Logarithmic modified gamma distribution and its application to disability data in Saudi Arabia

This study introduces the Logarithmic-Modified Gamma Distribution (LMGD), a flexible generalization of the classical Gamma distribution designed to effectively model skewed, asymmetric, and heavy-tailed disability data. The probability density function, cumulative distribution function, quantile function, survival function, hazard and reverse hazard rate functions, mean residual life function, and entropy measures are derived along with order statistics of the LMGD to establish its theoretical framework comprehensively. Parameter estimation for the LMGD was carried out using Maximum Likelihood Estimation and the Method of Moments. The performance of the estimators is evaluated through a simulation study in terms of bias and efficiency. Real disability data from Saudi Arabia are fitted to the proposed model to demonstrate its practical applicability. The results indicate that the LMGD provides a better fit than several well-known benchmark distributions, including the Exponential, Weibull, Gamma, Lindley, Noncentral F, Half-Cauchy, and Lévy distributions, thereby establishing it as a flexible and effective tool for modeling demographic and health-related data.