1. Copulas Univariate Functions Gary G Venter Conditioning with Copulas nLet C 1 (u,v) denote the first partial derivative of C(u,v). F(x,y) = C(F X (x),F Y (y)), distribution of Y|X=x is given by: nF Y|X (y) = C 1 (F X (x),F Y (y)) nC(u,v) = uv, the conditional distribution of V given that U=u is C 1 (u,v) = v = Pr(V<v|U=u). nIf C 1 is simple enough to invert algebraically, then the simulation of joint probabilities can be done using the derived conditional distribution. That is, first simulate a value of U, then simulate a value of V from C 1.

2. Copulas Univariate Functions Gary G Venter Tails of Copulas ASTIN 2001

3. Copulas Univariate Functions Gary G Venter Kendall correlation  is a constant of the copula  = 4E[C(u,v)] – 1  = 2 d E[C(u 1,...,u d )] – 1 2 d – 1 – 1

4. Copulas Univariate Functions Gary G Venter Frank’s Copula nDefine g z = e -az – 1 nFrank’s copula with parameter a  0 can be expressed as: nC(u,v) = -a -1 ln[1 + g u g v /g 1 ] nC 1 (u,v) = [g u g v +g v ]/[g u g v +g 1 ] nc(u,v) = -ag 1 (1+g u+v )/(g u g v +g 1 ) 2  (a) = 1 – 4/a + 4/a 2  0 a t/(e t -1) dt For a<0 this will give negative values of . nv = C 1 -1 (p|u) = -a -1 ln{1+pg 1 /[1+g u (1–p)]}

5. Copulas Univariate Functions Gary G Venter

6. Copulas Univariate Functions Gary G Venter Gumbel Copula nC(u,v) = exp{- [(- ln u) a + (- ln v) a ] 1/a }, a  1. nC 1 (u,v) = C(u,v)[(- ln u) a + (- ln v) a ] -1+1/a (-ln u) a-1 /u nc(u,v) = C(u,v)u -1 v -1 [(-ln u) a +(-ln v) a ] -2+2/a [(ln u)(ln v)] a- 1  {1+(a-1)[(-ln u) a +(-ln v) a ] -1/a }  (a) = 1 – 1/a nSimulate two independent uniform deviates u and v nSolve numerically for s>0 with ue s = 1 + as nThe pair [exp(-sv a ), exp(-s(1-v) a )] will have the Gumbel copula distribution

7. Copulas Univariate Functions Gary G Venter

8. Copulas Univariate Functions Gary G Venter Heavy Right Tail Copula nC(u,v) = u + v – 1 + [(1 – u) -1/a + (1 – v) -1/a – 1] -a a>0 nC 1 (u,v) = 1 – [(1 – u) -1/a + (1 – v) -1/a – 1] -a-1 (1 – u) -1-1/a nc(u,v) = (1+1/a)[(1–u) -1/a +(1– v) -1/a –1] -a-2 [(1–u)(1– v)] -1-1/a  (a) = 1/(2a + 1) nCan solve conditional distribution for v

9. Copulas Univariate Functions Gary G Venter

10. Copulas Univariate Functions Gary G Venter Joint Burr nF(x) = 1 – (1 + (x/b) p ) -a and G(y) = 1 – (1 + (y/d) q ) -a nF(x,y) = 1 – (1 + (x/b) p ) -a – (1 + (y/d) q ) -a + [1 + (x/b) p + (y/d) q ] -a nThe conditional distribution of y|X=x is also Burr: nF Y|X (y|x) = 1 – [1 + (y/d x ) q ] -(a+1), where d x = d[1 + (x/b) p/q ]

11. Copulas Univariate Functions Gary G Venter Partial Perfect Correlation Copula Generator nAssume logical values 0 and 1 are arithmetic also nh : unit square  unit interval nH(x) =  0 x h(t)dt nC(u,v) = uv – H(u)H(v) + H(1)H(min(u,v)) nC 1 (u,v) = v – h(u)H(v) + H(1)h(u)(v>u) nc(u,v) = 1 – h(u)h(v) + H(1)h(u)(u=v)

12. Copulas Univariate Functions Gary G Venter h(u) = (u>a) nH(u) = (u – a)(u>a)  (a) = (1 – a) 4

13. Copulas Univariate Functions Gary G Venter h(u) = u a nH(u) = u a+1 /(a+1)  (a) = 1/[3(a+1) 4 ] + 8/[(a+1)(a+2) 2 (a+3)]

14. Copulas Univariate Functions Gary G Venter The Normal Copula nN(x) = N(x;0,1) nB(x,y;a) = bivariate normal distribution function,  = a nLet p(u) be the percentile function for the standard normal: nN(p(u)) = u, dN(p(u))/du = N’(p(u))p’(u) = 1 nC(u,v) = B(p(u),p(v);a) nC 1 (u,v) = N(p(v);ap(u),1-a 2 ) nc(u,v) = 1/{(1-a 2 ) 0.5 exp([a 2 p(u) 2 -2ap(u)p(v)+a 2 p(v) 2 ]/[2(1- a 2 )])}  (a) = 2arcsin(a)/  n a: 0.156430.382680.707110.923880.98769  0.100000.250000.500000.750000.90000

15. Copulas Univariate Functions Gary G Venter

16. Copulas Univariate Functions Gary G Venter Tail Concentration Functions nL(z) = Pr(U<z,V<z)/z 2 nR(z) = Pr(U>z,V>z)/(1 – z) 2 nL(z) = C(z,z)/z 2 n1 - Pr(U>z,V>z) = Pr(U<z) + Pr(V<z) - Pr(U<z,V<z) n = z + z – C(z,z). nThen R(z) = [1 – 2z +C(z,z)]/(1 – z) 2 nGeneralizes to multi-variate case

17. Copulas Univariate Functions Gary G Venter

18. Copulas Univariate Functions Gary G Venter Cumulative Tau  –1+4  0 1  0 1 C(u,v)c(u,v) d v d u nJ(z) = –1+4  0 z  0 z C(u,v)c(u,v) d v d u/C(z,z) 2 nGeneralizes to multi-variate case

19. Copulas Univariate Functions Gary G Venter

20. Copulas Univariate Functions Gary G Venter Cumulative Conditional Mean nM(z) = E(V|U<z) = z -1  0 z  0 1 vc(u,v) d v d u nM(1) = ½ nA pairwise concept Copula Distribution Function nK(z) = Pr(C(u,v)<z) nGeneralizes to multi-variate case

21. Copulas Univariate Functions Gary G Venter

22. Copulas Univariate Functions Gary G Venter HRT Gumbel Frank Normal Parameter0.9681.674.920.624 Ln Likelihood124157183176 Tau0.340.400.450.43

23. Copulas Univariate Functions Gary G Venter