Gaussian Copula for Exponential Marginal Distributions

Summary

The question asks how to construct a bivariate Gaussian copula when each variable has an exponential marginal distribution. The answer defines the copula through a pair of correlated standard normal variables: one normal variable is compared with the inverse standard normal quantile at the first copula input, while a correlated combination of that variable and an independent normal variable is compared with the second quantile. The probability that both inequalities hold gives the Gaussian copula value.

The exponential rates determine the marginal distributions, but they do not enter this copula expression directly. To combine the copula with the exponential variables, the copula inputs are obtained by applying each exponential cumulative distribution function to its variable. The dependence parameter is the correlation of the latent normal pair. The response supplies the construction but does not discuss estimation, tail dependence, or whether the Gaussian dependence assumption is suitable for a particular application.

Key ideas

• A Gaussian copula is defined by the joint probability of threshold events for correlated standard normal variables.
• The latent normal correlation controls the dependence represented by the copula.
• Exponential marginal cumulative distributions transform observed variables into copula inputs.
• The copula formula specifies dependence but does not establish that Gaussian dependence fits a given dataset.

Tags

# Bivariate Gaussian copula with exponential margins


# Bivariate Gaussian copula with exponential margins












I got little bit lost in the formulas.

Assume to have two random variables distributed exponentially $X_i \sim Exp(\lambda_i)$ and $X_j \sim Exp(\lambda_j)$.

Thus, the distribution functions are $F_{X_i}(x_i)= 1-\exp(-\lambda_i x_i)$ and $F_{X_j}(x_j)=1-\exp(-\lambda_j x_j)$.

What is the formula for a Gaussian copula, $C(u,v)$, linking these exponential margins?

## Answer by M. Jeunesse (score 3, accepted)

https://quant.stackexchange.com/a/26140

$$C(u,v) = \mathbb{P}\left(X\leq N^{(-1)}(u),\quad \rho X + \sqrt{1-\rho^2}X^\perp \leq N^{(-1)}(v)\right)$$

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