A Simpson Based Estimation Approach for the Overlapping Coefficient of k>=2 Normal Distributions

1. Problem Statement

The Overlapping Coefficient (OVL), specifically the Weitzman coefficient ($\Delta$), quantifies the similarity between probability distributions by measuring the shared area under their density functions.

• Current Limitation: While the OVL for two distributions ($k=2$) is well-studied, extending this to $k \ge 2$ populations presents significant analytical and computational challenges.
• The Core Issue: For $k$ normal distributions, the OVL is defined as the integral of the minimum of the $k$ density functions:
  $$\Delta_k = \int_{-\infty}^{\infty} \min\{f_1(x), f_2(x), \dots, f_k(x)\} dx$$
  As $k$ increases, the number of intersection points between density functions grows rapidly, creating a piecewise function with complex boundaries. This makes analytical integration impractical and computationally expensive, necessitating a robust numerical approximation method.

2. Methodology

The authors propose a general estimation framework combining Maximum Likelihood Estimation (MLE) with Simpson's Numerical Integration Rule.

A. Plug-in Estimation

Since true parameters ($\mu_i, \sigma_i^2$) are unknown, the method uses Plug-in Maximum Likelihood Estimators (MLEs) derived from independent random samples:

• $\hat{\mu}_i = \bar{X}_i$ (Sample Mean)
• $\hat{\sigma}_i^2 = \frac{1}{n_i}\sum (X_{ij} - \bar{X}_i)^2$ (Sample Variance)
  These are substituted into the normal density functions to create estimated densities $\hat{f}_i(x)$.

B. Numerical Integration via Simpson's Rule

To compute the integral of the minimum function $g(x) = \min\{\hat{f}_1(x), \dots, \hat{f}_k(x)\}$, the authors address the infinite domain $(-\infty, \infty)$ using a smooth transformation:

1. Transformation: The infinite interval is mapped to the finite unit interval $[0, 1]$ using a generalized logistic cumulative distribution function:
   $$u = G(x) = 1 - (1 + e^x)^{-\alpha}$$
   where $\alpha > 0$ is a tuning parameter.
2. Jacobian Adjustment: The integral is rewritten in terms of $u$, incorporating the derivative $dx/du$ to preserve the area.
3. Simpson's Rule: The transformed integral over $[0, 1]$ is approximated using Simpson's rule with $r$ subintervals. The grid excludes endpoints ($u=0, 1$) to avoid divergence in the Jacobian term.

C. Tuning Parameter ($\alpha$)

The study evaluates three strategies for the transformation parameter $\alpha$:

1. Fixed $\alpha = 1$ (Symmetric logistic).
2. Fixed $\alpha = 2$.
3. Data-driven $\hat{\alpha}$: Estimated from the sample data to adapt to the specific distribution shape.

3. Key Contributions

• Generalization to $k \ge 2$: The paper provides a scalable framework for estimating OVL for any number of normal populations, overcoming the analytical intractability of high-dimensional intersections.
• Consistency Proof: The authors prove that the proposed Simpson-based estimator ($\hat{\Delta}_k^{(simp)}$) is consistent. As sample sizes $n_i \to \infty$, the estimator converges in probability to the true OVL value. The proof relies on the uniform convergence of the estimated densities and the properties of Simpson's rule error bounds ($O(h^4)$).
• Computational Efficiency: By transforming the domain and using a standard numerical integration rule, the method avoids complex root-finding algorithms required to determine intersection points analytically.
• Robustness to $\alpha$: The study demonstrates that the estimator's performance is largely insensitive to the specific choice of $\alpha$ (fixed or data-driven), provided $\alpha$ is not extreme.

4. Simulation Results

A Monte Carlo simulation study ($R=1000$ repetitions) was conducted comparing the proposed Simpson-based estimators against a benchmark estimator ($\hat{\Delta}_3^*$) by Eidous and Alsheyyab (2025).

Scenarios Tested:

• Four scenarios ($S_1$ to $S_4$) representing Large ($\Delta_3 \approx 0.93$), Moderate ($\approx 0.69$), Small ($\approx 0.47$), and Very Small ($\approx 0.07$) overlap.
• Sample sizes: Balanced $(50,50,50)$ and Unbalanced $(50,100,150)$.

Key Findings:

1. High Overlap ($\Delta_3 \approx 0.93$): All estimators perform similarly with negligible differences in bias and efficiency.
2. Moderate to Small Overlap: The Simpson-based estimators begin to outperform the benchmark, showing lower Relative Root Mean Squared Error (RRMSE) and higher efficiency.
3. Very Low Overlap ($\Delta_3 \approx 0.07$): The proposed method shows significant superiority.
   • Efficiency gains reached ~34% (Efficiency > 1.33) compared to the benchmark.
   • The data-driven $\hat{\alpha}$ version performed exceptionally well, confirming its adaptability.
4. Parameter Sensitivity: The three versions of the Simpson estimator ($\alpha=1, \alpha=2, \hat{\alpha}$) yielded nearly identical results, suggesting that a simple fixed $\alpha$ (e.g., 1 or 2) is sufficient for practical applications.

5. Significance and Conclusion

This paper offers a flexible, computationally efficient, and theoretically sound solution for estimating the overlapping coefficient in multi-population settings.

• Practical Impact: The method is particularly valuable in fields requiring the comparison of multiple groups (e.g., pharmacology for drug similarity, economics for income inequality across multiple demographics, or bioinformatics).
• Low-Overlap Advantage: The most critical contribution is the method's ability to maintain high accuracy and efficiency even when distributions have very little common support (low overlap), a scenario where traditional estimators often struggle.
• Future Directions: The framework is generalizable to other smooth parametric families and larger values of $k$, paving the way for broader applications in statistical inference and classification.