Theory and Properties of a New Pareto-Gumbel Distribution

A new family of continuous distribution called Pareto-Gumbel is developed using the transform-Transformer techniques. We theoretically combine two distributions (Pareto and Gumbel) to form a new probability distribution called Pareto-Gumbel Distribution (PGD) to checkmate the limitations of the two distribution in order to address the shortcomings of the respective distributions. The Pareto distribution is often used to model the tails of another distribution, and the shape parameter ξ relates to tail-behavior, distributions with tails that decrease exponentially are modeled with shape ξ = 0,while distributions with tails that decrease as a polynomial are modeled with a positive shape parameter, distributions with finite tails are modeled with a negative shape parameter and Gumbel Extreme Value distribution, are widely applied for extreme value analysis and this distribution has certain drawbacks, because it is a non-heavy-tailed distribution and is characterized by constant skewness and kurtosis. The Gumbel distribution remains one of the mostly used statistical distributions in the frequency analysis of extreme events. This aspect is mainly due to the simple parameter estimation expressions, as well as the simple and accessible expression of the Statistical properties, the main advantage of the Gumbel distribution is the simplicity and accessibility of expressions and relationships. It is essential to understand that most distributions described in the literature were developed using transformed transformer (T-X) method. This method was proposed by Alzaatreh, et al., (2013), Adewusi, et al., (2019) and Ajewole et al (2025). This study develops a new family of continuous distribution called the Pareto-Gumbel, which had been developed by combining Pareto and Gumbel distribution using T-X techniques to form Pareto-Gumbel distribution. Several expressions for distribution theory and properties were explored and obtained; the maximum likelihood estimation approach was used to estimate the distribution parameters, with simulations conducted to assess the asymptotic behavior of these estimates.