On the length of copula level curves

Motivated by the well-known fact that the surface of copulas is closely related to common dependence measures such as Spearman's rho, we investigate level curves of bivariate copulas and study their lengths. To this end, we establish the length profile LC(t) which maps each level t∈[0,1] to the length of the respective level curve. Some basic properties of the length profile, such as continuity and differentiability with respect to t, are examined. Based on the length profile, a measure ℓC is defined, which can be interpreted as the average level curve length. ℓC is a measure of association, it is, however, not a concordance measure in general. Some further, partially surprising properties, such as closed-form formulas of ℓC for completely dependent copulas, conclude the paper.