A Generalized Mixture of Geometric Distribution for Flexible Count Data Modelling

The geometric distribution is a fundamental model for count data and discrete-time lifetimes, but its memoryless property implies a constant hazard rate that is often too restrictive in practice. This paper introduces a new three-parameter discrete distribution, termed the generalized mixture of geometric (GMG) distribution, which extends the geometric model while preserving its tail behaviour. The proposed distribution allows controlled departures from memorylessness at early counts and admits clear parameter interpretations governing tail behaviour, shape, and perturbation intensity. Closed-form expressions for the cumulative distribution function, survival function, and hazard rate are fundamental for both theoretical analysis and reliability modelling. In addition to their practical usefulness, these quantities provide valuable insight into the structural differences between the GMG and geometric distributions. Shannon entropy generalizes the geometric baseline and its existence is established. Parameter estimation is addressed through a preliminary moment-based procedure and maximum likelihood estimation. The moment-based estimator is used to initialise numerical likelihood maximisation, while the inferential and numerical properties of the maximum likelihood estimator are investigated. Particular attention is paid to parameter configurations that may lead to weak numerical identification, and a stable optimisation strategy is discussed. A comprehensive Monte Carlo simulation study is conducted to assess the finite-sample performance of the maximum likelihood estimator. Finally, the practical usefulness of the GMG distribution is illustrated through four real-life count datasets, where its performance is compared with several established competing models.