A Framework for Constructing, Estimating, and Validating the Bivariate Generalized Log-Gamma Distribution

The generalized log-gamma (GLG) distribution provides a flexible framework for modelling continuous data with different directions and degrees of skewness. However, its univariate form cannot represent dependence between two related response variables. This method article develops a bivariate generalized log-gamma (BGLG) distribution from the three-parameter GLG model. The proposed procedure includes construction of the joint probability density function, derivation of its statistical properties, formulation of the likelihood function, parameter estimation, simulation, and distributional adequacy assessment. The validity of the proposed density is established by showing that its integral over the bivariate sample space equals one. Because the maximum likelihood score equations have no closed-form solutions, the parameters are estimated iteratively using the Berndt-Hall-Hall-Hausman algorithm. The adequacy of the BGLG distribution is assessed using a chi-square quantile plot. An illustrative simulation using 500 observations under a single parameter setting produced asymmetric marginal distributions and positive dependence, while the chi-square diagnostic closely followed the reference line, with minor upper-tail deviations, providing initial support for the proposed BGLG construction. Our approach involves:•Constructing the BGLG joint density and deriving its properties.•Estimating parameters by maximum likelihood using the BHHH algorithm.•Evaluating the distribution through simulation and chi square diagnostics.