Abstract
This article is devoted to the proposal of two relative stochastic orders namely the relative hazard rate and relative mean residual life orders. These stochastic orders are applied to provide some stochastic comparisons between two additive frailty models. Some closure properties of the model with respect to these relative stochastic orders are presented. In addition, we demonstrate how the variation of the baseline variable and the variation of the additive variable in the additive frailty model, each in one time, has an effect on the model. Finally, a possible extension of the concept of relative orders to the multivariate case is discussed.
Highlights
Introduction and preliminariesIn the context of reliability and survival analysis, various models have been introduced in the literature for modeling and analyzing failure time data
Most of the stochastic orders discovered and analyzed in the literature are the ones which compare the ‘location’ or the ‘magnitude’ of the random variables and there are other ones which compare the ‘variability’ or the ‘dispersion’ of the random variables. As another perspective of stochastic comparison between lifetime variables, we devoted our attention here to other kinds of stochastic orders which compare the random variables with respect to their ‘aging’ properties according to the well-known reliability measures of hazard rate and mean residual life
Because there is no theoretical base for choosing the distribution of the additive effect variable in the additive frailty model it is important to see how the overall variable is influenced by the variation of the additive effect and baseline variables in the model
Summary
In the context of reliability and survival analysis, various models have been introduced in the literature for modeling and analyzing failure time data. Some of these models are the proportional (additive) hazards model, the proportional (additive) reversed hazards model, and the proportional (additive) mean residual life model (cf [ – ]). The additive hazards model, which is well known in the literature, has played a prominent role in modeling survival data. For a non-negative random variable X with the probability density function (pdf ) f and the survival function (sf ) F , the random variable Xx = (X – x | X > x) for any x ∈ χ is known as the residual life of X at some fixed time x provided that X is greater than x, where χ = {x : F (x) > }.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
