The Copula Density Function

The copula density function is the joint density function of Xand Y,f{x, y). For independent variables, the joint density function is the product of the densities of the marginal distribution functions, using X= x and Y = y: F(x, y) =fXx)f(y). For dependent variables, it increases with the positive dependency between those variables. The copula density is the ratio of the joint density to the product of the two densities.

The copula density provides an intuitive view of copula functions. Whenever the variables are independent it is equal to 1. When the joint fx, y) is higher than the product, which happens when X and Y co-vary together, it is higher than 1. When the opposite holds, X and Y vary inversely, and the copula density is lower than 1.

The next paragraphs deal with the Gaussian copula example and detail all its forms, expanded or compact, starting with remainders of the bivariate normal distribution.

Generalization to Several Variables

When extending copula to more than two variables, it is convenient to designate variables using subscripts i, with / varying from 1 to N, N being the number of variables. If we use the values of the variables X — xp each following f(X^

The same copula is also expressed as a function of the percentiles u of each variable:

Throughout this chapter we deal with bivariate copula functions, and avoid subscripting the variables, using different letters for the two variables (w and v and x and y). However, we keep subscripts for the cumulative distribution functions and the probability density functions, respectively f^X) and f2(y) and ffa) and f2(y), because they can be other than normal distributions.

The easiest copula function to handle is the Gaussian copula because it takes the same familiar form of an integral of jointly normal functions. The Gaussian copula is detailed hereafter. It leads to formulas similar to those used to correlate normal distributions and serves as an example throughout this chapter. Alternative copula functions deal with student distributed variables or exponentially distributed variables. For convenience, we stick in this chapter to the Gaussian copula.

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