Comparison of different estimation methods for the inverse power perk distribution with applications in engineering and actuarial sciences

Introducing the Inverse Power Perk distribution, this paper presents a versatile probability distribution designed to model positively skewed data with unprecedented flexibility. Building upon the Perk distribution, it accommodates a wide range of shapes including right-skewed, J-shaped, reversed J-shaped, and nearly symmetric densities, as well as hazard rates exhibiting various patterns of increase and decrease. The paper delves into the mathematical properties of this novel distribution and offers a comprehensive overview of estimation techniques, including maximum likelihood estimators, ordinary least square estimators, percentile-based estimators, maximum product of spacing estimators, Cramer-von Mises, weighted least squares estimators, and Anderson-Darling estimators. To assess the performance of these estimation methods across different sample sizes, Monte Carlo simulations are conducted. Through comparisons of average absolute error and mean squared error, the efficacy of each estimator is evaluated, shedding light on their suitability for both small and large samples. In a practical application, three real datasets, including insurance data, are employed to demonstrate the versatility of the current model, when comparing to existing alternatives. The IPP distribution offers significant advantages over traditional distributions, particularly in its superior ability to model tail risks, making it an invaluable tool for practitioners dealing with extreme values and rare events. Its computational efficiency further sets it apart, enabling more robust and faster analysis in large-scale datasets.This empirical analysis further underscores the utility and adaptability of the Inverse Power Perk model in capturing the nuances of diverse datasets, thereby offering valuable insights for practitioners in various fields.