Modeling double bounded data based on correlated gamma random variables

1. Problem Statement

Many real-world phenomena involve data bounded within the unit interval $(0, 1)$, such as proportions, rates, and percentages (e.g., Human Development Index, recovery rates, financial ratios). The standard statistical approach for such data is the Beta distribution, which is derived from the ratio of two independent Gamma random variables ($X$ and $Y$):
$$Z = \frac{X}{X+Y}$$
However, the assumption of independence between $X$ and $Y$ is often unrealistic in practical applications. In many scenarios (e.g., economic income vs. expenditure, or disease cases vs. deaths), these underlying components are correlated due to shared mechanisms. Existing models for correlated Gamma variables (like the Kibble bivariate Gamma) often struggle to capture negative correlations or specific complex shapes (like bimodality) required by real data. There is a need for a flexible unit-interval model that explicitly accounts for correlation between the numerator and denominator components.

2. Methodology

The authors propose a new distribution called the Extended Bimodal Beta (EBB) distribution. The methodology involves the following key steps:

• Stochastic Representation: The model defines a random variable $Z$ as the ratio $Z = X / (X+Y)$, where $(X, Y)$ follows a Morgenstern's bivariate distribution with Gamma margins.
  • $X \sim \text{Gamma}(\alpha, \theta)$ and $Y \sim \text{Gamma}(\beta, \theta)$.
  • The joint Cumulative Distribution Function (CDF) is defined using the Farlie-Gumbel-Morgenstern (FGM) copula:
    $$F_{X,Y}(x, y) = F_X(x)F_Y(y)[1 + \rho(1 - F_X(x))(1 - F_Y(y))]$$
    where $\rho \in [-1, 1]$ is the dependence parameter.
• Derivation of PDF and CDF: Using the Jacobian transformation method and properties of special functions, the authors derive the closed-form Probability Density Function (PDF) and Cumulative Distribution Function (CDF) for $Z$.
  • The PDF involves Gauss hypergeometric functions (${}_2F_1$) and Appell double hypergeometric functions ($F_2$).
  • The resulting PDF is a linear combination of four distinct components, allowing for complex shapes.
• Parameter Estimation:
  • Maximum Likelihood Estimation (MLE): The log-likelihood function is constructed. Due to the complexity of the special functions involved, analytical solutions are impossible. The authors propose a two-stage estimation strategy:
    1. Estimate shape parameters $(\alpha, \beta)$ using the standard Beta model (assuming independence).
    2. Estimate the dependence parameter $\rho$ numerically (using gradient-based optimization) or via a method-of-moments approach based on the correlation of the underlying $X$ and $Y$.
• Simulation: A Monte Carlo simulation study was conducted using an algorithm based on the conditional distribution of the copula to generate random samples and evaluate the performance of the MLEs.

3. Key Contributions

• New Distribution (EBB): Introduction of a three-parameter unit distribution ($\alpha, \beta, \rho$) that extends the classical Beta distribution.
• Flexibility in Shape: Unlike the standard Beta distribution, the EBB model can exhibit:
  • Symmetry or asymmetry.
  • Unimodality or bimodality (specifically when $\rho \approx -1$).
  • U-shapes and J-shapes.
• Mathematical Rigor: The paper provides rigorous derivations of the PDF, CDF, moments, and moment-generating functions, utilizing advanced special functions (Appell $F_2$, Hypergeometric ${}_2F_1$).
• Handling Correlation: It explicitly models both positive and negative correlations between the underlying Gamma components, addressing a gap in existing literature where negative correlation in such ratios was difficult to model.
• Computational Framework: The authors provide R code implementations for the special functions (Appell $F_2$) and a specific algorithm for generating random samples and estimating parameters.

4. Results

• Theoretical Properties:
  • When $\rho = 0$, the EBB distribution reduces to the standard Beta distribution.
  • Symmetry is achieved when $\alpha = \beta$.
  • The model supports bimodality, particularly when $\rho$ is close to $-1$, a feature rarely captured by standard unit distributions.
• Simulation Study:
  • A Monte Carlo study with sample sizes $n \in \{30, 100, 300, 1000\}$ demonstrated that the MLEs are consistent and efficient.
  • Bias and RMSE: The Relative Bias (RB) and Root Mean Squared Error (RMSE) decreased as sample size increased, converging to zero.
  • The estimator for $\rho$ showed robustness, though RMSE was slightly higher for small sample sizes ($n=30$).
• Real-World Application (Household Financial Fragility):
  • Dataset: Analyzed 7,957 observations of household income vs. expenditure from the Bank of Italy (2008). The variable $Z$ represented the ratio of expenditure to total resources.
  • Model Comparison: The EBB model was compared against the standard Beta and Kumaraswamy distributions.
  • Performance: The EBB model yielded the lowest AIC (-15839.84) and BIC (-15818.90), significantly outperforming the Beta and Kumaraswamy models.
  • Hypothesis Testing: A Likelihood Ratio (LR) test rejected the null hypothesis ($H_0: \rho = 0$, i.e., the Beta model) in favor of the EBB model ($p < 0.01$), confirming that the correlation between income and expenditure is statistically significant and necessary for accurate modeling.

5. Significance

This paper makes a significant contribution to the field of statistical modeling for bounded data by:

1. Bridging the Gap: It provides a theoretically sound framework for modeling unit-interval data where the underlying components are correlated, a common but often ignored feature in real-world data.
2. Enhanced Flexibility: The ability to model bimodality and negative correlation allows for better fitting of complex economic and social data (e.g., financial fragility, recovery rates) that traditional Beta models fail to capture.
3. Practical Utility: The inclusion of R code, simulation algorithms, and a real-world case study demonstrates the model's applicability and ease of implementation for researchers in economics, reliability, and hydrology.
4. Future Directions: The work opens avenues for regression frameworks, Bayesian inference, and multivariate generalizations of the EBB distribution.

In conclusion, the Extended Bimodal Beta (EBB) distribution offers a versatile, analytically tractable, and empirically superior alternative to the standard Beta distribution for data arising from correlated positive random variables.