Empirical bivariate copula when one variable is restricted

Question

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 could really use some help.

Answer 1

For the empirical copula between $X$ and $Y$ as well as for the (estimate of the) probability $P(X\leq u, Y\leq 1)$ you would need additional assumptions before you restrict to $Y\leq 1$ or $Y\leq 2$. But you could calculate $P(X\leq u\mid Y\leq 1)$, i.e. the empirical distribution of $X$ conditional on $Y\leq 1$, just using data with $Y\leq 1$. This is proportional to $P(X\leq u, Y\leq 1)$ and sometimes all one needs.