Copula density

swtzang · December 8th, 2005, 1:01 am

Dear all:How to derive the density of Frank copula, i.e. 2 variates? I know that the C(u,v)=-(1/a)*ln(1+(exp(-au)-1)(exp(-av)-1)/(exp(-a)-1)), what is its univariate density function? Thanks for help.SWT

AlanS · December 8th, 2005, 1:05 pm

To find the density c(u,v) = d^2/dudv [C(u,v)] you can either:1) just differentiate C(u,v) wrt u and v directly; 2) use the result that c(u,v) = (-phi''(C(u,v)) * phi'(u) * phi'(v)) / (phi'(C(u,v)))^3 where phi is the generator, in this case phi(t) = -ln((exp(-at)-1) / (exp(-a)-1));3) cheat and differentiate C(u,v) numerically.Don't know what you mean by "univariate density function".

genkideska · December 8th, 2005, 2:14 pm

(exp(t*(1+x[,1]+x[,2]))*(-1+exp(t))*t)/(exp(t)-exp(t+t*x[,1])+exp(t*(x[,1]+x[,2]))-exp(t+t*x[,2]))^2

swtzang · December 8th, 2005, 9:52 pm

Thanks to AlanS. Method 1 works fine. However, I don't know about method 2. How to get the generator of the frank copula? Or any sources I can refer to? Thanks a lot.SWT

swtzang · December 9th, 2005, 2:28 am

Dear All:I want to draw the density of Gaussian copula with student's t marginals. I use the matlab program in the following:p=zeros(num,num);for i = 1:num for j = 1:num x1(i) = tinv(u(i),v); x2(j)= tinv(u(j),v); tmp1=1/(2*pi*(1-R^2)^0.5); tmp2=(x1(i).^2-2*R.*x1(i).*x2(j)+x2(j).^2); p(i,j)=tmp1*exp(-tmp2./(2*(1-R^2))); endendfigure;surf(x1,x2,p);However, the graph doesn't seem to match right. Can anyone fix the problem for me? Thanks for help.

swtzang · December 11th, 2005, 10:19 pm

Dear AlanS:I found the reference to the method 2. Nelson(1999), p105.Thanks.