The Generalized Weibull Poisson G Family of Continuous Probability Distributions with Copulas, Mathematical Properties and Applications to Real Data Sets

Abstract

Abstract This scientific report aims to provide a comprehensive understanding of new compound G family of probability distributions with copulas, shedding light on their mathematical properties and demonstrating their applicability in various real data settings. The insights from this study can benefit researchers, statisticians, and practitioners in their efforts to model complex dependencies and make informed decisions based on multivariate data analysis. The versatility of compound families of continuous probability distributions makes them useful for modelling a variety of events. They can be used to simulate the distribution of a system's time to failure, the amount of money lost in a financial transaction, or the number of accidents in a particular year, for instance. One of the main advantages of using compound families of continuous probability distributions is that they are flexible. This means that they can be used to model a wide range of phenomena, even those that are not well-described by other distributions. Additionally, compound families of continuous probability distributions are often easy to understand and use. Another advantage of using compound families of continuous probability distributions is that they can be used to model the dependence between two or more random variables. In this work, a novel G family called the generalized Weibull Poisson G family is derived and analyzed. The new compound G family of distributions is derived based on the zero-truncated-Poisson G and the generalized Weibull G families. Its statistical characteristics are mathematically studied. The Clayton copula, Archimedean-Ali-Mikhail-Haq copula, Renyi's entropy copula, copula of Farlie, Gumbel and Morgenstern and its modified version which contains four minor types are used to construct some new bivariate type G families. The Lomax model with one parameter receives special attention. The relevance of the new family is demonstrated through two examples from everyday life.