A New Family of Continuous Probability Distributions

Area of Science:

• Statistics
• Probability Theory
• Mathematical Modeling

Background:

• Continuous probability distributions are fundamental in statistical modeling.
• Existing distributions may not capture the complexity of all real-world phenomena.
• There is a continuous need for flexible and adaptable probability models.

Purpose of the Study:

• To derive and investigate a new parametric compound G family of continuous probability distributions: the Poisson generalized exponential G (PGEG) family.
• To explore bivariate G families using various copula theorems.
• To illustrate the practical importance of the new PGEG family through real-life data applications.

Main Methods:

• Derivation of the Poisson generalized exponential G (PGEG) family.
• Application of Farlie-Gumbel-Morgenstern, modified Farlie-Gumbel-Morgenstern, Clayton, and Renyi's entropy copulas for bivariate extensions.
• Estimation of model parameters using the maximum likelihood method.
• Graphical simulation to assess the finite sample behavior of estimators.

Main Results:

• Successful derivation and mathematical characterization of the PGEG family.
• Development of novel bivariate G families based on established copula theories.
• Demonstration of the utility of the PGEG family and its special members (e.g., exponential, Pareto type II) through simulations and real-data analysis.

Conclusions:

• The proposed PGEG family offers a flexible new tool for statistical modeling.
• The bivariate extensions provide enhanced capabilities for analyzing dependent random variables.
• The study confirms the practical relevance and effectiveness of the new distribution family in applied scenarios.