Copulas, Characterization, Correlation, and Counterexamples

Abstract

1. Copulas Copulas are functions that join univariate distribution functions to form multivariate distribution functions. They were first introduced in 1959 by A. Sklar [10] to answer some questions posed by M. Frechet concerning the relationship between a multidimensional probability distribution function and its lower dimensional marginals. Over the past 30 years or so, copulas have played an important role in several areas of probability and statistics, including multivariate distribution theory, nonparametric statistics, and Markov processes; but only recently have they come to the attention of the general statistical and mathematical communities. In this paper, we will survey some of the important properties of copulas, and demonstrate ways in which copulas can be employed in probability and mathematical statistics courses to enhance and illuminate the presentation of a number of topics. Specifically, we will show how copulas can be used in statistics i) to characterize some dependence concepts for two random variables; ii) to obtain a geometric interpretation of the population version of a nonparametric correlation coefficient, and to illustrate how that coefficient measures dependence; and iii) to facilitate the generation of counterexamples.