Parametric and semiparametric estimation methods for bivariate copula in rainfall application

Abstract

Hydrological phenomena such as drought, flood, and rainfall are one of the natural phenomena that often provide dependent multivariate variables. The correlation of the hydrologic dependent variables can be described by using copula. To determine a specified copula structure that fitted with the marginal variables, the copula dependence parameter needs to be estimated. This study focuses on the application of parametric and semiparametric approaches in estimating the copula dependence parameter. The performance of seven parameter estimation methods namely, maximum likelihood (ML) estimation, inference function of margins (IFM), maximization by parts (MBP), pseudo maximum likelihood (PML), the inversion of rank correlation coefficient approach based on Kendall’s tau and Spearman’s rho and maximum likelihood based on kernel density estimation (MLKDE) are compared in the simulation and empirical studies. The simulation and empirical studies are limited to the case of bivariate copulas. The result from the simulation study shows that the parametric approaches are inefficient when the marginal distributions are misspecified. Among the parametric approaches, MBP performs better than MLE and IFM. While, for semiparametric approaches, PML performs well and consistent for any correlation and sample size. The PML can be efficient and consistent with the parametric once the sample size is large. The empirical study is done by applying the estimation methods to identify the dependence of the daily rainfall at two rain gauge stations Station Kuala Krai and Station Ulu Sekor. The result from the empirical study is consistent with the result from the simulation study. Thus, it can be concluded that MBP is preferred when the copula and the marginal distributions are known. While, PML is preferred when the marginal distribution is unknown, where the situation is common in a real data application.