Dependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixDependence of non-continuous random variablesJohanna NešlehováDepartment of MathematicsETH ZurichSwitzerlandwww.math.ethz.ch/~johannaAugust 31, 2005J. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixMotivationConsider• Two possibly dependent loss types andN i (T) = number of losses of type i within the year T• Dependence structures of the pair of non continuous randomvariables(N 1 (T),N 2 (T))Other examples• Counting random variables like claim/loss frequencies ornumber of defaults in a portfolio• Variables with jumps like losses censored by a certain thresholdJ. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixDistributions with continuous marginals• Notation: (X,Y ) ∼ F with marginals F 1 and F 2F 1 and F 2 continuous1. There exists a unique copula C such thatF(x,y) = C(F 1 (x),F 2 (y)) (Sklar’s theorem)2. Modeling of the marginals and the copula can be doneseparately3. C captures dependence properties which are invariat under a.s.strictly increasing transformations of the marginals4. Scale and translation invariant measures of dependence arefunctions of C alone➠ C is the (scale and location invariant) dependence structureJ. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixPathological Example2/3q1/31/3 p 2/3J. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixPathological Example2/3q1/3• (p,q) ∈ [0,1/3] × [0,1/3]:perfect positive dependence• (p,q) = (1/ √ 3,1/ √ 3):independence• (p,q) ∈ [2/3,1] × [2/3,1]:perfect negative dependence1/3 p 2/3J. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixFurther Pitfalls• The copula is not unique; i.e. several possible copulas exist• The dependence structures of F are generally not the same asthe dependence structures of the possible copulasF(x,y) ≤ F ∗ (x,y) ∀x,y ∈ R ⇏ C(u,v) ≤ C ∗ (u,v) ∀u,v ∈ [0,1]• Possible copulas remain invariant under strictly increasing andcontinuous transformations, but do not necessarily change inthe same way as the unique copula in the “continuous” case ifat least one of the transformations is decreasing• Weak convergence of F n does not imply the point-wiseconvergence of the corresponding possible copulas• Any measure of association which depends only on the copulaof F is a constantJ. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixKendall’s Tau and Spearman’s rho in the”Non-Continuous” Case?”Naive” Idea: work with some of the non unique fitting copulaBut: fitting copulas can differ considerably.• For independent Bernoulli random variables X and Y◮ τ(X, Y ) ∈ [−3/4, 3/4]◮ ρ(X, Y ) ∈ [−13/16, 13/16]• for comonotonic Bernoulli random variables X and Y◮ τ(X, Y ) ∈ [0, 1]◮ ρ(X, Y ) ∈ [1/2, 1]Because: Difference between the probability of concordance anddiscordance ≠ 4 ∫ C(u,v)dC ∗ (u,v) − 1J. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixSome Starting Points• Investigation of the family of possible copulas correspondingto a fixed bivariate distribution◮ In particular search for a suitable extension strategy whichwould produce a copula capturing the dependence structuresof the joint distribution function• Investigation of possible bivariate distributions obtained froma fixed copula and marginals which follow some specifieddistribution (up to parameters), such as Poisson, binomial orBernoulliJ. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixThe ”Standard” Extension StrategyC(0.4, 0.4) = 0C(0.4, 0.4) = 0.16C(0.4, 0.4) = 0.31110.80.80.80.60.60.6yyy0.40.40.40.20.20.2000.20.40.60.81000.20.40.60.81000.20.40.60.81xxxThe Standard Extension C S• The standard extension copula of Schweizer and Sklar• C S corresponds to the linear interpolation of the uniquesubcopula as well as the unique copula of the smoothed jdfJ. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixSome Selected Properties of C SPros• X and Y are independent if and only if C S is theindependence copula• (X ∗ ,Y ∗ ) more concordant than (X,Y ) if and only ifC S (u,v) ≤ CS ∗ (u,v) for all u,v ∈ [0,1]• C S reacts on monotone transformations of the marginals asthe unique copula in the “continuous” caseCons• If X and Y are perfect monotonic dependent, C S does notcoincide with the Fréchet-Hoeffding bounds• Weak convergence does not imply the point-wise convergenceof the C S ’sJ. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixTowards Kendall’s Tau and Spearman’s RhoDifference between the probabilities of concordance anddiscordance equals 4 ∫ C S (u,v)dC S ∗ (u,v) − 1Consider• Kendall’s tau: ˜τ(C S ) = 4• Spearman’s rho: ˜ρ(C S ) = 12However∫ 1 ∫ 100∫ 1 ∫ 1• ˜τ and ˜ρ do not reach the bounds 1 and −10C S (u,v) dC S (u,v) − 10[C S (u,v) − uv] du dv• Exact bounds of ˜τ and ˜ρ are complicated and do not have thesame absolute valueJ. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixExact Bounds for Kendall’s Tau and Spearman’s Rho1.51.5110.50.5000.20.4x0.60.8000.20.4x0.60.8-0.5-0.5-1-1|˜τ(M S )/˜τ(W S )| for Binomialdistributions F 1 = B(n, 0.4) andF 2 = B(n, x) with n =1,4 and 10.|˜ρ(M S )/˜ρ(W S )| for Binomialdistributions F 1 = B(n, 0.4) andF 2 = B(n, x) with n =1,4 and 10.J. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixLess Sharp Bounds for Kendall’s Tau and Spearman’s Rho1. |˜τ(X,Y )| ≤ √ 1 − E ∆F 1 (X) √ 1 − E∆F 2 (Y ). The boundsare attained if Y = T(X) a.s. where T is a strictly monotoneand continuous transformation on the range of X.2. |˜ρ(X,Y )| ≤ √ 1 − E ∆F 1 (X) 2√ 1 − E ∆F 2 (Y ) 2 . The boundsare attained if Y = T(X) a.s. where T is a strictly monotoneand continuous transformation on the range of X.J. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixKendall’s Tau and Spearman’s Rho for Non-ContinuousRandom Variables1∫ ∫ 14 C S (u,v) dC S (u,v) − 10 0τ(X,Y ) = √[1 − E ∆F1 (X)][1 − E∆F 2 (Y )](∫1∫ 1 (12 CS (u,v) − uv ) )du dv0 0ρ(X,Y ) = √[1 − E ∆F1 (X) 2 ][1 − E∆F 2 (Y ) 2 ]• τ and ρ satisfy (modified) axioms of concordance measures• Bounds are attained if X a.s. = T(Y ) for T continuous andstrictly monotone• τ and ρ are the sample versions for empirical distributionsJ. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References Appendixτ, ρ and ϱ for Binomial Distributions1110.80.80.80.60.60.60.40.40.40.20.20.200.20.40.60.8 10 0.2 0.40.60.8100.20.40.60.8 1xxxF 1 = B(2, 0.4), F 2 = B(2, x).x0 0.2 0.4 0.6 0.8 1F 1 = B(4, 0.4), F 2 = B(4, x).x0 0.2 0.4 0.6 0.8 1F 1 = B(10, 0.4), F 2 = B(10, x).x0 0.2 0.4 0.6 0.8 1-0.2-0.2-0.2-0.4-0.4-0.4-0.6-0.6-0.6-0.8-0.8-0.8-1-1-1J. Nešlehová F 1 = B(2, 0.4), F 2 = B(2, x). F 1 = B(4, 0.4), F 2 = B(4, x). F 1 = B(10, 0.4), F 2 = B(10, ETHx).ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixSome Problems• Modeling: dependence properties of a familyH = {C(F,G) : F ∈ F,G ∈ G}Kendall’s tau0.0 0.1 0.2 0.3 0.4 0.5Kendall’s tau0.0 0.1 0.2 0.3 0.4 0.5Kendall’s tau0.0 0.1 0.2 0.3 0.4 0.50.0 0.2 0.4 0.6 0.8 1.0q0.0 0.2 0.4 0.6 0.8 1.0q0 100 200 300qKendall’s tau for binomial marginals and a Gauss copula (solidline), Frank copula (short-dashed line), Gumbel copula (dottedline), and Fréchet copula (long-dashed line).J. Nešlehová ETH ZurichDependence of non-continuous random variables

Fallacies Dependence Concordance Measures Problems & References AppendixSome Theory Behind: Dependence Structure in theNon-Continous CaseIn the continuous case: C is the cdf. of the transformed vector(F 1 (X),F 2 (Y )).Idea: In the non-continuous case, use a different transformation ofthe marginals:ψ(x,u) := P[X < x] + uP[X = x] = F(x−) + u∆F(x)Result: for a random vector (U,V) with uniform marginals whichis independent of (X,Y ), (ψ(X,U),ψ(Y ,V)) has uniformmarginals and the corresponding unique copula is a possible copulaof (X,Y ).Moreover: different dependence structure of (U,V) leads todifferent possible copulas of (X,Y).J. Nešlehová ETH ZurichDependence of non-continuous random variables