A semiparametric estimation of copula models based on the method of moments
aa r X i v : . [ s t a t . M E ] J a n A semiparametric estimation of copula models basedon the method of moments
Brahim Brahimi, Abdelhakim Necir ∗ Laboratory of Applied Mathematics, Mohamed Khider University of Biskra,Biskra 07000, Algeria
Abstract
Using the classical estimation method of moments, we propose a new semiparametric estima-tion procedure for multi-parameter copula models. Consistency and asymptotic normality ofthe obtained estimators are established. By considering an Archimedean copula model, anextensive simulation study, comparing these estimators with the pseudo maximum likelihood,rho-inversion and tau-inversion ones, is carried out. We show that, with regards to the othermethods, the moment based estimation is quick and simple to use with reasonable bias androot mean squared error.
MSC classification:
Primary 62G05; Secondary 62G20.
Keywords:
Archimedean copulas; Asymptotic distribution; Copula models; Measures of as-sociation; Method of moments; Semiparametric models; Statistical inference; Z-estimator.
Recently, considerable attention has been paid to the problem of inference about copulas. Themonographs of Cherubini et al. (2004), Nelsen (2006) and Joe (1997) summarize to some extent theactivities in this area. Roughly speaking, a copula function is a multivariate distribution functionwith uniform margins. It is used as a linking block between the joint distribution function (df) F of a vector of random variables X = ( X , ..., X d ) and its marginal df’s F , ..., F d . This probabilisticinterpretation of copulas is justified by the famous Sklar’s theorem (Sklar, 1959) which states that,under some mild conditions, there exists a unique copula function C, such that F ( x , ..., x d ) = C ( F ( x ) , .., F d ( x d )) . ∗ Corresponding author: [email protected]
1n other words, the copula C is the joint df of the random vector U = ( U , ..., U d ) , with U j = F j ( X j ) . That is, for u = ( u , ..., u d ) , we have C ( u ) = F (cid:0) F − ( u ) , ..., F − d ( u d ) (cid:1) , where F − j ( s ) := inf { x : F j ( x ) ≥ s } denotes the generalized inverse function (or the quantilefunction) of F j . A parametric Archimedean copula model arises for X when the copula C belong to a class C := { C θ , θ ∈ O} , where O is an open subset of R r for some integer r ≥ . Statistical inferenceon the dependence parameter θ is one of the main topics in multivariate statistical analysis.Several methods of copula parameter estimation have been developed, including the methodsof concordance (Oakes, 1982, Genest, 1987), fully maximum likelihood (ML), pseudo maximumlikelihood (PML) (Genest et al. , 1995), inference function of margins (IFM) (Joe, 1997, 2005),and minimum distance (MD) (Tsukahara, 2005). The performance of the PML procedure vis-a-visto the other methods has been discussed by several authors. For example, the simulation studycarried out by Kim et al . (2007) has concluded that the PML method is conceptually almost thesame as the IFM one. It overcomes its non robustness against misspecification of the marginaldistributions. Moreover, by using the PML method, one would not lose any important statisticalinsights that would be gained by applying the IFM. An advantage of the PML over the IFM is thatthe former does not require modeling the marginal distributions explicitly. Therefore, the PMLestimator is better than those of the ML and IFM in most practical situations. However, in time-consuming point of view the ML, PML, IFM and MD methods require intensive computations,notably when the copula dimension increases. Moreover, when using these methods the copuladensity has to be involved, therefore a serious inaccuracy at boundary points arises. Severalnumerical methods are proposed to solve this problem, but they are still inefficient when dealingwith high dimensional copula models, more precisely for d > τ -inversion and ρ -inversion, which are based, respectively, on Kendall’s τ and Spear-man’s ρ rank correlation coefficients, used to estimate parametric copula models with at more twoparameters. Indeed, the τ -inversion and ρ -inversion methods use the functional representations2f τ and ρ in terms of the underling copula C (Schmid et al. , 2010), given by τ = τ ( C ) = 12 d − − ( d Z [0 , d C ( u ) dC ( u ) − ) ,ρ = ρ ( C ) = d + 12 d − ( d + 1) ( d Z [0 , d C ( u ) d u − ) . More precisely, suppose that copula C is a parametric model, i.e. C = C θ , then both τ and ρ become functions in θ as well, that is τ = τ ( θ ) and ρ = ρ ( θ ) . Let b τ and b ρ be, respectively,empirical versions of τ and ρ pertaining to the sample ( X , ..., X n ) from the random vector X andsuppose that C θ is one-parameter copula model (i.e. r = 1) . Then, the estimators of θ obtainedby τ -inversion or ρ -inversion methods are defined by b θ := τ − ( b τ ) or b θ := ρ − ( b ρ ) , where τ − and ρ − are the inverses, if they exist, of functions θ → τ ( θ ) and θ → ρ ( θ ) respectively. In the casewhen r = 2 , that is when θ = ( θ , θ ) , we have to use jointly the two inversion methods, called( τ , ρ )-inversion, to have a system of two equations τ ( θ , θ ) = b τ , ρ ( θ , θ ) = b ρ. (1)The consistency of such estimators is discussed in the Appendix section A.3. In conclusion, whenthe dimension of parameter θ equals r, we have to use r measures of association, for exampleBlomqvist’s beta β, Gini’s gamma γ, ... (see, Nelsen, 2006, page, 207) which, in general, isnot convenient on the choice of measures point of view. More precisely, suppose that we aredealing with a parameter θ = ( θ , θ ) of a copula model C θ , then one has the right to ask thefollowing question: What couple among all measures of association have to be chosen to geta better estimation for θ ? On the other hand, it is worth mentioning that often there existdifficulties while using Spearman’s rank correlation coefficient. One such difficulty is when usingvery large or very small samples. For example, in the case of very large samples, it is very timeconsuming to perform Spearman’s coefficient since it requires ranking of the data of all variables.Then we have to look for an alternative more convenient class of measures providing estimatorswith nice properties. A solution to this problem may be given by applying the classical methodof moments to random variable (rv) C ( U ) . Indeed, let us define the k th-moment M k ( C ) , called copula moment, of rv C ( U ) as the expectation of ( C ( U )) k , that is M k ( C ) := E h ( C ( U )) k i = Z [0 , d ( C ( u )) k dC ( u ) , k = 1 , , ... (2)Notice that the case k = 1 corresponds to M ( C ) = E [ C ( U )] = (cid:0) d − − (cid:1) τ + 12 d .
3n other words, M k ( C ) may be considered as a generalization of Kendall’s rank correlation τ . Toour knowledge, the method of moments is only used in one-parameter copula models, also knownby the τ -inversion method (see for instance, Tsukahara, 2005). Note that, since 0 ≤ C ( u ) ≤ , then M k ( C ) are finite for every integer k. Now we are in position to present a new estimationmethod that we call copula moment (CM) estimation . Suppose that, for unknown parameter θ ∈ O ⊂ R r , we have C = C θ , then M k ( C ) = M k ( θ ) , where M k ( θ ) := Z [0 , d ( C θ ( u )) k dC θ ( u ) , k = 1 , , ... (3)From equations (3) , we may consider M : θ → ( M ( θ ) , ..., M r ( θ )) as a mapping from O ⊂ R r to R r , that will be used as a means to estimate the parameter θ . More precisely, for a givensample ( X , ..., X n ) of the random vector X , let us denote b θ CM as the estimator of θ defined by( M k ) ≤ k ≤ r . That is b θ CM := M − (cid:16) c M , ..., c M r (cid:17) , (4)where c M k is the empirical version of M k ( C ) and M − is the inverse of the mapping M , providedthat it exists. The rest of the paper is organized as follows. In Section 2, we present the mainsteps of the copula moment estimation procedure and establish the consistency and asymptoticnormality of the proposed estimator. In Section 3, an application to multiparameter Archimedeancopula models is given. In Section 4, an extensive simulation study is carried out to evaluate andcompare the CM based estimation with the PML and ( τ , ρ )-inversion methods. Comments andconclusion are given in Section 5. The proofs are relegated to the Appendix. In this section we present a semiparametric estimation procedure for the copula models based onthe CM’s (3) . First suppose that the underlying copula C belongs to a parametric family C θ , with θ = ( θ , · · · , θ r ) , and satisfies the concordance ordering condition of copulas (see, Nelsen, 2006,page 135), that is: for every θ , θ ∈ O : θ = θ = ⇒ C θ ( > or < ) C θ . (5)It is clear that this condition implies the well-known identifiability condition of copulas:for every θ , θ ∈ O : θ = θ = ⇒ C θ = C θ . X , ..., X n ) from random vector X = ( X , ..., X d ) , we define the correspondingjoint empirical df by F n ( x ) = n − n X i =1 { X i ≤ x , ..., X di ≤ x d } , with x := ( x , ..., x d ) , and the marginal empirical df’s pertaining to the sample ( X j , ..., X jn ) , from rv X j , by F jn ( x j ) = n − n X i =1 { X ji ≤ x j } , j = 1 , ..., d. (6)According to Deheuvels (1979), the empirical copula function is defined by C n ( u ) := F n (cid:0) F − n ( u ) , ..., F − dn ( u d ) (cid:1) , for u ∈ [0 , d , where F − jn ( s ) := inf { x : F jn ( x ) ≥ s } denotes the empirical quantile function pertaining to df F jn . We are now in position to present, in three steps, the semiparametric CM-based estimation: • Step 1 : For each j = 1 , ..., d, compute b U ji := F jn ( X ji ) , then set b U i := (cid:16) b U i , ..., b U di (cid:17) , i = 1 , ..., n. • Step 2 : For each k = 1 , ..., r, compute c M k := n − n X i =1 (cid:16) C n (cid:16) b U i (cid:17)(cid:17) k . (7)as the natural estimators of CM’s M k given in equation (2) . • Step 3 : Solve the following system M ( θ , ..., θ r ) = c M M ( θ , ..., θ r ) = c M ... M r ( θ , ..., θ r ) = c M r . (8)The obtained solution ˆ θ CM := (cid:16)b θ , ..., b θ r (cid:17) is called the CM estimator for θ . ˆ θ CM are stated in Theorem 1 below whose proof isrelegated to the Appendix A.1. For convenience we set L k ( u ; θ ) := ( C θ ( u )) k − M k ( θ ) and L ( u ; θ ) = ( L ( u ; θ ) , ..., L r ( u ; θ )) . (9)Let θ be the true value of θ and assume that the following assumptions [ H. − [ H.
3] hold. • [ H. θ ∈ O ⊂ R r is the unique zero of the mapping θ → R [0 , d L ( u ; θ ) dC θ ( u ) which isdefined from O to R r . • [ H. L ( · ; θ ) is differentiable with respect to θ with the Jacobian matrix denoted by • L ( u ; θ ) := (cid:20) ∂L k ( u ; θ ) ∂θ ℓ (cid:21) r × r , • L ( u ; θ ) is continuous both in u and θ , and the Euclidian norm (cid:12)(cid:12)(cid:12)(cid:12) • L ( u ; θ ) (cid:12)(cid:12)(cid:12)(cid:12) is dominated by a dC θ -integrable function h ( u ) . • [ H.
3] The r × r matrix A := R [0 , d • L ( u ; θ ) dC θ ( u ) is nonsingular. Theorem 1
Assume that the concordance ordering condition (5) and assumptions [ H. − [ H. hold. Then with probability tending to one as n → ∞ , there exists a solution b θ CM to the system (8) which converges to θ . Moreover √ n (cid:16)b θ CM − θ (cid:17) D → N (cid:16) , A − D (cid:0) A − (cid:1) T (cid:17) , as n → ∞ , where D := var { L ( ξ ; θ ) + V ( ξ ; θ ) } and V ( ξ ; θ ) = ( V ( ξ ; θ ) , ..., V r ( ξ ; θ )) with V k ( ξ ; θ ) := d X j =1 Z [0 , d ∂ ( C θ ( u )) k ∂u j (cid:0) (cid:8) ξ j ≤ u j (cid:9) − u j (cid:1) dC θ ( u ) , k = 1 , ..., r, where ξ := ( ξ , ..., ξ d ) is a (0 , d -uniform random vector with joint df C θ . Remark 1
The asymptotic variance A − D (cid:0) A − (cid:1) T may be consistently estimated by the samplevariance of of the sequence of rv’s (cid:18) b A − i b D i (cid:16) b A − i (cid:17) T , i = 1 , ..., n (cid:19) where b A i := Z [0 , d • L (cid:16) u ; b θ CM (cid:17) dC b θ CM ( u ) and b D i := L (cid:16) b U i ; b θ CM (cid:17) + V (cid:16) b U i ; b θ CM (cid:17) , as is done, in Genest et al. (1995) and Tsukahara (2005) in the case of PML’s estimator andZ-estimator respectively. For more details on the Z-estimation theory, one refers to van der Vaart(1998), page 41. Application: Archimedean copula models
As application to the CM estimation method, we consider the Archimedean copula family definedby C ( u ) = ϕ − (cid:16)P dj =1 ϕ ( u j ) (cid:17) , where ϕ : [0 , → R is a twice differentiable function called thegenerator, satisfying: ϕ (1) = 0 , ϕ ′ ( x ) < , ϕ ′′ ( x ) ≥ x ∈ (0 , . The notation ϕ − standsfor the inverse function of ϕ. Archimedean copulas are easy to construct and have nice properties.A variety of known copula families belong to this class, including the models of Gumbel, Clayton,Frank, ... (see, Table 4.1 in Nelsen, 2006, page 116). Let K C ( s ) := P ( C ( U ) ≤ s ) , s ∈ [0 , , bethe df of rv C ( U ) , then equation (2) may be rewritten into: M k ( C ) = Z s k d K C ( s ) , k = 1 , , .... Suppose now, for unknown θ ∈ O , that ϕ = ϕ θ , it follows that C = C θ , K C = K θ and M k ( C ) = M k ( θ ) , that is M k ( θ ) = Z s k d K θ ( s ) , k = 1 , , ..., Notice that, one of the nice properties of Archimedean copula is that the df K C of C ( U ) may berepresented in terms of the first and second derivatives of the generator. Indeed from Theorem 4.3.4in Nelsen (2006), for any s ∈ [0 , , K θ ( s ) = s − ϕ θ ( s ) /ϕ ′ θ ( s ) , it follows that the correspondingdensity is K ′ θ ( s ) = ϕ ′′ θ ( s ) ϕ θ ( s ) / ( ϕ ′ θ ( s )) . Therefore the k th CM, defined in (2) , may be rewritteninto M k ( θ ) = Z s k ϕ ′′ θ ( s ) ϕ θ ( s ) (cid:0) ϕ ′ θ ( s ) (cid:1) ds, k = 1 , , ... (10)In terms of K θ , the assumptions [ H. − [ H.
3] and Theorem 1 may be rephrased, respectively, to[ H. ′ ] − [ H. ′ ] and Theorem 2 below. For convenience, we set L ( t ; θ ) = ( L ( t ; θ ) , ..., L r ( t ; θ )) with L k ( t ; θ ) := t k − M k ( θ ) . • [ H. ′ ] θ ∈ O ⊂ R r is the unique zero of the mapping θ → R L ( t ; θ ) d K θ ( t ) that is definedfrom O to R r . • [ H. ′ ] L ( · ; θ ) is differentiable with respect to θ with the Jacobian matrix denoted by • L ( t ; θ ) := (cid:20) ∂M k ( θ ) ∂θ ℓ (cid:21) r × r , • L ( t ; θ ) is continuous both in t and θ , and the Euclidian norm (cid:12)(cid:12)(cid:12)(cid:12) • L ( t ; θ ) (cid:12)(cid:12)(cid:12)(cid:12) is dominated by a d K θ -integrable function h ( t ) . [ H. ′ ] The r × r matrix A := R • L ( t ; θ ) d K θ ( t ) is nonsingular. Theorem 2
Assume that concordance ordering condition (5) and assumptions [ H. ′ ] − [ H. ′ ] hold.Then with probability tending to one as n → ∞ , there exists a solution b θ CM to the system (8) which converges to θ . Moreover √ n (cid:16)b θ CM − θ (cid:17) D → N (cid:16) , A − D (cid:0) A − (cid:1) T (cid:17) , as n → ∞ , where D := var (cid:26) L ( ξ ; θ ) + Z g ( t ) ( { ξ ≤ t } − t ) d K θ ( t ) (cid:27) , where ξ is a (0 , -uniform rv and g ( t ) := (cid:0) kt k − (cid:1) ≤ k ≤ r is r -dimensional vector. The Gumbel family is an Archimedean copula defined by C β ( u ) = exp − d X j =1 ( − ln u j ) β /β , β ≥ , with generator ϕ β ( t ) = ( − ln t ) β , β ≥ . For the sake of flexibility in data modeling, it is betterto use the multi-parameters copula models than the one-parameter ones. To have a copula withmore than one parameter, we use, for instance, the transformed (or distorted) copula defined by C Γ ( u ) = Γ − ( C (Γ ( u ) , ..., Γ ( u d ))) , where Γ : [0 , → [0 ,
1] is a continuous, concave and strictly increasing function with Γ (0) = 0 andΓ (1) = 1 . As an example, suppose that Γ = Γ α , with Γ α ( t ) = exp (1 − t − α ) , α > C β , then the transformed copula C α,β ( u ) = Γ − α ( C β (Γ α ( u ) , ..., Γ α ( u d ))) isgiven by C α,β ( u ) := d X j =1 (cid:16) u − αj − (cid:17) β /β + 1 − /α , (11)which is also a two-parameter Archimedean copula with generator ϕ α,β ( t ) := ( t − α − β . Notethat C α,β verifies the concordance ordering condition (5) ( see, Nelsen, 2006, page, 145 ) . By anelementary calculation we get the k th CM: M k ( α, β ) = ( k + 1) β + αβ − k ( k + 1) β + ( k + 1) αβ .
8n particular the first two CM’s are M ( α, β ) := 2 β + αβ − β + 2 αβ and M ( α, β ) := 3 β + αβ − β + 3 αβ . Let ( X , ..., X n ) be a sample of random vector X = ( X , ..., X d ) , then the CM estimator (cid:16)b α, b β (cid:17) of ( α, β ) is the unique solution of the system M ( α, β ) = c M M ( α, β ) = c M . That is b α = 8 c M − c M − − c M + 3 c M , b β = 1 − c M + 3 c M (cid:16) − c M (cid:17) (cid:16) − c M (cid:17) . (12) First notice that all numerical computations are performed on a personal computer with a micro-processor speed of 2.4 GHz. To evaluate and compare the performance of CM’s estimator with thePML and ( τ , ρ )-inversion estimators, a simulation study is carried out by considering the trans-formed bivariate Gumbel copula family C α,β defined above. The evaluation of the performance isbased on the bias and the RMSE defined as follows: Bias = 1 N N X i =1 ˆ θ i − θ, RMSE = N N X i =1 (cid:16) ˆ θ i − θ (cid:17) ! / , (13) where ˆ θ i is an estimator (from the considered method) of θ from the i th samples for N generatedsamples from the underlying copula. In both parts, we selected N = 1000 . The procedure out-lined in Section 2 is repeated for different sample sizes n with n = 30 , , ,
200 to assess theimprovement in the bias and RMSE of the estimators with increasing sample size. Furthermore,the simulation procedure is repeated for a large set of parameters of the true copula C α,β . For eachsample, by using formulas (12) , we obtain the CM-estimator (cid:16)b α i , b β i (cid:17) of ( α, β ) for i = 1 , ..., N, andthe estimators b α and b β are given by b α = N P Ni =1 b α i and b β = N P Ni =1 b β i . The choice of the truevalues of the parameter ( α, β ) have to be meaningful, in the sense that each couple of parametersassigns a value of one of the dependence measure, that is weak, moderate and strong dependence.In other words, if we consider Kendall’s τ as a dependence measure, then we should select valuesfor copula parameters that correspond to specified values of τ by means of the equation τ ( α, β ) = 4 Z [0 , C α,β ( u , u ) dC α,β ( u , u ) − . (14)The selected values of the true parameters are summarized in Table 1:9 α β .
01 0 . . . . . . . . . . . τ = 0 . τ = 0 . τ = 0 . α = 0 . β = 1 . α = 0 . β = 1 . α = 0 . β = 3 . n Bias RMSE Bias RMSE Bias RMSE Bias RMSE Bias RMSE Bias RMSE CPU − .
081 0 .
330 0 .
032 0 . − .
051 0 .
654 0 .
039 0 . − .
073 0 . − .
372 1 .
130 22 . sec − .
046 0 .
253 0 .
022 0 . − .
043 0 .
487 0 .
018 0 . − .
032 0 .
723 0 .
261 0 .
916 49 . sec − .
026 0 .
173 0 .
009 0 . − .
023 0 .
350 0 .
012 0 . − .
027 0 . − .
089 0 .
733 2 . mins − .
011 0 .
122 0 .
002 0 . − .
009 0 .
243 0 .
006 0 .
180 0 .
003 0 . − .
056 0 .
506 10 . mins − .
005 0 .
075 0 .
000 0 . − .
007 0 .
155 0 .
003 0 . − .
007 0 . − .
026 0 .
323 1 . hours Table 2: Bias and RMSE of CM estimator of two-parameter transformed Gumbel copula. = 0 . τ = 0 . τ = 0 . τ = 0 . α = 0 . β = 1 . α = 0 . β = 1 . α = 0 . β = 1 . α = 0 . β = 3 . Bias RMSE Bias RMSE Bias RMSE Bias RMSE Bias RMSE Bias RMSE Bias RMSE Bias RMSE n = 30 CM − .
082 0 .
313 0 .
039 0 . − .
064 0 .
642 0 .
033 0 . − .
068 0 .
567 0 .
074 0 . − .
072 0 . − .
359 1 . PML − .
067 0 . − .
485 0 . − .
117 0 . − .
561 0 .
568 0 .
078 0 . − .
426 0 . − .
043 0 .
421 0 .
253 1 . ρ - τ .
236 2 . − .
213 1 .
775 1 .
101 2 . − .
913 1 . − .
556 2 . − .
975 1 . − .
439 0 .
691 0 .
919 1 . n = 50 CM − .
046 0 . − .
021 0 . − .
052 0 .
468 0 .
029 0 . − .
037 0 .
506 0 .
015 0 . − .
033 0 .
732 0 .
206 0 . PML − .
060 0 . − .
478 0 .
482 0 .
102 0 . − .
516 0 . − .
072 0 . − .
472 0 .
556 0 .
025 0 .
315 0 .
330 0 . ρ - τ .
115 2 . − .
289 1 .
378 1 .
035 2 . − .
354 1 . − .
478 2 . − .
952 1 . − .
392 0 .
508 0 .
801 0 . n = 100 CM − .
022 0 .
171 0 .
009 0 . − .
019 0 .
342 0 .
016 0 .
258 0 .
029 0 . − .
005 0 . − .
025 0 .
551 0 .
155 0 . PML − .
058 0 . − .
483 0 . − .
109 0 . − .
524 0 . − .
071 0 . − .
403 0 . − .
023 0 . − .
017 0 . ρ - τ .
973 1 . − .
176 1 .
273 0 .
923 2 . − .
340 1 . − .
365 1 . − .
852 1 . − .
255 0 .
397 0 .
708 0 . n = 200 CM − .
016 0 .
122 0 .
002 0 . − .
011 0 .
244 0 .
001 0 . − .
014 0 .
245 0 .
005 0 . − .
001 0 . − .
060 0 . PML − .
041 0 . − .
503 0 . − .
099 0 . − .
514 0 . − .
050 0 . − .
333 0 . − .
059 0 .
144 0 .
038 0 . ρ - τ .
874 1 . − .
235 1 . − .
890 2 . − .
330 1 . − .
321 0 . − .
786 0 . − .
239 0 .
331 0 .
580 0 . τ - ρ estimators of two-parameter transformed Gumbel copula. Comments and conclusions
From Table 2, we conclude that by considering three dependence cases: weak ( τ = 0 . , moderate( τ = 0 .
5) and strong ( τ = 0 . , the performance, in terms of bias and RMSE, of the CM basedestimation is well justified. In each case, for small and large samples, the bias and RMSE aresufficiently small. Moreover, in time-consuming point of view, we observe that for a sample size n = 30 and for N = 1000 replications, the central processing unit (CPU) time to process CM’smethod took 22 .
013 seconds, which is relatively small. For one replication N = 1 , the CPUtime (in seconds) for different sample sizes are summarized as follows: ( n, CP U ) = (30 , . , (100 , . , (200 , . , (500 , . . Table 3 shows that both the PML and the CM basedestimation perform better than the ( τ , ρ )-inversion method. However, in weak dependence case τ = 0 . , the CM method provides better results than the PML one, mainly when the samplesize increases. On the other hand, it is worth mentioning that our method is quick with respectto the PML one. The main advantage of our method is that it provides estimators with explicitforms, as far as Archimedean copula models are concerned. This is not the case of the othermethods which require numerical procedures leading to eventual problems in execution time andinaccuracy issues. In conclusion, the CM based estimation method performs well for the chosenmodel. Furthermore, its usefulness in the weak dependence case particularly makes it a goodcandidate for statistical tests of independence. Acknowledgement.
The authors are indebted to an anonymous referee for valuable remarksand suggestions.
References [1] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A., 1993. Efficient and adaptiveestimation for semiparametric models. Johns Hopkins Series in the Mathematical Sciences.
Johns Hopkins University Press, Baltimore, MD. [2] Cherbini, U., Luciano, E. and Vecchiato, W., 2004. Copula methods in finance.
Wiley FinanceSeries. John Wiley & Sons, Ltd., Chichester. [3] Deheuvels, P., 1979. La fonction de d´ependance empirique et ses propri´et´es.
Acad. Roy. Belg.Bull. Cl. Sci. , 274-292. 124] Genest, C., 1987. Frank’s family of bivariate distributions. Biometrika, , 549-555.[5] Genest, C., Ghoudi, K. and Rivest, L. P., 1995. A semiparametric estimation procedure ofdependence parameters in multivariate families of distributions. Biometrika , 543-552.[6] Joe, H., 1997. Multivariate Models and Dependence Concepts , Chapman & Hall, London.[7] Joe, H., 2005. Asymptotic efficiency of the two-stage estimation method for copula-basedmodels.
J. Multivariate Anal. , 401-419.[8] Kim, G., Silvapulle, M. J. and Silvapulle, P., 2007. Comparison of semiparametric and para-metric methods for estimating copulas. Comm. Statist. Simulation Comput. , 2836-2850.[9] Nelsen, R. B., 2006. An Introduction to Copulas , second ed. Springer, New York.[10] Oakes, D., 1982. A model for association in bivariate survival data.
J. Roy. Statist. Soc. Ser.B , no. 3, 414-422.[11] Schmid, F., Schmidt, R., Blumentritt, T., Gaißer, S. and Ruppert, M., 2010. Copula-BasedMeasures of Multivariate Association. Lecture Notes in Statistics, , Volume 198, CopulaTheory and Its Applications, Part 1, Pages 209-236.[12] Sklar, A., 1959. Fonctions de r´epartition `a n dimensions et leurs marges, Publ. Inst. Statist.
Univ. Paris 8, 229-231.[13] Tsukahara, H., 2005. Semiparametric estimation in copula models. Canad. J. Statist. ,357-375.[14] van der Vaart, A. W. and Wellner, J. A., 1996. Weak Convergence and Empirical Processes:With applications to Statistics . Springer, New York.[15] van der Vaart, A. W., 1998.
Asymptotic Statistics , Cambridge University Press.[16] Yan, J., 2007. Enjoy the Joy of Copulas: With a Package copula.
Journal of StatisticalSoftware, (4), 1-21. 13 Appendix
A.1 Proof of Theorem 1
By considering CM’s estimator as a Z-estimator (van der Vaart, 1998, page 41), a straight ap-plication of Theorem 1 in Tsukahara (2005) leads to the consistency and asymptotic normalityof the considered estimator. Indeed, the existence of a sequence of consistent roots b θ CM to (4) , may be verified by using similar arguments as the proof of Theorem 1 in Tsukahara (2005). Moreprecisely, we have to check only the conditions in Theorem A.10.2 in Bickel et al. (1993). Indeed,first recall (9) and setΦ ( θ ) := Z I d L ( u ; θ ) dC θ ( u ) , and Φ n ( θ ) := n − n X i =1 L (cid:16) b U i ; θ (cid:17) , where b U i = ( F n ( X i ) , ..., F dn ( X di )) , with ( X j , ..., X jn ) is a given random sample from the rv X j . In view of assumption [ H.
2] the following derivatives exist • Φ ( θ ) = Z I d • L ( u ; θ ) dC θ ( u ) , • Φ n ( θ ) = 1 n n X i =1 • L (cid:16) b U i ; θ (cid:17) . Next, we verify that sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) • Φ n ( θ ) − • Φ ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) : | θ − θ | < ǫ n (cid:27) P → , as n → ∞ , (15)for any real sequence ǫ n → . Indeed, since • L is continuous in θ , thensup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) • L (cid:16) b U i ; θ (cid:17) − • L (cid:16) b U i ; θ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) : | θ − θ | < ǫ n (cid:27) = o P (1) , i = 1 , ..., n, and the fact that (cid:12)(cid:12)(cid:12)(cid:12) • Φ n ( θ ) − • Φ n ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n n X i =1 (cid:12)(cid:12)(cid:12)(cid:12) • L (cid:16) b U i ; θ (cid:17) − • L (cid:16) b U i ; θ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . implies sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) • Φ n ( θ ) − • Φ n ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) : | θ − θ | < ǫ n (cid:27) P → , as n → ∞ . (16)On the other hand, in view of the law of the large number, we have1 n n X i =1 • L ( U i ; θ ) P → • Φ ( θ ) , as n → ∞ , where U i = { F j ( X ji ) } ≤ j ≤ d . Moreover, in view of the continuity of function • L in u and Glivenko-Cantelli theorem, that issup x j | F jn ( x j ) − F j ( x j ) | → , j = 1 , ..., d, almost surely, as n → ∞ ,
14e have n − n X i =1 (cid:12)(cid:12)(cid:12)(cid:12) • L (cid:16) b U i ; θ (cid:17) − • L ( U i ; θ ) (cid:12)(cid:12)(cid:12)(cid:12) P → . It follows that (cid:12)(cid:12)(cid:12)(cid:12) • Φ n ( θ ) − • Φ ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) P → , which together with (16), implies (15). Conditions (MG0)and (MG3) in Theorem A.10.2 in Bickel et al. (1993) are trivially satisfied by our assumptions[ H − [ H . In view of the general theorem for Z-estimators (see, van der Vaart and Wellner, 1996,Theorem 3.3.1), it remains to prove that √ n (cid:18) • Φ n − • Φ (cid:19) ( θ ) converges in law to the appropriatelimit. But this follows from Proposition 3 in Tsukahara (2005), which achieves the proof ofTheorem 1. (cid:3) A.2 Proof Theorem 2
The proof of Theorem 2 is straightforward by using similar argument as the proof of Theorem 1,therefore the details are omitted. (cid:3)
A.3 Consistency of ( τ , ρ ) -inversion estimators In this section we give assumptions on copula models C θ , satisfying condition (5), that allowconsistency of ( τ , ρ )-inversion estimators of θ = ( θ , θ ) defined in (1) . On the other terms, wepropose some conditions of copula family C θ ensuring, for large sample sizes, both existence anduniqueness of system (1) . The idea is to express ( τ , ρ )-inversion estimators in terms of Z-estimators(van der Vaart, 1998, page 41) and then we use similar assumptions allowing consistency of theseestimators (see, van der Vaart and Wellner, 1996, Theorem 3.3.1). Indeed, recall that Kendall’stau and Spearman’s rho corresponding to the couple of rv’s ( X , X ) of dependence function C θ are defined, respectively, by τ ( θ ) = 4 Z [0 , C θ ( u , u ) dC θ ( u , u ) − ρ ( θ ) = 12 Z [0 , u u dC θ ( u , u ) − . It is easy to verify that (1) is equivalent to the following system n X i =1 L τ ( F n ( X i ) , F n ( X i ) ; θ ) = 0 n X i =1 L ρ ( F n ( X i ) , F n ( X i ) ; θ ) = 0 , (17)where L τ ( u , u ; θ ) := 4 C θ ( u , u ) − − τ ( θ ) , L ρ ( u , u ; θ ) := 12 u u − − ρ ( θ ) , with F jn denotes the empirical df pertaining to the sample ( X j , ..., X jn ) defined in (6) . Thisimplies, by representation (17) , that ( τ , ρ )-inversion estimators of the true value θ are, indeed,Z-estimators. Therefore, by using the Z-estimation theory, we conclude that consistency of suchestimators may be established provided that the following two assumptions hold: • [ A. θ , element of an open O ⊂ R , is the unique zero of the mapping θ → Z [0 , L ( u , u ; θ ) dC θ ( u , u ) , defined from O to R , with L ( u , u ; θ ) := ( L τ ( u , u ; θ ) , L ρ ( u , u ; θ )) . • [ A. L ( · ; θ ) is differentiable with respect to θ with the Jacobian matrix denoted by • L ( u , u ; θ ) := ∂L τ ( u ,u ; θ ) ∂θ ∂L τ ( u ,u ; θ ) ∂θ ∂L ρ ( u ,u ; θ ) ∂θ ∂L ρ ( u ,u ; θ ) ∂θ , • L ( u , u ; θ ) is continuous both in ( u , u ) and θ , and the Euclidian norm (cid:12)(cid:12)(cid:12)(cid:12) • L ( u , u ; θ ) (cid:12)(cid:12)(cid:12)(cid:12) isdominated by a dC θ -integrable function g ( u , u ) ..