DOI: 10.14457/tu.the.2024.88
title: Interval estimation of the dependence parameter in bivariate clayton copulas: non-covariate and incorporating covariate models

description: The objective of this dissertation is to develop and evaluate interval estimation methods for the dependence parameter of Clayton copula (????), considering both known and unknown marginal distributions. In scenarios without covariates, the dependence structure between two variables is modeled using the Clayton copula and evaluated through three methods: Wald confidence intervals, likelihood-based confidence intervals, and Bayesian credible intervals. An explicit formula for the observed Fisher information is derived, enabling precise variance estimation necessary for Wald confidence intervals. The study examines Bayesian intervals using three distinct prior distributions: a uniform non-informative prior, a low-variance gamma prior, and a high-variance gamma prior.For known marginal distributions, the simulation results indicate that most methods perform robustness across various sample sizes and ???? values, except for Bayesian credible intervals with a uniform prior, which performs optimally only at high ???? and larger sample sizes. The Bayesian credible intervals with a low-variance gamma prior consistently outperforms the other methods. The impact of sample size on interval lengths is significant, with larger samples resulting in narrower intervals. Additionally, an increase in ???? correlates with increased interval lengths across all methods.When the marginal distributions are unknown, two main approaches are utilized in the study: a nonparametric method, where empirical cumulative distribution functions are applied to transform bivariate samples to a unit scale, and a parametric method that fits distributions to the marginal data, estimating parameters via maximum likelihood and then applying probability integral transformation to achieve a unit scale. For small sample sizes, Wald and likelihood-based confidence intervals, along with Bayesian credible intervals using a uniform prior in parametric settings, consistently outperform nonparametric methods across all values of ???? values. Bayesian intervals with both low-variance and high-variance gamma priors exhibit stable coverage probability behavior across parametric and nonparametric methods. While nonparametric methods generate wider intervals, increasing sample sizes result in the convergence of interval lengths between parametric and nonparametric approaches across all ???? values. Additionally, when marginal distributions are misspecified in parametric transformations, Bayesian intervals with a low-variance gamma prior maintain robustness for small ???? values and sample sizes of 100 or fewer. The inclusion of a covariate in the model prompts an investigation into Wald and likelihood-based confidence intervals, assuming known conditional marginal distributions. It is assumed that the covariate influences only the marginal distributions through regression models, without impacting the copula-based dependence structure. Both nonparametric and parametric approaches are considered for transforming adjusted residuals to a unit scale. These methods perform well only at ???? = 0.2222, but coverage probabilities decline sharply as θ increases across all sample size. The study also examines the impact of misspecified marginal regression models on the robustness of interval estimation for the Clayton copula’s dependence parameter. Wald and likelihood-based confidence intervals show similar performance, with coverage probability increasing at ???? = 0.2222, as sample sizes approach 100. However, as ???? increases, coverage probabilities decline sharply, approaching zero across all sample sizes. The parametric approach also demonstrates a similar pattern in coverage probability behavior under these situations.