The Extended Gompertz Model: Applications, Mean of Order P Assessment and Statistical Threshold Risk Analysis Based on Extreme Stresses Data

Abstract

Threshold risk analysis based on extreme stress data assess the probability of events that exceeds a certain threshold in a dataset characterized by extreme values or stresses. This type of analysis is often used in finance, insurance, environmental science, and engineering, where understanding and managing extreme events are crucial. This article proposed the extended Gompertz (ExGo) model for analyzing Mean of Order <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> (MOOP<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$_{[ P] }$</tex-math></inline-formula>) and statistical threshold risk analysis based on real extreme stress data. This study examined the statistical properties of the proposed model and its efficiency. Several estimation methods (maximum likelihood, Anderson–Darling, ordinary least squares, Cramér–von Mises, weighted least squares, and right-tail Anderson–Darling), were evaluated through simulations and experiments to determine their effectiveness. The comparisons were based on bias and the root mean square error. Additionally, proposed distribution was applied to reliability and stress data, demonstrating its applicability and flexibility in modeling. This study focused on MOOP<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$_{ [ P] }$</tex-math></inline-formula> analysis using the ExGo model to determine the optimal parameter <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i>, crucial for extreme stress data analysis in engineering and reliability contexts. Furthermore, proposed distribution is employed in various risk measurement and analysis indicators, including value-at-risk, tail-value-at-risk, tail variance, and tail mean-variance indicators, with emphasis on their application in reliability and engineering, particularly concerning change in stress data and breaking stress data.