On Discrete Mixture of Moment Exponential Using Lagrangian Probability Model: Properties and Applications in Count Data with Excess Zeros

In this paper, we introduce a new distribution for modeling count datasets with some unique characteristics, obtained by mixing the generalized Poisson distribution and the moment exponential distribution based on the framework of the Lagrangian probability distribution, so-called generalized Poisson moment exponential distribution (GPMED). It is shown that the Poisson-moment exponential and Poisson-Ailamujia distributions are special cases of the GPMED. Some important mathematical properties of the GPMED, including median, mode and non-central moment are also discussed through this paper. It is shown that the moment of the GPMED do not exist in some situations and have increasing, decreasing, and upside-down bathtub shaped hazard rates. The maximum likelihood method has been discussed for estimating its parameters. The likelihood ratio test is used to assess the effectiveness of the additional parameter included in the GPMED. The behaviour of these estimators is assessed using simulation study based on the inverse tranformation method. A zero-inflated version of the GPMED is also defined for the situation with an excessive number of zeros in the datasets. Applications of the GPMED and zero-inflated GPMED in various fields are presented and compared with some other existing distributions. In general, the GPMED or its zero-inflated version performs better than the other models, especially for the cases where the data are highly skewed or excessive number of zeros.