When observing dependence between random variables we often use functions that have special characteristics, called copulas. They link distribution functions of one variable with their joint distribution function. Sklar’s theorem tells us exactly how they are linked together, and this presents theoretical background for using copulas in practice. They are helpful in statistics: we can use them to estimate distribution of random vector by estimating distribution functions of random variables.
When measuring dependence between random variables, it can be helpful to use measures of dependence. Two of the most widely used are Kendall’s $\tau$ and Spearman’s $\rho$, and because they meet the criteria of Scarsini’s definition, they are both measures of concordance.
In case of continuous random variables, copula linked with joint distribution functions is unique. However in discrete case that is no longer true. Therefore we run into troubles, as a lot of characteristics do not translate from continuous to discrete case. Nevertheless, copula theory can be useful in discrete case as well, but it has to be used with caution.
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