# Stochastic ordering results in parallel and series systems with Gumble distributed random variables

## TL;DR

This paper investigates stochastic ordering relations in parallel and series systems with independent Gumble distributed components, providing conditions for various orderings based on system parameters.

## Contribution

It introduces new conditions for stochastic orderings of systems with Gumble components, utilizing likelihood ratio and majorization techniques.

## Key findings

- Likelihood ratio ordering condition for parallel systems.
- Comparison of systems using majorization and hazard rate orderings.
- Analysis of system uncertainty and dispersiveness based on location parameters.

## Abstract

The stochastic comparisons of parallel and series system are worthy of study. In this paper, we present some stochastic comparisons of parallel and series systems having independent components from Gumble distribution with two parameters (one location and one shape). Here, we first put a condition for the likelihood ratio ordering of the parallel systems and second we use the concept of vector majorization technique to compare the systems by the reversed hazard rate ordering, the hazard rate ordering, the dispersive ordering, and the less uncertainty ordering with respect to the location parameter.