The Application of Copula Continuous Extension Technique for Bivariate Discrete Data: A Case Study on Dependence Modeling of Seismicity Data

Abstract

The Copula approach for continuous variables is highly developed, while discrete ones are underdeveloped due to computational difficulties and sometimes algorithm failure to convergent. Therefore, providing an alternative method for discrete variables becomes an essential issue. In this paper, a simple method is proposed to answer the problem by applying the Continuous Extension Technique (CET). This is carried out by adding random independent perturbations in the form of either Uniform distribution U(0,1) or (U(0,1) – 1), and the discrete variables are treated as continuous. Subsequently, a Copula model for resulted variables is estimated based on the Copula theory for continuous variables. This method is called a Copula continuous extension technique. Our analytic and simulation approaches show that both random perturbation forms produce the same Kendall’s Tau measure and the selected Copula bivariate model. As illustrations, the proposed method is applied to the seismicity data obtained from the annual frequencies of earthquakes that occurred in the Sumatra megathrust of Indonesia, from January 1971 to December 2018, with magnitudes (Mw) of at least 4.6. Based on the selected Copula models, our simulations confirm the evidence of dependence seismic activity in each of the two adjacent large earthquake sources. These results provide new information regarding the seismicity behavior in the Sumatra megathrust.