Probability density functions of quotient-product of two generalized order statistics from the generalized exponential distribution

Abstract

A major area of statistical research, especially in fields like dependability analysis and stochastic modeling, is the investigation of the Quotient-Product (Q\&P) of generalized order statistics. These functions are crucial for determining the properties of different statistical distributions and for comprehending the dynamics of complex systems. The generalized exponential distribution (GED) is one of them that is particularly adaptable and can handle a variety of data kinds. The purpose of this work is to obtain explicit equations for the Quotient-Product of two generalized order statistics and its probability density function. We provide a thorough framework for these derivations by using the Mellin transform and its inverse. The findings advance theoretical understanding of generalized order statistics and their relationships by offering insightful examples and useful insights, like Quotient-Product of extreme generalized order statistics and consecutive generalized order statistics, which are pertinent to applications in engineering reliability, risk assessment, and survival analysis. The consistency of the derived results with existing literature is confirmed through several corollaries that recover well-known special cases. The paper concludes with a discussion of the theoretical and practical limitations of the current work and outlines promising directions for future research, including extensions to dependent GOS models, other lifetime distributions, and Bayesian estimation frameworks.