A measure of radial asymmetry for bivariate copulas based on Sobolev norm

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1 A measure of radial asymmetry for bivariate copulas based on Sobolev norm Ahmad Alikhani-Vafa Ali Dolati Abstract The modified Sobolev norm is used to construct an index for measuring the degree of radial asymmetry of a copula. We study various aspects of this index discuss its rank-based estimator. Through simulation a real data example we compare the proposed index with the other radial asymmetry measures. Keywords: norm. Bivariate symmetry; Copula; Empirical process; Radial asymmetry; Sobolev 2 AMS Classification: 62H5 62H2 1. Introduction Let X Y ) be a pair of continuous rom variables with the joint distribution function Hx y) = P X x Y y) univariate marginal distributions F x) = P X x) Gy) = P Y y) at each x y R. Let C be the unique copula associated with X Y ) through the relation Hx y) = CF x) Gy)) x y R in view of Sklar s Theorem [19]. In fact C is the cumulative distribution function of the pair U V ) = F X) GY )) of uniform 1) rom variables. For a given copula C let Ĉu v) = u + v 1 + C1 u 1 v) be the survival copula or reflected copula associated with C or equivalently the cumulative distribution function of the pair 1 U 1 V ). A copula C is said to be radially symmetric [12] if 1.1) Cu v) = Ĉu v) for all u v [ 1]. This concept is also called reflection symmetry or tail symmetry in literature see e.g [12 15]. When the condition 1.1) fails for some u v [ 1] the copula C is said to be radially asymmetric. Nelsen [14] argued that any suitably normalized distance between the surfaces z = Cu v) z = Ĉv u) in particular any L p distance would yield a measure of radial asymmetry. Diagnostics such as asymmetry measures are useful during data analysis. For instance the presence of radial asymmetry in a set of data rejects the null hypothesis of bivariate normality other models with more flexibility should be considered. Recently several copula-based measures of radial asymmetry have been proposed the desirable properties for such measures are addressed in [3 15]. Some tests for identifying radial symmetry of bivariate copulas are discussed in [1 1]. The asymmetry considered here is also distinguished from the issue of whether a copula is exchangeable i.e. for all u v [ 1] Cu v) = Cv u). For discussion on this kind of symmetry we refer to [ ]. The purpose of this paper is to introduce study another copula based measure of radial Department of Statistics School of Mathematics Yazd University Yazd Iran. Department of Statistics School of Mathematics Yazd University Yazd Iran adolati@yazd.ac.ir Corresponding Author.

2 2 asymmetry based on the modified Sobolev norm [18]. The paper is organized as the following: the proposed index its properties are discussed in Section 2. In Section 3 we compare the new radial asymmetry measure with the other measures. The estimation of the proposed asymmetry measure is given in Section 4. Sample properties are studied through simulation a real data example in Sections 5 6. Concluding comments are given in Section Sobolev measure of radial asymmetry Since copulas are Lipschitz continuous functions from [ 1] 2 to [ 1] with Lipschitz constant equal to 1 then they are absolutely continuous in each argument so that it can be recovered from any of its partial derivatives by integration. The partial derivatives of a copula C can be seen as conditional distribution functions Ċ1u v) = Cu v)/ u = P V v U = u) [ 1] Ċ2u v) = Cu v)/ v = P U u V = v) [ 1]. For more details on copulas we refer to [13]. Let C be the class of all bivariate copulas. The differentiability properties of copulas imply that C is a subset of any stard Sobolev space W 1p I 2 R) for p [1 ); see [2]. Among these spaces the Sobolev space W 12 [ 1] 2 R) is a Hilbert space. Let spanc) denote the vector space generated by C. Obviously spanc) W 12 [ 1] 2 R). For A B spanc) let 2.1) A B = {Ȧ1 u v)ḃ1u v) + A } 2 u v)ḃ2u v) dudv 2.2) A 2 = { A 1 u v)) 2 + Ȧ2u v)) 2 }dudv. As shown in [17]... define a scalar product a norm on spanc) respectively. For every copula C it is easy to see that C 2 = Ĉ 2. Thus for a radially symmetric copula C we have that C Ĉ = C 2. A natural measure of radial asymmetry in a copula C based on the modified Sobolev norm could be constructed by using the quantity C Ĉ / C Definition. Consider the functional λ : C R + defined by ) C Ĉ 2.3) λc) = 2 1 C 2. We call λc) as the Sobolev measure of radial asymmetry for copula C. The following result shows that the index λc) satisfies a set of reasonable properties of a measure of radial asymmetry proposed in [3] Theorem. For every C C the functional λc) satisfies i) λc) [ 1]; ii) λc) = if only if C = Ĉ; iii) λc) = λĉ). iv) If {C n } n IN C are in C if lim n C n C = then lim n λc n ) = λc). Proof. Part i) follows from the fact that 1/2 C Ĉ 1 C 2 1 see Theorem 14 in [17]) the identity 2.4) C Ĉ 2 = C 2 + Ĉ 2 2 C Ĉ = 2 C 2 C Ĉ ). For part ii) if C = Ĉ then λc) =. Conversely if λc) = then C Ĉ = C 2 or equivalently C Ĉ 2 =. Let Du v) = Cu v) Ĉu v). If D 2 = then udu v) =

3 3 v Du v) = a.e. in [ 1]2. Since D ) = in view of absolutely continuity of D in each arguments we have that Du v) = for almost all u v [ 1]. Part iii) follows from the fact that C 2 = Ĉ 2 Ĉ = C. For part iv) first note that the inequality C n C C n C implies that C n C. Since C n Ĉn) C Ĉ) C n C + Ĉn Ĉ from the identity C n Ĉn = C n C n Ĉn 2 we have that C n Ĉn C Ĉ which gives the required result Definition. We say that a copula C is maximally radially asymmetric with respect to λ if only if λc) = Theorem. For every C C the following are equivalent: i) C is maximally radially asymmetric with respect to λ. ii) C 2 = 1 C Ĉ = 1 2. Proof. From 2.3) 2.4) we have that λc) = 1 if only if C Ĉ 2 = C 2 = 2 C Ĉ. The statement ii) follows from the fact that 1/2 C Ĉ 1 C 2 1 see Theorem 14 in [17]) Example. Let θ [ 1] C θ be a family of copulas given by 2.5) C θ u v) = minu v u 1 + θ) + + v θ) + ) where t + = maxt ). C θ is a copula whose mass is distributed uniformly on the line segments joining the points θ) to 1 θ 1) 1 θ ) to 1 θ). Easy calculations show that Ĉθ = C 1 θ for each θ [ 1 2 ] C θ 2 = 1 C θ Ĉθ = 8θ 2 4θ + 1. For this copula we have that λc 1 ) = 1. Therefore C 1 is a maximally radially asymmetric 4 4 copula with respect to λ. Note that for θ = C θ u v) = minu v) which is radially symmetric. 3. Comparing with the other asymmetry measures In this section we compare the Sobolev measure of radial asymmetry with two other alternatives 3.1) Ψ C) = 3 sup Cu v) Ĉu v) 3.2) Ψ 2 C) = uv) [1] 2 Cu v) Ĉu v) ) 2 dudv which are constructed based on the L L 2 distance between C its survival copula Ĉ. Both measures take values in [1] were first discussed by Dehgani et al. [3]. If we consider the family of copulas given by 2.5) then we have Ψ C θ/3 ) = θ thus C 1/3 is a maximally radially asymmetric copula with respect to Ψ while Ψ 2 C 1/3 ).46. As explained in [3] there are several relationships between radial asymmetry dependence. For example the Spearman s rho coefficient [13] for maximally radially asymmetric copulas with respect to Ψ takes values in [ 5/9 1/3]; see [3] for details.

4 4 4. Sample version Given a rom sample X 1i X 2i ) i = n from an unknown distribution H with unique copula C we derive a non-parametric estimator for the measure of asymmetry λc) defined in 2.3). For i { n} k = 1 2 consider the pseudo-observations Ûki = R ki n+1 where R 1i R 2i ) i = n are the corresponding vectors of ranks. The natural estimators of C Ĉ are then given by the empirical copula C n Ĉn [7] defined at each u v [ 1] by 4.1) C n u v) = 1 II{Û1i u n Û2i v} 4.2) Ĉ n u v) = 1 n i=1 II{1 Û1i u 1 Û2i v} i=1 where II{A} denotes the indicator function of A. A plug-in rank-based estimator of λc) is given by ) 4.3) λ = λcn ) = 2 1 C n Ĉn C n 2. To approximate the partial derivatives of a copula we proceed as in [16]. Note that the derivative Ċ1 will fail to be continuous on 1) [ 1] if the distribution of V given U = u has atoms. for instance This phenomenon occurs for the Fréchet lower upper bound copulas W u v) = maxu + v 1 ) Mu v) = minu v). A copula C is said to be regular [16] if i) the partial derivatives Ċ1 Ċ2 exist everywhere on [ 1] 2 ii) Ċ1 is continuous on 1) [ 1] Ċ2 is continuous on [ 1] 1). For a regular copula E let Ė1n Ė2n be the estimates of the partial derivatives Ė1 Ė2 defined by 4.4) Ė 1n u v) = E nu + l n v) E n u l n v) 4.5) Ė 2n u v) = E nu v + l n ) E n u v l n ) where l n is a bwidth parameter typically l n = 1/ n; see [16] for details. We could also use a kernel based estimate of the derivative [6] but this would limit the writing of explicit expressions for λ. By using 4.4) 4.5) natural estimators of C Ĉ C 2 are given by 4.6) C n Ĉn = {Ċ1n u v) Ĉ 1n u v) + Ċ2nu v) Ĉ 2n u v)} dudv 4.7) C n 2 = {Ċ1nu v)) 2 + Ċ2nu v)) 2 }dudv. Although theses statistics are defined by integrals they reduce to sums which make it possible to compute λc n ) explicitly Theorem. Let X 1j X 2j ) j = n be a sample of size n from a vector X 1 X 2 ) of continuous rom variables having a regular copula C let R 1j R 2j ) j = 1... n be the corresponding vectors of ranks. Then 4.8) C n Ĉn = 1 A 1) ij B2) ij + A 2) ij B1) ij )

5 5 4.9) C n 2 = 1 where for k = 1 2 A k) ij = max 4.1) B k) ij 2 max D 1) ij E2) ij + D 2) 1 R kj ij E1) ij ) l n 1 R ) kj + l n ) D k) ij = max l n 2 max R kj R kj + l n ) ) + max + l n 1 R ) kj l n ) + max + l R kj n l n = min 1 R ) ki R kj E k) ij = min 1 R ki 1 R ) kj. Proof. For k = 1 2 h = n let Ûkh = R kh /) A h u 1 u 2 ) = II{Û1h u 1 }II{Û2h u 2 } B h u 1 u 2 ) = II{1 Û1h u 1 }II{1 Û2h u 2 }. Let C n be the associated empirical copula. Then one may write Ċ 1n u v) Ĉ 1n u v) = 1 Ċ1nu v)) 2 = 1 Ċ 2n u v) Ĉ 2n u v) = 1 Ċ2nu v)) 2 = 1 It is easy to see that {A i u+l n v) A i u l n v)}{b j u+l n v) B j u l n v)} {A i u+l n v) A i u l n v)}{a j u+l n v) A j u l n v)} {A i u v+l n ) A i u v l n )}{B j u v+l n ) B j u v l n )} {A i u v+l n ) A i u v l n )}{A j u v+l n ) A j u v l n )}. A i u + l n v)b j u + l n v) = II{maxÛ1i l n 1 Û1j l n ) u}ii{maxû2i 1 Û2j) v} A i u + l n v)b j u l n v) = II{maxÛ1i l n 1 Û1j + l n ) u}ii{maxû2i 1 Û2j) v} A i u l n v)b j u + l n v) = II{maxÛ1i + l n 1 Û1j l n ) u}ii{maxû2i 1 Û2j) v} A i u l n v)b j u l n v) = II{maxÛ1i + l n 1 Û1j + l n ) u}ii{maxû2i 1 Û2j) v} A i u + l n v)a j u + l n v) = II{maxÛ1i l n Û1j l n ) u}ii{maxû2i Û2j) v} A i u + l n v)a j u l n v) = II{maxÛ1i l n Û1j + l n ) u}ii{maxû2i Û2j) v} A i u l n v)a j u + l n v) = II{maxÛ1i + l n Û1j l n ) u}ii{maxû2i Û2j) v} A i u l n v)a j u l n v) = II{maxÛ1i + l n Û1j + l n ) u}ii{maxû2i Û2j) v}. Similar expressions hold for Ĉ Ċ2n 2n Ċ2n) 2. Upon integrating letting Ûkh = R kh n+1 k = 1 2 h = 1... n one gets the required result. The following result shows that λc n ) is a consistent estimator of λc).

6 6 Table 1. Average mean square errors estimated values for measure ˆλ from 1 iterations with sample size n {2 5 1} from Clayton Gumbel Joe copulas with τ { } n=2 n=5 n=1 τ Copula ˆλ Bias RMSE ˆλ Bias RMSE ˆλ Bias RMSE Clayton Gumbel Joe Clayton Gumbel Joe Clayton Gumbel Joe Clayton Gumbel Joe Clayton Gumbel Joe Theorem. If C is a regular copula then λc n ) defined by 4.3) converges in probability to λc) as n. Proof. If C 1 C 2 are continuous on [ 1] 2 then C n = nc n C) converges weakly to a continuous Gaussian process C; see e.g. [7]. For u v [ 1] l n = 1/ n we may write Ċ 1n = C nu + l n v) C n u l n v) = Cu + l n v) Cu l n v) + C nu + l n v) C n u l n v) n then sup Ċ1nu v) Ċ1u v) = sup C nu + l n v) C n u l n v) uv [1] uv [1] 2l Ċ1u v) n sup Cu + l n v) Cu l n v) uv [1] 2l Ċ1u v) n sup C n u + l n v) C n u l n v) uv [1] which tends to as n ; that is the partial derivative estimates of a regular copula are uniformly convergent. By using dominated convergence theorem we have that C n C the continuity property iv) in Theorem 2.2 gives the required result.

7 7 Figure 1. Histogram of simulated estimates for Clayton copula with τ =.25 top left) τ =.5 top right) τ =.75 bottom left) τ =.9 bottom right). 5. Simulation study In this section we compare the sample version of the radial asymmetry measure λ with those of Ψ Ψ 2 given in [3] by Ψ 2 = n 2 min 1 R 1i 1 R ) 1j min 1 R 2i 1 R ) 2j 5.1) 2 min 1 R ) 1i R 1j min 1 R ) 2i R 2j ) ) R1i + min R 1j R2i min R 2j 5.2) Ψ = 3 n max C i n ij {12...n} j ) Ĉn i j ) see e.g. [1] by simulation. For specific sample size n {2 5 1} a total of 1 Monte Carlo replications generated from three radially asymmetric copulas Clayton Gumbel Joe with five different degrees of dependence in terms of Kendall s tau: τ { }. For each case 1 rom samples are generated the estimates of λ are computed. The average values of the estimates from 1 iterations are reported in Table 1 together with their corresponding RMSEs. The absolute bias is reduced by increasing the sample size in all cases. It can be observed that the three measures perform well in the high dependence case in fact bias decreases slightly) when dependence increases. We also see that the RMSEs decrease as n increases.

8 8 V V U U Figure 2. Scatter plot of uniform scores of the loss U) ALAE V) left panel) Scatter plot of a rom sample of size 15 from Gumbel copula with the parameter θ = right panel) Remark. We have not addressed the important problem of finding the asymptotic distribution of the estimator of λ which is useful for testing symmetry of bivariate data. However histograms of the estimates generated from above simulation for Clayton copula with different degree of dependence presented in Figure 1 support the asymptotic normality. Table 2. Estimates of λ Ψ Ψ 2 for Loss-ALAE data ˆλ ˆΨ2 ˆΨ Estimate Stard error.56.1 NA 6. Example with real data For the application we present an example with a real data set on 15 insurance claims. Each claim consists of an indemnity payment the loss) an allocated loss adjustment expense ALAE). For details of the data set; see [8]. The scatter plot for uniform scores of data presented in Figure 2 left panel). We can see that there is some asymmetry a positive dependence in this data set. For these two variables the Spearman s rho Kendall s tau coefficients are respectively. The estimates of the radial asymmetry measures λ Ψ Ψ 2 for Loss-ALAE data are shown in Table 2. For ˆΨ 2 ˆλ the stard errors are obtained using the jackknife method which is not applicable for calculating the stard error of ˆΨ. Choosing a copula for this data set has been examined by means of various model selection methods in [4 8 9]. For instance in [4] the Gumbel Clayton Frank Joe copulas were fitted reported that the Gumbel copula with the parameter θ = is the best fits. The approximate values of three radial asymmetry measures for Gumbel copula with the parameter θ = are given Ψ =.744 Ψ 2 =.685 λ =.23. The right panel of Figure 2 displays the scatter plot of a rom sample of size 15 from Gumbel copula with the parameter θ =

9 9 7. Concluding remarks Based on a modified version of the Sobolve norm an index is introduced for measuring the radial asymmetry of a bivariate copula. The proposed index satisfies desirable properties of a measure of radial asymmetry addressed in [3]. The measure λ is also shown to be continuous therefore it is possible to use empirical method to estimate the value of this measure. The rank based estimator of λ has a closed form can be used to detecting radial asymmetry in a set of bivariate data. Acknowledgements The authors wish to thank the two anonymous reviewers for their careful reading insightful comments on an earlier version of this paper. References [1] Bouzebda S. Cherfi M. Test of symmetry based on copula function Journal of Statistical Planning Inference [2] Darsow W. F. Olsen E. T. Norms for copulas International Journal of Mathematics Mathematical Sciences [3] Dehgani A. Dolati A. Úbeda-Flores M. Measures of radial asymmetry for bivariate rom vectors Statistical Papers [4] Denuit M. Purcaru O. Van Keilegom I. Bivariate Archimedean copula modelling for loss-alae data in non-life insurance Working paper UCL 24. [5] Durante F. Mesiar R. L -measure of non-exchangeability for bivariate extreme value Archimax copulas Journal of Mathematical Analysis Applications [6] Fermanian J. D. Scaillet O. Nonparametric estimation of copulas for time series Journal of Risk [7] Fermanian J. D. Radulovic D. Wegkamp M. Weak convergence of empirical copula processes Bernoulli 1 5) [8] Frees E. Valdez E. Understing relationships using copulas North American Actuarial Journal [9] Klugman S. Parsa R. Fitting bivariate loss distributions with copulas Insurance: Mathematics Economics [1] Genest C. Nešlehová J. G. On tests of radial symmetry for bivariatecopulas Statistical Papers [11] Lojasiewicz S. Introduction to the Theory of Real Functions Wiley Chichester [12] Nelsen R.B. Some concepts of bivariate symmetry Journal of Nonparametric Statistics [13] Nelsen R.B. An Introduction to Copulas. Second Edition Springer New York 26. [14] Nelsen R.B. Extremes of nonexchangeability Statistical Papers [15] Rosco J. F. Joe H. Measures of tail asymmetry for bivariate copulas Statistical Papers [16] Segers J. Asymptotics of empirical copula processes under non-restrictive smoothness assumptions Bernoulli [17] Siburg K.F. Stoimenov P. A. A scalar product for copulas Journal of Mathematical Analysis Applications 3441) [18] Siburg K. F. Stoimenov P. A. Symmetry of functions exchangeability of rom variables Statical Papers [19] Sklar A. Functions de répartition à n dimensions et leurs marges Publ. Inst. Statistique Univ. Paris