Some Results on Reversed Hazard Rate Ordering (original) (raw)
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Journal of Applied Probability, 2006
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We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k-out-of-n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions.
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Let X 1 ,. .. , X n be independent exponential random variables with respective hazard rates λ 1 ,. .. , λ n , and Y 1 ,. .. , Y n be independent and identically distributed random variables from an exponential distribution with hazard rate λ. Then, we prove that X 2:n , the second order statistic from X 1 ,. .. , X n , is larger than Y 2:n , the second order statistic from Y 1 ,. .. , Y n , in terms of the dispersive order if and only if λ ≥ 1 n 2 1≤i< j≤n λ i λ j. We also show that X 2:n is smaller than Y 2:n in terms of the dispersive order if and only if λ ≤ n i=1 λ i − max 1≤i≤n λ i n − 1. Moreover, we extend the above two results to the proportional hazard rates model. These two results established here form nice extensions of the corresponding results on hazard rate, likelihood ratio, and MRL orderings established recently by Pǎltǎnea
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Advances in Applied Probability, 1993
Shanthikumar and Yao (1991) introduced some new stochastic order relations to compare the components of a bivariate random vector (X 1, X 2). As they point out in their paper, even if according to their hazard rate (or likelihood ratio) ordering, the marginal distributions may not be ordered accordingly. We introduce some new concepts where the marginal distributions preserve the corresponding stochastic orders. Also a relation between the bivariate scale model and the introduced bivariate hazard rate ordering is established.
Stochastic ordering properties for systems with dependent identically distributed components
Applied Stochastic Models in Business and Industry, 2013
In this paper, we obtain ordering properties for coherent systems with possibly dependent identically distributed components. These results are based on a representation of the system reliability function as a distorted function of the common component reliability function. So, the results included in this paper can also be applied to general distorted distributions. The main advantage of these results is that they are distribution-free with respect to the common component distribution. Moreover, they can be applied to systems with component lifetimes having a non-exchangeable joint distribution.
Preservation of some partial orderings under the formation of coherent systems
Statistics & Probability Letters, 1998
The reversed (backward) hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their reversed hazard rate functions. In this paper, we have given some sufficient conditions under which the ordering between the components with respect to the reversed hazard rate is preserved under the formation of coherent systems. We have also shown that these sufficient conditions are satisfied by k-out-of-n systems. Both the cases when components are identically distributed and not necessarily identically distributed are discussed. Some results for likelihood ratio order are also obtained. The parallel (series) systems of not necessarily iid components have been characterized by means of a relationship between the reversed hazard rate (hazard rate) function of the system and the reversed hazard rate (hazard rate) functions of the components. (~)
arXiv (Cornell University), 2019
This paper considers the joint distribution of elements of a random sample and an order statistic of the same sample. The motivation for this work stems from the important problem in reliability analysis, to estimate the number of inspections we need in order to detect failed components in a coherent system. We consider an (n − r + 1)-out-of-n system, which is intact until at least n − r + 1 of the components are alive, and it fails if the number of failed components exceeds r. The life time of the system is the rth order statistic. Assuming that some of the components failed but the system is still functioning, using the results presented in this paper it is possible to find an expected value of the number of inspections we need to do for detecting certain number of failed components.