# How are copulas used in the real world?

I have been reading about copula models. Essentially, copula models seem to be a creative method for creating a joint probability distribution from several variables, in which each individual variable can belong to a different probability distribution.

My Question: How are copula models actually used in the real world?

For example, suppose you have data corresponding to 5 variables $$X_1, X_2, X_3, X_4$$ and $$X_5$$. Let's say that you fit a copula model to these data.

Given values of $$X_2, X_3, X_4$$ and $$X_5$$, suppose you want to find the most probable value of $$X_1$$. At first glance, it seems like you could consider this as a conditional probability distribution $$X_1 | X_2 = x_2, X_3 = x_3, X_4 = x_4, X_5 = x_5\dots$$ therefore, you could use the Metropolis Hastings algorithm to draw many random samples from the above conditional probability distribution function, and then take the expected value of these samples as your answer?

Have I understood this correctly? Is this the main application of copula models?

Note: I have read about copula regression that relates more closely to GLM regression models – but it seems like the approach I outlined above might be closer to the intended use for copulas?

• I think to call it the main application is likely too strong, but it is a possible application. Dec 18, 2021 at 7:56

If you are interested in a feature of the distribution of $$X_1$$ (e.g., the mean, a percentile, etc.) given values of $$X_2$$ through $$X_5$$, these other variables can be included as covariates in a univariate model for $$X_1$$. When it comes to inference one can apply a link function to the estimator so that it is approximately normally distributed and use the delta method to estimate the standard error and invert a Wald test to produce a confidence interval.
• That is certainly a viable numerical approach for estimating the unknown fixed true population mean, $E[X_1|X_2=x_2,...,X_5=x_5]$, given the observed data. Alternatively, you can construct an estimator for $E[X_1|X_2=x_2,...,X_5=x_5]$ given the observed data, e.g. the maximum likelihood estimator. This value should be in close agreement with your numerical approach. Dec 21, 2021 at 13:17