Are copulas good tool to model the dependence between the two uncorrelated variables. I have X and Y datasets with 260 data points each with Pearson's correlation=-0.06 and Kendall rank correlation=0.1093. Will copulas be able to capture the dependence between the two variables? I tried fitting Arhimedian copulas and found that the Gumbel copula is the best fit. When I am finding conditional copula distribution to obtain Function C(P<=y|X=x), the results are not good. I could not figure out where the problem lies? Is it because I chose copula for uncorrelated variables or I am missing something in copula analysis.
Uncorrelated does not imply independent, hence a copula could capture the dependence for very small correlations if there is any. As an example, the student t copula with a degree of freedom of 0.4 and a correlation of 0 even has a tail-dependence of 0.4 and you can see the structure in the scatter plot of a sample. However, if there is not any structure in your data, the conditional copula
C(v|u=u_0) will result in a uniform distribution on [0,1] (= no matter which value
v can still be anything). You can use the copulatheque.org to play around with several copula families to better assess and understand their properties.