# Iterated failure rate monotonicity and ordering relations within Gamma and Weibull distributions

## TL;DR

This paper investigates stochastic ordering within Gamma and Weibull distributions by introducing an iterative method that simplifies the verification of order relations, providing a comprehensive characterization of these orderings.

## Contribution

It presents a novel iterative approach for verifying stochastic orderings, specifically applied to Gamma and Weibull distributions, with explicit descriptions of their order relations.

## Key findings

- Complete description of order relations within Gamma distributions
- Complete description of order relations within Weibull distributions
- Method simplifies verification of stochastic orderings

## Abstract

Stochastic ordering of distributions of random variables may be defined by the relative convexity of the tail functions. This has been extended to higher order stochastic orderings, by iteratively reassigning tail-weights. The actual verification of those stochastic orderings is not simple, as this depends on inverting distribution functions for which there may be no explicit expression. The iterative definition of distributions, of course, contributes to make that verification even harder. We have a look at the stochastic ordering, introducing a method that allows for explicit usage, applying it to the Gamma and Weibull distributions, giving a complete description of the order relations within each of those families.