Fit to a T - Estimation, Application and Limitations of the T-copula

Copulas provide a convenient way to represent joint distributions. In fact the joint distribution function can be expressed as the copula function applied to the separate individual distributions. That is, F(x1, x2, . . .xm) = C[F1(x1), F2(x2), . . . Fm(xm)] where C is the copula function. Background information on copulas is covered in a number of papers, and will be largely assumed here.

The starting point of this topic is bivariate copulas, but most of these do not extend well into higher dimensions. For a multivariate copula for insurance related variates you would like to be able to feed in a correlation matrix of the variates as well as have some control over the degree of correlation in the tails of the distributions. Often more than two related variates are needed, such as losses in several lines of insurance.

This paper focuses on the t-copula, which meets these minimum requirements, but just barely. You can input a correlation matrix and you do have control over the tail behavior, but you only have one parameter to control the tail, so all pairs of variates will have tail correlation that is determined by that parameter. The normal copula is a limiting case, in which the tails are ultimately uncorrelated if you go out far enough.

The structure of the paper is to jump right in to a discussion of the t-copula in the bivariate case, then extend this to higher dimensions. A trivariate example is given using cat model output for three lines of insurance. Methods for selecting parameters and testing goodness of fit are discussed in this context, using descriptive functions.

Keywords: Copula, correlation, multi-variate