Interval Estimation of the Dependence Parameter in Bivariate Clayton Copulas

Abstract

In various disciplines, discerning dependencies between variables remains a crucial undertaking. While correlation measures like Pearson, Spearman, and Kendall provide insight into the degree of two-variable relationships, they fall short of revealing the intricate structure of dependencies between these variables. The Clayton copula, known for its flexible attributes, becomes instrumental in unveiling this dependency structure. This paper aims to advance knowledge by providing an explicit formula for creating Wald confidence intervals (CIs) for the dependence parameter in a bivariate Clayton copula, along with a mathematical derivation of the observed Fisher information. In comparison, we also propose likelihood CIs, whose performance we examine in simulation studies using both coverage probability and average length of CIs as performance indicators. Our findings reveal that in scenarios characterized by small sample sizes, likelihood-based CIs, despite their slightly more complex computational requirements, outperform Wald CIs, yielding a coverage probability more proximate to the nominal confidence level of 0.95. However, in situations involving large samples and a dependence parameter distant from zero, both Wald and likelihood-based CIs demonstrate comparable utility. For real-world data applications, the daily closing prices of two cryptocurrencies are analyzed using the proposed CIs.