I am trying to find the empirical copula linking two random variables $X$ and $Y$. I have some data available but it's limited with respect to the variable $Y$ and I am not convinced it's enough data and will lead to the right copula.
The variable $Y$ can attain any value greater than 0 and I am interested in the probability
$$\mathbb{P}(X\leq u, Y\leq 1)=C(F_{X}(u),F_{Y}(1))$$ for different $u$. I have data pairs for $Y\leq 2$, but no data pairs for $Y$ greater than 2. As I am only interested in the copula linking the probability of $\mathbb{P}(Y\leq 1)$ and $\mathbb{P}(X\leq u)$ and not interested in probabilities of $Y$ greater than 2, can I use the data with values of $Y$ up to 2 and not greater or do I need data for all possible values of $Y$?
I've been stuck on this for a few weeks now and would really appreciate some help.
The empirical Copula (also 'Deheuvels Copula'): http://en.wikipedia.org/wiki/Copula_(probability_theory)#Empirical_copulas
This Copula is based on its given sample, it needs a full sample for a good estimate, so you can only use it in your case if the actual probability for the unavailable data is close to zero aswell.