Introduction to Bayesian Estimation and Copula Models of Dependence

6 Copula Models of Dependence

6.1 Introduction

The previous chapter contained a brief description of models of dependence between two random variables X and Y. We have agreed that modeling entire joint distribution of X and Y described by a function H(x, y) = P(X ≤ x, Y ≤ y) for all possible values x and y would make it possible to address all sorts and sources of dependence.

However, simple bivariate distributions such as bivariate normal or bivariate exponential would rarely satisfy the needs of good data fit. Geometrically speaking, scatterplots of real bivariate data such as black bears’ biometrics and joint mortality discussed in Section 5.1 exhibit types of behavior (skewness, asymmetry, multimodality, fat tails) which are impossible to describe within the confinements of most common bivariate distribution families.

Additional consideration should be given to our observation that in many applications researchers tend to obtain much more data and prior information regarding individual variables X and Y taken separately and possess relatively little information related to their interdependence. Therefore, as in the joint mortality example, there exist many good models for separate male and female life lengths based on extensive mortality databases; at the same time very few reliable models have been developed for joint mortality since little is known about dependent lives, and only a few paired datasets are commonly available.

Ideally, a statistical model for joint distribution ...