Abstract
In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.
Highlights
The comparison of random quantities in terms of the so called “stochastic orders”has received great attention along the last 50 years and we can find applications of this topic in different fields such as reliability, insurance, risks, finance, epidemics, and so on
An interesting topic in this context is that of finding sufficient conditions for several stochastic orders when there are no closed expressions for some of the functions used in the comparison
We show that there are situations where the unimodality of the likelihood ratio is a sufficient condition for the hazard rate order or the mean residual life order
Summary
The comparison of random quantities in terms of the so called “stochastic orders”. has received great attention along the last 50 years and we can find applications of this topic in different fields such as reliability, insurance, risks, finance, epidemics, and so on (see [1,2,3,4,5,6]). Despite being a powerful tool to compare the residual lifetime of two random lifetimes, the checking of these criteria requires the computation of series which usually do not have a closed expression, which is an important disadvantage in real problems It is well-known that a sufficient condition for both criteria is the likelihood ratio order, which is satisfied when the ratio of the mass probability (density) functions is monotone. We show that there are situations where the unimodality of the likelihood ratio is a sufficient condition for the hazard rate order (which is stronger than the stochastic order) or the mean residual life order These results will be applied to the comparison of some of the most important discrete families of distributions in the context of reliability in discrete time.
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