Parameter Estimation for the Bivariate Compound Zero-Truncated Poisson-Gamma Model under Different Data Scenarios

Abstract

The bivariate compound zero-truncated Poisson-gamma distribution models the sum of a random number of bivariate Gamma variables, where the count follows a zero-truncated Poisson distribution, which makes it well-suited for applications in actuarial science, climatology, and reliability engineering, where zero outcomes are inherently absent. Owing to the intractable nature of the probability density function, which involves an infinite sum, the direct maximum likelihood estimation is computationally challenging. In this study, we used a standard (exact) maximum likelihood estimation when event counts were observed (complete data and Scenario~A) and employed the saddle-point approximation only when counts were latent (Scenario~B). We developed a stable maximum likelihood estimation based on saddle-point approximation. We derived the cumulative distribution function from the cumulant generating function and obtained the probability density function using numerical differentiation. Detailed derivations, implementation guidelines in the \textsf{R} programming language, and a parameter initialization strategy using the method of moments are provided. A simulation study using various sample sizes demonstrated the accuracy, consistency, and superiority of this method over the moment-based estimators. Computational challenges and limitations are discussed, along with potential extensions to model the dependence structures using copulas. In addition, we develop a likelihood ratio test and a formal symmetry test (for example, $H_0:\alpha_1=\alpha_2,\ \beta_1=\beta_2$) to compare nested specifications, enabling principled inference on symmetry and overall model adequacy.