Rational-Power Shifted Lagrangian Distribution for Count Data with Flexible Dispersion

The rational-power shifted Lagrangian distribution and its corresponding regression model are discrete Lagrange probability distributions. The proposed model is constructed from a shifted Lagrangian framework with a rational-power component that introduces an additional shape parameter and provides greater flexibility in modeling dispersion and tail behavior. The derivation of the distribution is presented, and its main statistical properties are discussed, together with maximum likelihood estimation based on the expected Fisher information matrix. Using this distribution, a rational-power shifted Lagrangian regression model is formulated for analyzing count data. Simulation results are used to examine the performance of the parameter estimators and to compare the proposed model with the Poisson and modified Sunil models. A real data application using domestic violence data is also provided to illustrate its practical usefulness. The proposed model has a better fit and lower information criteria than the competing models, suggesting it could be used to model overdispersed count data.