Modeling double bounded data based on correlated gamma random variables

Abstract

Many types of bounded data defined on the unit interval arise naturally as ratios of the form $X/(X + Y)$. In the existing literature, the main statistical models proposed for this type of bounded data typically based on the assumption that the random variables $X$ and $Y$ are independent. However, this assumption is often unrealistic in practical applications, where $X$ and $Y$ tend to be correlated due to shared underlying mechanisms or common sources of variability. In this paper, we overcome such limitations and propose a model in which the marginal distributions of the two components are linked by a copula, leading to a more flexible and realistic representation of unit-interval data. In particular, in the proposed model, $X$ and $Y$ are correlated gamma random variables linked by the Farlie-Gumbel-Morgenstern (FGM) copula, allowing for positive and negative correlations between the components. The mathematical properties and practical applications are rigorously investigated. Depending on the choice of parameters, the new model is very versatile, showing symmetric or asymmetric behavior, as well as unimodality or bimodality. A Monte Carlo simulation study is carried out that shows the good performance of the maximum likelihood estimator in several scenarios of parameter choices. The potential and limitations of efficient likelihood-based computations are also discussed. We evaluate the effectiveness of the new model and its estimates in modeling real-world datasets related to economics.

Figures (9)

• Figure 1: Plot of the PDF $f_Z$ with varying parameters $\alpha$ (left), $\beta$ (middle) and $\rho$ (right).
• Figure 2: Plot of the PDF $f_Z$ with varying parameters $\alpha=\beta$ for $\rho\in\{-0.5, -0.75, 0.5, 0.75\}$.
• Figure 3: Plot of the PDF $f_Z$ with varying parameters $\alpha$, $\beta$ and $\rho$, illustrating bimodality.
• Figure 4: Joint PDF of Morgenstern's bivariate distribution with Gamma margins and contour lines of the joint density for $\rho=-0.99$ in (a) and (b), $\rho=0$ in (c) and (d), and $\rho=0.75$ in (e) and (f). $\alpha=1.1$ and $\beta=1.5$.
• Figure 5: RB (left) and RMSE (right) for the ML estimates of $\alpha$ (top), $\beta$ (middle), and $\rho$ (bottom), with varying $(\alpha, \beta)$ and fixing negative $\rho=-0.5$.

Theorems & Definitions (15)

• Remark 2.1
• Remark 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3
• Remark 5.1
• Proposition A.1
• proof