amount O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.114/138 Application 1 : Reinsurance premiums (ctd) Loss Loss Loss (uncensored) Total N ALAE (censored) 1,500 1,500 1,466 34 10 15 10 5,000 4,000 2,333 3,750 50,000 Mean 41,208 12,588 37,110 217,941 Median 12,000 5,471 11,049 100,000 3rd Qu. 35,000 12,577 32,000 300,000 2,173,595 501,863 2,173,595 1,000,000 102,748 28,146 92,513 258,205 Min 1st Qu. Max Std Dev. • Censorship cannot be ignored: the mean Loss of censored claims was 217,491$, whereas the corresponding mean for uncensored claims was 37,110$. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.115/138 Application 1 : Reinsurance premiums (ctd) 13 logALAE 9 5 1 2 6 10 14 logLoss O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.116/138 Application 1 : Reinsurance premiums (ctd) • Marginal distibutions : Pareto Pr[Loss ≤ x1 ] = 1 − λ1 λ 1 + x1 Pr[ALAE ≤ x2 ] = 1 − λ2 λ 2 + x2 θ1 and θ2 are estimated via ML, taking censorship into account. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.117/138 0.6 0.4 0.2 0.0 Survival function 0.8 1.0 Application 1 : Reinsurance premiums (ctd) 0.01 0.10 1.00 10.00 100.00 ALAE O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.118/138 0.6 0.4 0.2 0.0 Survival function 0.8 1.0 Application 1 : Reinsurance premiums (ctd) 0.01 0.10 1.00 10.00 100.00 1000.00 LOSS O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.119/138 Application 1 : Reinsurance premiums (ctd) • Copula : Frank (partly due to its convenient analytical properties). • If the Loss variable is not censored then the contribution of the observation x to the likelihood is ¡ f1 (x1 )f2 (x2 )C (1,1) F1 (x1 ), F2 (x2 ) . ¡ f2 (x2 ) 1 − C (0,1) F1 (x1 ), F2 (x2 ) ¡ • If the Loss variable is censored then the contribution of the observation x to the likelihood is . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.120/138 Application 1 : Reinsurance premiums (ctd) Parameter Dependence λ1 14,453 14,558 1.135 1.115 λ2 15,133 16,678 θ2 ALAE Bivariate θ1 Loss Univariate 2.223 2.309 α - 3.158 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.121/138 Application 1 : Reinsurance premiums (ctd) cdf 0.6 0.8 1.0 • To have an idea of the behavior of ALAE for some given Loss level, the next figure displays the graph of x2 → Pr[ALAE ≤ x2 |Loss]: 0.0 0.2 0.4 Loss=10 000 Loss=100 000 Loss=1 000 000 0 50000 100000 150000 200000 ALAE O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.122/138 Application 1 : Reinsurance premiums (ctd) 50000 100000 Loss=10 000 Loss=100 000 Loss=1 000 000 0 quantile 150000 200000 • Another interesting graph is the quantile function of ALAE for some given Loss level: 0.0 0.2 0.4 0.6 0.8 1.0 q O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.123/138 Application 1 : Reinsurance premiums (ctd) 95th percentile 75th 50th 25th 5th 8 6 log(ALAE) 10 12 • We also provide the quantile regression curves (i.e. the qth quantiles of ALAE for some given Loss level): 8 10 12 14 log(Loss) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.124/138 Application 1 : Reinsurance premiums (ctd) ¡ ¢ ¡ Assume pro-rata sharing of expenses and let £ ¢ g(loss, ALAE) = 0 , if loss < R loss − R + LR − R + loss−R ALAE loss LR−R ALAE LR , if R ≤ loss < LR , if loss ≥ LR. be the reinsurer’s payment on a policy with limit LR and insurer’s retention R. Of interest ⇒ the expected payment : E[g(loss, ALAE)] O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.125/138 Application 1 : Reinsurance premiums (ctd) lossi Li LR reinsurer R ( insurer ’s indemnity limit ) ( reinsurer ’s intervention limit ) ( insurer ’s retention ) O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.126/138 Application 1 : Reinsurance premiums (ctd) • Computing E[g(loss, ALAE)] = π ⇒ by numerical integration, if the joint density is available : g(x1 , x2 )f1 (x1 )f2 (x2 )cφ F1 (x1 ), F2 (x2 ) dx1 dx2 . π= x1 ∈R+ x2 ∈R+ O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.127/138 Application 1 : Reinsurance premiums (ctd) • Computing E[g(loss, ALAE)] = π ¢ £ ¢ λ1 (1 − u1 )−1/θ1 − 1 , λ2 (1 − u2 )−1/θ2 − 1 £ ¡ ¢ ⇒ by simulation : generate a large number nsim (100,000, say) of realizations u from Frank’s copula and then build realizations x of (Loss, ALAE) as . The estimated value for the reinsurer’s expected payment is 1 π= nsim nsim g(x1i , x2i ). i=1 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.128/138 Application 2 : Multilife insurance premiums Ages at death of 533 couples buried in two random cemeteries in Brussels. (see D ENUIT & all (2001)) • Descriptive statistics: “Age at death/Man" “Age at death/Woman" Mean 73.083 78.340 Mode 81.000 81.000 Std. deviation 12.268 11.074 Std. error 0.531 0.480 Minimum 24.000 22.000 Maximum 98.000 103.000 Skewness -0.752 -0.949 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.129/138 Application 2 : Multilife insurance premiums (Ctd) • Dependence measures: Spearman’s rank Kendall’s rank correlation correlation Rank correlation 0,139 0,092 Z-value 3,199 3,169 p-value 0,0014 0,0015 corrected for ties 0,138 0,094 Tied Z-value 3,180 3,254 Tied p-value 0,0015 0,0011 Rank correlation O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.130/138 Application 2 : Multilife insurance premiums (Ctd) 0.6 0.4 0.2 empirical distribution Cook Frank Gumbel 0.0 cumulative distribution function 0.8 1.0 • Graphical procedure for selecting the appropriate archimedean copula: 0.0 0.2 0.4 0.6 0.8 1.0 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.131/138 Application 2 : Multilife insurance premiums (Ctd) ¡ 1 Cα ( ) = exp − {(− ln u1 )α + (− ln u2 )α } α ¢ • The Gumbel’s copula was selected : . • The parameter α involved in the Gumbel model is estimated via τ ˆ ML, starting from the following initial value 1−ˆ = 0, 104 ⇒ τ α = 0.1015378. t qxy ¡ α ˆ = Cφα (t qx , t qy ) = exp − (− ln t qx ) + (− ln t qy ) ˆ α ˆ 1 α ˆ ¡ • The probability that both remaining lifetimes Tx and Ty fail before t can be estimated by . O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.132/138 Application 2 : Multilife insurance premiums (Ctd) • Probabilities t pxy and t pxy can then be estimated by t pxy and t pxy ˆ ˆ given by ˆ t pxy = 1 − t q x − t qy + Cφα (t qx , t qy ), t ∈ R+ , ˆ and ˆ t pxy = 1 − Cφα (t qx , t qy ), t ∈ R+ . ˆ O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.133/138 Application 2 : Multilife insurance premiums (Ctd) 9.5 11.0 12.5 14.0 15.5 17.0 independence minimum maximum Gumbel 8.0 n-year joint-life annuities • For joint-life annuities axy;n| : 10 15 20 25 30 35 40 45 50 55 60 45 50 55 60 x = y = 40 12.0 7.5 8.5 9.5 10.5 n-year joint-life annuities 13.5 15.0 n independence minimum maximum Gumbel 10 15 20 25 30 35 n 40 x = y = 50 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.134/138 Application 2 : Multilife insurance premiums (Ctd) 18 16 14 12 independence maximum minimum Gumbel 8 10 n-year last-survivor annuities 20 • For last-survivor annuities axy;n| : 10 20 30 40 50 60 50 60 x = y = 40 16 14 12 10 independence maximum minimum Gumbel 8 n-year last-survivor annuities 18 n 10 20 30 40 n x = y = 50 O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.135/138 Application 2 : Multilife insurance premiums (Ctd) 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Widow’s pension • For widow’s pensions ax|y : independence minimum maximum Gumbel 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 age x=y O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.136/138 References • D ENUIT, M., D HAENE , J., L E B AILLY DE T ILLEGHEM , C. and T EGHEM , S.(2001) Measuring the impact of a dependence among insured lifelenghts. Belgian Actuarial Bulletin 1, 19-39 • F REES, E.W., and VALDEZ , E.A. (1998) Understanding relationships using copulas. North American Actuarial Journal 2, 1-25. • G ENEST, C. and M AC K AY, J. (1986) The Joy of Copulas : Bivariate Distributions with Uniform Marginals. The American Statistician 40, 4, 280-283. • G ENEST, C. and R IVEST, L. (1993) Statistical inference procedures for bivariate Archimedean copulas Journal of the American Statistical Association 88, 1034-1043 • G ENEST, C. and W ERKER , B.J.M. (2001) Conditions for the asymptotic semiparametric efficiency of an omnibus estimator of dependence parameters in copula models. Distributions With Given Marginals and Statistical Modelling, Kluwer, 103-112. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.137/138 References • J OE , H. (1997) Multivariate Models and Dependence Concepts. Chapman & Hall, London. • K LUGMAN , S.A. and PARSA , R. (1999) Fitting bivariate loss distributions with copulas. Insurance: Mathematics and Economics 24, 139-148. • L EHMANN , E.L. and C ASELLA , G. (1998) Theory of Point Estimation. Springer. • N ELSEN , R.B.(1999) An Introduction to Copulas. Springer-Verlag, New York. • OAKES, D.(1994) Multivariate Survival Distributions. Journal of Nonparametric Statistics 3, 343-354. • S HIH , J.H. and L OUIS, T.A.(1995) Inferences on the Association Parameter in Copula Models for Bivariate Survival Data Biometrics 51, 1384-1399. O. Purcaru & Y. Goegebeur - Ubatuba, Sept 2003 – p.138/138 ...
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