Copulas: An Introduction I - Fundamentals

1 Copulas: An IntroductionI - FundamentalsJohan SegersUniversit catholique de Louvain (BE)Institut de statistique, biostatistique et sciences actuariellesColumbia University, New York City9 11 Oct 2013 Johan Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 20131 / 74 The starting point:Margins versus dependenceDecomposition of a multivariate cdfFintoIunivariate marginsF1,..,FdIcopulaCIdea: the copulaCcaptures the dependence among thedvariables,irrespective of their marginal Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 20132 / 74 Course aimIntroduction to the basic concepts and main principlesIFundamentalsIIModelsIIII nferenceCaveats:IPersonal selection of topics in a wide and fast-growing fieldISpeaker s bias towards (practically useful) theoryIReferences are a random selection from an ocean of literatureJohan Segers (UCL)Copulas.

2 I - FundamentalsColumbia University, Oct 20133 / 74 Some references to start withJaworski, P., F. Durante, W. H rdle, and T. Rychlik (2010).Copula Theory and ItsApplications: Proceedings of the Workshop Held in Warsaw, 25-26 September2009. Lecture Notes in Statistics. Berlin: , H. (1997).Multivariate Models and Dependence Concepts. London: Chapman& , I. and J. Yan (2010). Modeling multivariate distributions withcontinuous margins using the copula R of StatisticalSoftware 34(9), 1 , A. J., R. Frey, and P. Embrechts (2005).Quantitative Risk Management:Concepts, Techniques and Tools. Princeton: Princeton University Press. Chapter 5, Copulas and Dependence .Nelsen, R. B. (2006).

3 An Introduction to Copulas. New York: , P. K. and D. M. Zimmer (2005). Copula modeling: an Introduction and Trends in Econometrics 1(1), 1 111.+ books on the use of copulas in specific domains, notably financeJohan Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 20134 / 74 Copulas: An IntroductionI - FundamentalsSklar s theoremDensities and conditional distributionsCopulas for discrete variablesMeasures of associationJohan Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 20135 / 74 Copulas: An IntroductionI - FundamentalsSklar s theoremDensities and conditional distributionsCopulas for discrete variablesMeasures of associationJohan Segers (UCL)Copulas.

4 I - FundamentalsColumbia University, Oct 20136 / 74 Generalized inverse functionsThe left-continuous generalized inverse function of a univariate cdfFisdefined asF (u) =inf{x R:F(x) u},0<u< a picture ofF (u) =xin continuous and increasing continuous but flat an atom outF ifFis the cdf of a rvXwithP(X=1) =p=1 P(X=0).Johan Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 20137 / 74 Properties of generalized inverse functionsLetFbe a univariate cdf, not necessarily (F (u)) uIF(x) uiffx F (u)IIfUis uniform(0,1), thenX=F (U)has these properties.[Hint:Fis right continuous.] would the second result help you to generate random numbersfromF?Johan Segers (UCL)Copulas.

5 I - FundamentalsColumbia University, Oct 20138 / 74 Probability integral transform:Reduction to uniformityIfXis a random variable with continuous cdfF, then the distribution ofU=F(X)is Uniform(0,1), [F(X) u] =u,u [0,1] goes wrong ifFis not continuous? Take for instanceXBernoulli(p). the above property.[Hint: Justify the equalities inP[F(X) u] =P[X F (u)] =1 F(F (u)) =1 u.] a pseudo-random sampleX1,..,Xnfrom your favouritecontinuous distributionF. ComputeF(X1),..,F(Xn)and assess its uniformity ( histogram, kernel density estimate, QQ-plot, .. ).Johan Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 20139 / 74So what s a copula?Ad-variate copulaC: [0,1]d [0,1]is the cdf of a random vector(U1.)

6 ,Ud)with Uniform(0,1)margins:C(u) =P[U1 u1,..,Ud ud]whereP[Uj uj] =ujforj {1,..,d}and 0 uj : Alternative definition possible, in terms of properties ofCas a Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201310 / 74 The representation of a copula as a cdfimplies a number of propertiesC(u) =P[U1 u1,..,Ud ud],Uj Uniform(0,1) some componentujis 0, thenC(u) = (1,..,1,uj,1,..,1) =ujif 0 uj , ifd=2 andaj bj,0 C(b1,b2) C(a1,b2) C(b1,a2) +C(a1,a2) nondecreasing in each of Lipschitz and hence continuous:|C(u) C(v)| |u1 v1|+ +|ud vd| these Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201311 / 74 Sklar s theorem I:How to construct a multivariate cdfLetCbe ad-variate copula and letF1.

7 ,Fdbe univariate cdf s. Then thefunctionF(x) =C(F1(x1),..,Fd(xd))(Skl)is ad-variate cdf with marginsF1,.., (U1,..,Ud) Cand putXj=F j(Uj) Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201312 / 74 Sklar s theorem II:Any multivariate cdf has a copulaIfFis ad-variate cdf with univariate cdf sF1,..,Fd, then there exists acopulaCsuch that (Skl) the margins are continuous, thenCis unique and is equal toC(u) =F(F 1(u1),..,F d(ud)) the margins are continuous. LetX Fand putUj=Fj(Xj) Uniform(0,1).ThenU CwithCas given in the display, and (Skl) Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201313 / 74 Elementary examplesLet(X,Y)be a random vector with continuous margins and independent if and only if their copula isC(u,v) =uvIIfY=g(X)withgincreasing, thenC(u,v) =min(u,v) =:M(u,v)IIfY=g(X)withgdecreasing, thenC(u,v) =max(u+v 1,0) =:W(u,v) Show the above Show thatMis the cdf of(U,U).

8 What is its support?3. Show thatWis the cdf of(U,1 U). What is its support?Johan Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201314 / 74Fr chet Hoeffding upper and lower bounds:Supported on the (anti) = = 1 UUVM(u,v) =min(u,v)W(u,v) =max(u+v 1,0)Johan Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201315 / 74Fr chet Hoeffding boundsAny bivariate copulaCverifiesmax(u+v 1,0) C(u,v) min(u,v) these : use the Bonferroni inequalitiesP(A) +P(B) 1 P(A B) min{P(A),P(B)} the bounds tod-variate upper bound is the copula of the random vector(U,..,U).IThe lower bound is not a copula ifd Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201316 / 74 Invariance under monotone transformationsIfICis a copula ofX FIT1.

9 ,Tdare increasing functionsthenICis also a copula of(T1(X1),..,Td(Xd)) the above property.[Hint: the cdf ofTj(Xj)isFj(T 1j). Calculate the joint cdf of(T1(X1),..,Td(Xd)), using Sklar s representation ofF.]Johan Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201317 / 74 Survival Copulas: Linking joint and marginal survival functionsAssume continuous margins. IfX= (X1,..,Xd)andUj=Fj(Xj),then 1 Ujis uniform on(0,1) cdf Cof(1 U1,..,1 Ud)is the survival copula ofX, andP[X1>x1,..,Xd>xd] = C( F1(x1),.., Fd(xd))linking the joint survival function with the marginal ones, Fj(xj) =1 Fj(xj) =P[Xj>xj]This way of modelling dependence is popular in survival Segers (UCL)Copulas.

10 I - FundamentalsColumbia University, Oct 201318 / 74 Example: the Ali Mikhail Haq (survival) copulaC (u,v) =uv1 (1 u) (1 v), [ 1,1) random sample, theta = AMH random sample, theta = Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201319 / 74 Survival copulas are copulas dimensiond=2, show that C(u,v) =u+v 1 C(1 u,1 v) that ifCis the copula of(X1,..,Xd),then Cis the copula of( X1,.., Xd),or more generally of(T1(X1),..,Td(Xd)) (U,V) C, calculate the cdf s (copulas) of(1 U,V)and(U,1 V).More generally, to ad-variate copulaC, one can associate2dcopulas byconsidering transformations(T1,..,Td) Segers (UCL)Copulas. I - FundamentalsColumbia University, Oct 201320 / 74 SymmetriesLetU copulaCis called symmetric or exchangeableif, for any permutation, , of{1.]}