Empirical Copula when there are thresholds for no data pairs

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 would really appreciate some help.

Answer 1

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.

Answer 2

Estimate the law $\hat{F}_X$ of X from all the data avaliable for X, say $x_1,...x_{n_X}$.

Estimate the law $\hat{F}_Y$ of Y from all the data avaliable for Y, say $y_1,...,y_{n_Y}$.

Estimate the law of the copula $\hat{C}$ from all the paired data avaliable for $(X,Y)$, say ${(x_i,y_i)}_{i \in {1,...,min(n_X,n_Y)}}$.

Then the joint probability $\mathbb{P}(X<a, Y<b) = F_{X,Y}(a,b)$ can be estimated through :

$$\hat{C}(\hat{F}_X(a),\hat{F}_Y(b))$$

This is the beauty of copula modeling.

The fact that the empirical distributions were not computed on the same data as the copula or on paired data is not a problem. Furthermore, the values of your tresholds and the values taken by the marginals are irrelevant to the problem of "Which data can i use".

More precisely, estimations can be done as you like -- Using e.g ecdf and deuheuvel's copulas, see https://en.wikipedia.org/wiki/Copula_(probability_theory)#Empirical_copulas and https://en.wikipedia.org/wiki/Empirical_distribution_function .