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

Technical Summary: Bivariate Frank Copula: Point Estimation and Goodness of Fit with Groundwater Application

Problem Statement
This work addresses two primary challenges in the statistical analysis of bivariate data using the Frank Copula. First, while previous research (Pham et al., 2025) established the superiority of the Maximum Likelihood Estimator (MLE) over method-of-moments estimators for the association parameter $\theta$, the performance of Bayesian estimators under noninformative priors remained unexplored. Second, while Genest et al. (2006) introduced Goodness of Fit (GoF) tests for Frank Copulas, their critical values were based on limited simulations (1,000 replications) and lacked comprehensive tables for varying sample sizes and parameter ranges. Additionally, the authors identified a need to apply these statistical tools to real-world environmental data, specifically groundwater contamination in the Mekong Delta, where marginal distributions are often non-normal.

Methodology
The paper is divided into two major components:

1. Bayesian Point Estimation:
   The authors investigate two Bayes estimators for the association parameter $\theta$ under squared error loss:

   • Bayes Flat Prior Estimator (BFPE): Derived using a noninformative generalized flat prior ($\pi_{FP}(\theta) = d > 0$).
   • Bayes Jeffreys Prior Estimator (BJPE): Derived using the invariant Jeffreys prior ($\pi_{JP}(\theta) = \sqrt{I(\theta)}$), where $I(\theta)$ is the Fisher Information Per Observation.

   These estimators are compared against the MLE through a comprehensive simulation study involving $M = 4 \times 10^4$ replications. The study evaluates Bias and Mean Squared Error (MSE) across sample sizes $n$ ranging from 15 to 100 and parameter values $\theta$ from 0.1 to 10. The numerical integration required for the Bayes estimators is performed using adaptive quadrature in R, with careful attention to computational precision for small sample sizes.

2. Goodness of Fit (GoF) Testing:
   The authors revisit the Cramér–von Mises ($S_n$) and Kolmogorov–Smirnov ($T_n$) test statistics proposed by Genest et al. (2006). They conduct an extensive simulation study with 10,000 replications to generate expanded tables of critical values for $S_n$ and $T_n$ at the 90th and 95th percentiles. These tables cover sample sizes $n \in [25, 1000]$ and association parameters $\theta \in [-25, 25]$. The study also analyzes the distributional properties of these statistics, noting their lack of symmetry with respect to $\theta$.

3. Application to Groundwater Data:
   The methodology is applied to a dataset from Dong Thap province, Vietnam, comprising groundwater samples from two subregions (North, $n=23$; South, $n=44$). The study models the association between Arsenic (As) and three benign elements: Chloride (Cl), Redox level (Eh), and pH. The authors first verify the non-normality of the marginal distributions and then test the hypothesis that the bivariate pairs follow a Frank Copula distribution using the newly generated critical values and bootstrap p-values.

Key Results

• Estimator Performance:

  • Small Samples ($n \le 25$): The BJPE uniformly dominates both the BFPE and the MLE in terms of MSE. The MLE performs well but is slightly inferior to the BJPE in this range.
  • Moderate to Large Samples ($n > 25$): The performance gap narrows significantly. For $n > 25$, the BJPE, BFPE, and MLE exhibit nearly identical performance in terms of both Bias and MSE, becoming statistically equivalent.
  • Computational Nuance: The authors note that for very small samples ($n \le 10$), solving the normal equation via the nleqslv command in R yields less accurate MLE results compared to maximizing the log-likelihood directly via optim.

• GoF Test Properties:

  • Asymmetry: Contrary to intuition, the critical values for $S_n$ and $T_n$ are not symmetric with respect to $\theta$ (i.e., values for $\theta$ and $-\theta$ differ). This is attributed to the lack of symmetry in the distribution function $K(t, \theta)$.
  • Monotonicity: The critical values for both statistics decrease as the magnitude of $\theta$ increases (moving away from 0).
  • Implementation: The authors propose a practical rule for implementing the GoF test: use the absolute value of the estimated parameter, $|\hat{\theta}|$, to select critical values from the tables, or reorient the data $(U, 1-V)$ if $\hat{\theta}$ is negative and $|\hat{\theta}| \ge 2.5$.

• Groundwater Application:

  • The GoF tests (both $S_n$ and $T_n$) and bootstrap p-values overwhelmingly support the hypothesis that the bivariate pairs (As-Cl, As-Eh, As-pH, Eh-pH) in both subregions can be modeled by the Frank Copula.
  • The estimated association parameters ($\hat{\theta}$) reveal distinct dynamics: As and pH show positive association in the North but weak association in the South; As and Eh show strong negative association in the South.
  • Parametric correlation estimates derived from the Frank Copula ($\hat{\hat{\rho}}_K, \hat{\hat{\rho}}_S$) align closely with nonparametric estimates, validating the model fit.

Significance and Claims
The paper claims to make three specific contributions:

1. Bayesian Extension: It provides the first comprehensive comparison of Bayes estimators (BFPE and BJPE) against the MLE for the Frank Copula. It establishes that the BJPE is the preferred estimator for small sample sizes ($n \le 25$) due to its superior MSE performance.
2. Refinement of GoF Tests: It complements and refines the seminal work of Genest et al. (2006) by providing high-precision critical value tables based on 10,000 replications (versus 1,000 previously) and uncovering non-intuitive asymmetry in the test statistics' distributions.
3. Practical Application: It demonstrates the utility of the Frank Copula in modeling complex, non-normal environmental data, specifically groundwater dynamics in Vietnam, offering a robust framework for understanding the association between carcinogenic and benign elements.

The authors maintain a modest tone, noting that while the GoF tests support the Frank Copula model, this application serves as a starting point for future studies involving nonlinear regression and multivariate extensions. They also highlight that the computational aspects of the MLE in R require careful handling for very small samples to avoid bias.