A New Discrete Generator with Mathematical Characterization, Properties, Count Statistical Modeling and Inference with Applications to Reliability, Medicine, Agriculture, and Biology Data

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In this piece of work, we examine and present a completely new discrete family of distributions that we have created. Our investigation into the relevant mathematical properties and characterizations of the system makes use of both analytical and numerical methods. We focus on a particular member of this family so that we can study its theoretical foundations as well as its graphical and numerical representations. This new model contains a few different hazard rate functions, some of which are referred to as "increasing constant", "decreasing-constant-increasing (U)", "constant", "U-constant", "decreasing", and "J-shape" In a similar vein, the model's probability mass function provides a variety of forms, all of which are helpful and practical. These forms include "asymmetric left skewed," "right skewed with wide peak," "right skewed," "bimodal," "symmetric," and "right skewed," amongst others. Each of these forms is valuable and applicable in their own way. These forms might be discovered in the probability mass function that the model generates. In this investigation, in addition to the Bayesian estimating technique under the traditional loss function of squared errors, we investigate and make use of a total of eight estimate strategies that are not founded on Bayesian theory (classical methods). Simulations employing the Markov Chain Monte-Carlo method are run for comparing the Bayesian way of estimation with the more traditional approach of estimating values. According to the findings that we've compiled, the estimation strategy that is referred to as maximum likelihood yields the most accurate results across the board and for all different types of sample sizes. In addition, we evaluate and contrast the various methods of estimation by making use of six distinct real dataset sets; this indicates the versatility of the unique model that we have developed.