An Extended Lagrangian Zero-Truncated Poisson Distribution: Theory and Applications

An extended Lagrangian zero-truncated Poisson (ELZTP) distribution is proposed as a new member of the Lagrangian distributions of the second kind. The model provides a flexible framework for capturing diverse dispersion patterns and tail behaviors. Key statistical properties are derived, including the probability mass function, moments, entropy, and the hazard rate function. Parameters are estimated using the maximum likelihood method, and the finite-sample behavior of the estimators is investigated through a simulation study. The effectiveness of the model is demonstrated using a real-life air-pollution dataset, highlighting the adaptability of the ELZTP distribution to environmental count data applications.