Estimating a Copula with Truncated Observations of One Variable

Summary

The document considers estimating the dependence between two random variables when observations of one variable are available only below a cutoff. The target is a joint probability involving an event for each variable, expressed through their marginal distributions and a copula. The response cautions that restricting the sample to observations below a limit does not, by itself, provide enough information to estimate the full empirical copula or the requested joint probability; additional assumptions would be needed.

It distinguishes that target from a conditional distribution that can be estimated directly from the restricted data. With observations satisfying the tighter event of interest, one can estimate the distribution of the other variable conditional on that event. This conditional probability is proportional to the joint probability, but does not alone identify it unless the event probability is also known. The short exchange offers no specific model for extrapolating beyond the observed range or evidence that the available truncated sample supports a full dependence estimate.

Key ideas

• Estimating the full empirical copula from observations truncated in one variable requires additional assumptions.
• The desired joint probability cannot generally be inferred from the restricted sample alone.
• Data selected by the event of interest can estimate the other variable’s conditional distribution within that event.
• The conditional probability is proportional to the joint probability, but the excerpt gives no extrapolation method beyond the observed range.

Tags

# Empirical bivariate copula when one variable is restricted


# Empirical bivariate copula when one variable is restricted












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 by g g (score 1)

https://quant.stackexchange.com/a/11179

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.

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