Bivariate Frank Copula: Some More Results on Point Estimation of the Association Parameter from a Bayesian Perspective and Revisiting the Goodness of Fit Tests with an Application to Model Groundwater Data from Dong Thap, Vietnam
Abstract
This work consists of two major parts. (i) The first part extends the recent comprehensive study of Pham et al. (2025), where three classical point estimators of the association parameter of a bivariate Frank Copula were compared and the maximum likelihood estimator (MLE) was established to have the best overall performance in terms of bias and mean squared error (MSE). We investigate two Bayes estimators under two natural priors, namely the noninformative (generalized) flat prior and the invariant Jeffreys prior, and find that the latter uniformly dominates the former as well as the MLE in terms of MSE for small sample sizes (n <= 25). For moderate to large sample sizes (n > 25), all three estimators have almost identical performances in terms of bias and MSE. We also point out computational aspects in R that may have important implications for computing the MLE and its bias and/or MSE for very small samples. (ii) The second part uses a recent dataset from Vietnam and applies the Frank Copula to analyze the association between groundwater arsenic concentration and three other benign elements that are easy to monitor. In this application, we revisit the two goodness-of-fit (GoF) tests introduced by Genest et al. (2006), explore non-intuitive behavior of the two test statistics, and provide extensive tables of their critical values through a comprehensive simulation study. This complements and refines the computational results of Genest et al. (2006). Using the groundwater data, we also demonstrate that nonparametric marginals can offer benefits over parametric marginals in predicting arsenic concentration from a benign element using the Frank Copula-based regression model.
Related
- Power-posterior implicit sampling for efficient Bayesian experimental design: Application to contaminant source identification in groundwater flow
- Revisiting entropy and entropy-related magnitudes of a symbolic sequence and their estimation
- Asymptotic approximation of sensitivities in finite dimensional continuous data assimilation with application to parameter estimation
- A New Extended Exponential Rayleigh Distribution: Estimation and Application
- Additive results of the dual Moore--Penrose generalized inverse with an application to rigid-body motion estimation
BibTeX
@misc{pham2026bivariate,
title = {{Bivariate Frank Copula: Some More Results on Point Estimation of the Association Parameter from a Bayesian Perspective and Revisiting the Goodness of Fit Tests with an Application to Model Groundwater Data from Dong Thap, Vietnam}},
author = {Thi-Yen-Anh Pham and Dung T. Nguyen and Nabendu Pal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.06496},
eprint = {2605.06496},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.06496},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bivariate Frank Copula: Some More Results on Point Estimation of the Association Parameter from a Bayesian Perspective and Revisiting the Goodness of Fit Tests with an Application to Model Groundwater Data from Dong Thap, Vietnam
AU - Thi-Yen-Anh Pham
AU - Dung T. Nguyen
AU - Nabendu Pal
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2605.06496
UR - https://arxiv.org/abs/2605.06496
AB - This work consists of two major parts. (i) The first part extends the recent comprehensive study of Pham et al. (2025), where three classical point estimators of the association parameter of a bivariate Frank Copula were compared and the maximum likelihood estimator (MLE) was established to have the best overall performance in terms of bias and mean squared error (MSE). We investigate two Bayes estimators under two natural priors, namely the noninformative (generalized) flat prior and the invariant Jeffreys prior, and find that the latter uniformly dominates the former as well as the MLE in terms of MSE for small sample sizes (n <= 25). For moderate to large sample sizes (n > 25), all three estimators have almost identical performances in terms of bias and MSE. We also point out computational aspects in R that may have important implications for computing the MLE and its bias and/or MSE for very small samples. (ii) The second part uses a recent dataset from Vietnam and applies the Frank Copula to analyze the association between groundwater arsenic concentration and three other benign elements that are easy to monitor. In this application, we revisit the two goodness-of-fit (GoF) tests introduced by Genest et al. (2006), explore non-intuitive behavior of the two test statistics, and provide extensive tables of their critical values through a comprehensive simulation study. This complements and refines the computational results of Genest et al. (2006). Using the groundwater data, we also demonstrate that nonparametric marginals can offer benefits over parametric marginals in predicting arsenic concentration from a benign element using the Frank Copula-based regression model.
ER -