On the Product and Ratio of two Generalized Order Statistics from the Generalized Extreme Value Type-II Distribution

Abstract

In this study, we derive the probability density functions (PDFs) for the product and quotient of two generalized order statistics (GOS) from the exponentiated Fréchet distribution (EFD), also known as the generalized extreme value type-II distribution. Utilizing the Mellin transform and its inverse, along with the Fox H-function, we provide explicit expressions for these PDFs. We analyze special cases, including extreme GOS, consecutive GOS, and reductions to the standard Fréchet distribution. The motivation stems from the EFD's applications in modeling extreme events such as earthquakes, floods, and wind speeds, where products and quotients of order statistics are crucial for reliability analysis and stress-strength models. Our contributions include novel analytical derivations extending prior works on Weibull and Pareto distributions, enhanced by numerical simulations and real-data applications to earthquake magnitudes. These results offer insights for fields like extreme value theory, with illustrative examples demonstrating practical utility.