Introduction to Bayesian Estimation and Copula Models of Dependence

5 Statistical Dependence Structures

5.1 Introduction

In this chapter we will briefly survey common statistical tools for modeling dependence between two or more random variables. In order to explain the problem in its most general form, let us consider a dataset consisting of n pairs (x_i, y_i), where x_i and y_i represent values of two random variables X and Y corresponding to the ith case (observation). A classical example introduced by Gary Alt, a famous black bear expert [1], contains measurements of length (X, measured in inches) and weight (Y, measured in pounds) of a sample of 143 black bears taken in the state of Pennsylvania. Figure 5.1 shows the scatterplot of this dataset.

Figure 5.1 Black bears in Pennsylvania.

Another example deals with a joint survival or joint mortality problem. We consider data on the length of human life, studying married couples, where X represents the wife's age at death, and Y represents the husband's. Figure 5.2 shows the scatterplot for about 11,000 pairs of spouses observed for a period of 5 years introduced in [7]. Evidently, not all of the observed died by the end of the period, so the actual number of points on the graph is much smaller than 11,000.

Figure 5.2 Joint mortality data.

Why can X and Y in these two examples be considered ...