A stochastic comparison result about hazard rate ordering of two parallel systems comprising of geometric components

2015 ◽  
Vol 106 ◽  
pp. 86-90 ◽  
Author(s):  
Jiantian Wang
Keyword(s):  
Hazard Rate ◽  
Parallel Systems ◽  
Stochastic Comparison ◽  
Comparison Result ◽  
Hazard Rate Ordering

MONOTONICITY PROPERTIES OF RESIDUAL LIFETIMES OF PARALLEL SYSTEMS AND INACTIVITY TIMES OF SERIES SYSTEMS WITH HETEROGENEOUS COMPONENTS

2011 ◽  
Vol 26 (1) ◽  
pp. 61-75 ◽  
Author(s):  
Weiyong Ding ◽  
Xiaohu Li ◽  
Narayanaswamy Balakrishnan
Keyword(s):  
Hazard Rate ◽  
Parallel Systems ◽  
Stochastic Comparison ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Monotonicity Properties ◽  
Series Systems ◽  
Residual Lifetimes

Here, we discuss the stochastic comparison of residual lifetimes of parallel systems and inactivity times of series systems by means of the reversed hazard rate order when the components of the systems are independent but not necessarily identically distributed. We also establish some monotonicity properties of such residual lifetimes of parallel systems and inactivity times of series systems. These results extend some of the recent results in this direction due to Zhao, Li, and Balakrishnan [21], Kochar and Xu [12], and Saledi and Asadi [16].

Applications of the hazard rate ordering in reliability and order statistics

1994 ◽  
Vol 31 (1) ◽  
pp. 180-192 ◽  
Author(s):  
Philip J. Boland ◽  
Emad El-Neweihi ◽  
Frank Proschan
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Parallel Systems ◽  
Rate Function ◽  
Hazard Rate Function ◽  
Two Component ◽  
Rate Functions ◽  
Hazard Rate Ordering ◽  
Component Parallel

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T1, · ··, Tn) to the vector (T′1, · ··, T′n), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (λ1r(t), λ2r(t))), the more diverse (λ1, λ2) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering.The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if τ k|n is the lifetime of a k-out-of-n system, then τ k|n is greater than τ k+1|n in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T1, · ··, Tn, we let Tk:n represent the kth order statistic (in increasing order). Then it is shown that Tk +1:n is greater than Tk:n in the hazard rate ordering for all k = 1, ···, n − 1. The result does not, however, extend to the stronger likelihood ratio order.

Applications of the hazard rate ordering in reliability and order statistics

1994 ◽  
Vol 31 (01) ◽  
pp. 180-192 ◽  
Author(s):  
Philip J. Boland ◽  
Emad El-Neweihi ◽  
Frank Proschan
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Parallel Systems ◽  
Rate Function ◽  
Hazard Rate Function ◽  
Two Component ◽  
Rate Functions ◽  
Hazard Rate Ordering ◽  
Component Parallel

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T 1, · ··, Tn ) to the vector (T′ 1, · ··, T′n ), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (λ 1 r(t), λ 2 r(t))), the more diverse (λ 1, λ2 ) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering. The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if τ k|n is the lifetime of a k-out-of-n system, then τ k|n is greater than τ k+ 1|n in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T 1, · ··, Tn , we let Tk:n represent the kth order statistic (in increasing order). Then it is shown that Tk + 1:n is greater than Tk:n in the hazard rate ordering for all k = 1, ···, n − 1. The result does not, however, extend to the stronger likelihood ratio order.

Stochastic comparison of parallel systems with Pareto components

2021 ◽  
pp. 1-13
Author(s):  
Sameen Naqvi ◽  
Weiyong Ding ◽  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Extreme Value Theory ◽  
Pareto Distribution ◽  
Parallel Systems ◽  
Value Theory ◽  
Extreme Value ◽  
Stochastic Comparison ◽  
Shape Parameters ◽  
Scale Parameters ◽  
Dispersive Ordering

Abstract Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$ , $a > 0$ . As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

Stochastic comparisons of series and parallel systems with independent heterogeneous Gumbel and truncated Gumbel components

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Fatih Kızılaslan
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Design Methodology ◽  
Parallel Systems ◽  
Parallel System ◽  
Series System ◽  
Stochastic Comparisons ◽  
Content Type ◽  
Reversed Hazard Rate ◽  
The Relationship

PurposeThe purpose of this paper is to investigate the stochastic comparisons of the parallel system with independent heterogeneous Gumbel components and series and parallel systems with independent heterogeneous truncated Gumbel components in terms of various stochastic orderings.Design/methodology/approachThe obtained results in this paper are obtained by using the vector majorization methods and results. First, the components of series and parallel systems are heterogeneous and having Gumbel or truncated Gumbel distributions. Second, multiple-outlier truncated Gumbel models are discussed for these systems. Then, the relationship between the systems having Gumbel components and Weibull components are considered. Finally, Monte Carlo simulations are performed to illustrate some obtained results.FindingsThe reversed hazard rate and likelihood ratio orderings are obtained for the parallel system of Gumbel components. Using these results, similar new results are derived for the series system of Weibull components. Stochastic comparisons for the series and parallel systems having truncated Gumbel components are established in terms of hazard rate, likelihood ratio and reversed hazard rate orderings. Some new results are also derived for the series and parallel systems of upper-truncated Weibull components.Originality/valueTo the best of our knowledge thus far, stochastic comparisons of series and parallel systems with Gumbel or truncated Gumble components have not been considered in the literature. Moreover, new results for Weibull and upper-truncated Weibull components are presented based on Gumbel case results.

Optimal allocation of a coherent system with statistical dependent subsystems

2021 ◽  
pp. 1-20
Author(s):  
Bin Lu ◽  
Jiandong Zhang ◽  
Rongfang Yan
Keyword(s):  
Hazard Rate ◽  
Optimal Allocation ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Parallel Systems ◽  
Coherent System ◽  
Allocation Policy ◽  
Usual Stochastic Order ◽  
Series Systems ◽  
Parallel Series

Abstract This paper studies the optimal allocation policy of a coherent system with independent heterogeneous components and dependent subsystems, the systems are assumed to consist of two groups of components whose lifetimes follow proportional hazard (PH) or proportional reversed hazard (PRH) models. We investigate the optimal allocation strategy by finding out the number $k$ of components coming from Group A in the up-series system. First, some sufficient conditions are provided in the sense of the usual stochastic order to compare the lifetimes of two-parallel–series systems with dependent subsystems, and we obtain the hazard rate and reversed hazard rate orders when two subsystems have independent lifetimes. Second, similar results are also obtained for two-series–parallel systems under certain conditions. Finally, we generalize the corresponding results to parallel–series and series–parallel systems with multiple subsystems in the viewpoint of the minimal path and the minimal cut sets, respectively. Some numerical examples are presented to illustrate the theoretical findings.

scholarly journals Comparisons of Parallel Systems According to the Convex Transform Order

2009 ◽  
Vol 46 (2) ◽  
pp. 342-352 ◽  
Author(s):  
Subhash Kochar ◽  
Maochao Xu
Keyword(s):  
Hazard Rate ◽  
Parallel Systems ◽  
Sufficient Condition ◽  
Proportional Hazard ◽  
Rate Model ◽  
Hazard Rate Model ◽  
Convex Transform Order ◽  
Right Spread Order ◽  
The Right ◽  
Equivalent Conditions

A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples in the sense of the dispersive order and the right-spread order are established. A sufficient condition is also given for the proportional hazard rate model.

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