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

TL;DR

This paper extends Bayesian point estimation methods for the bivariate Frank copula's association parameter, compares their performance with MLE, and applies the copula to groundwater data, revisiting goodness-of-fit tests.

Contribution

It introduces a comparison of Bayesian estimators under different priors with MLE and revisits goodness-of-fit tests with extensive simulations and real data application.

Findings

Bayes estimator under Jeffreys prior outperforms others for small samples.

All estimators perform similarly for larger samples.

Revisiting goodness-of-fit tests reveals non-intuitive behaviors and provides simulated critical values.

Abstract

This work has two major parts. First, we extend the recent study of Pham et al. (2025) on point estimation of the association parameter of a bivariate Frank copula. We investigate two Bayes estimators under the generalized flat prior and the Jeffreys prior, and compare them with the maximum likelihood estimator (MLE). Simulation results show that, for small sample sizes (n <= 25), the Bayes estimator under the Jeffreys prior uniformly outperforms both the generalized flat prior estimator and the MLE in terms of mean squared error (MSE). For moderate and large sample sizes, all estimators have very similar performances in terms of bias and MSE. We also discuss computational issues in the R package implementation that may significantly affect the computation of the MLE for very small samples. In the second part, we apply the Frank copula to analyze the association between groundwater arsenic concentration and other hydrochemical variables using a recent dataset from Vietnam. We revisit the goodness-of-fit tests proposed by Genest et al. (2006), investigate several non-intuitive behaviors of the test statistics, and provide extensive simulated critical value tables. Our results complement and refine the computational findings reported in the earlier literature.