Erratum to: Mixture representations of residual lifetimes of used systems

We have found a mistake in the proofs of Navarro (2008, Theorem 2.3(b) and 2.3(c)) due to misapplication of properties of hazard rate and likelihood ratio orders. In this paper we show with an example that the stated results do not hold. This example is interesting since it proves some unexpected properties for these orderings under the formation of coherent systems. The result stated in Navarro (2008, Theorem 2.3(a)) for the usual stochastic order is correct.

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STOCHASTIC COMPARISONS OF LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER EXPONENTIAL MODELS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964811000313 ◽

2012 ◽

Vol 26 (2) ◽

pp. 159-182 ◽

Cited By ~ 24

Author(s):

Peng Zhao ◽

N. Balakrishnan

Keyword(s):

Order Statistics ◽

Likelihood Ratio ◽

Hazard Rate ◽

Stochastic Order ◽

Likelihood Ratio Order ◽

Stochastic Comparisons ◽

Hazard Rate Order ◽

Reversed Hazard Rate ◽

Exponential Models ◽

Usual Stochastic Order

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

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LIKELIHOOD RATIO AND HAZARD RATE ORDERINGS OF THE MAXIMA IN TWO MULTIPLE-OUTLIER GEOMETRIC SAMPLES

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996481200006x ◽

2012 ◽

Vol 26 (3) ◽

pp. 375-391 ◽

Cited By ~ 6

Author(s):

Baojun Du ◽

Peng Zhao ◽

N. Balakrishnan

Keyword(s):

Likelihood Ratio ◽

Hazard Rate ◽

Stochastic Order ◽

Likelihood Ratio Order ◽

Stochastic Comparisons ◽

Hazard Rate Order ◽

Usual Stochastic Order ◽

Likelihood Ratio Ordering ◽

Special Case ◽

Two Parameter

In this paper, we study some stochastic comparisons of the maxima in two multiple-outlier geometric samples based on the likelihood ratio order, hazard rate order, and usual stochastic order. We establish a sufficient condition on parameter vectors for the likelihood ratio ordering to hold. For the special case whenn= 2, it is proved that thep-larger order between the two parameter vectors is equivalent to the hazard rate order as well as usual stochastic order between the two maxima. Some numerical examples are presented for illustrating the established results.

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ORDERING RESULTS ON EXTREMES OF EXPONENTIATED LOCATION-SCALE MODELS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964819000408 ◽

2019 ◽

pp. 1-24

Author(s):

Sangita Das ◽

Suchandan Kayal ◽

Debajyoti Choudhuri

Keyword(s):

Likelihood Ratio ◽

Hazard Rate ◽

Stochastic Order ◽

Scale Model ◽

Likelihood Ratio Order ◽

Hazard Rate Order ◽

Number Of Components ◽

Usual Stochastic Order ◽

Scale Models ◽

Unequal Number

AbstractIn this paper, we consider exponentiated location-scale model and obtain several ordering results between extreme order statistics in various senses. Under majorization type partial order-based conditions, the comparisons are established according to the usual stochastic order, hazard rate order and reversed hazard rate order. Multiple-outlier models are considered. When the number of components are equal, the results are obtained based on the ageing faster order in terms of the hazard rate and likelihood ratio orders. For unequal number of components, we develop comparisons according to the usual stochastic order, hazard rate order, and likelihood ratio order. Numerical examples are considered to illustrate the results.

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Optimal allocation of relevations in coherent systems

Journal of Applied Probability ◽

10.1017/jpr.2021.23 ◽

2021 ◽

Vol 58 (4) ◽

pp. 1152-1169

Author(s):

Rongfang Yan ◽

Jiandong Zhang ◽

Yiying Zhang

Keyword(s):

Optimal Allocation ◽

Necessary Conditions ◽

Stochastic Order ◽

Coherent Systems ◽

Stochastic Comparisons ◽

Hazard Rate Order ◽

Usual Stochastic Order ◽

Special Cases ◽

Sufficient And Necessary Conditions ◽

Allocation Strategies

AbstractIn this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

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Optimal allocation of a coherent system with statistical dependent subsystems

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964821000437 ◽

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.

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Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Journal of Applied Probability ◽

10.1017/s0001867800012222 ◽

2015 ◽

Vol 52 (01) ◽

pp. 102-116 ◽

Cited By ~ 12

Author(s):

Nuria Torrado ◽

Subhash C. Kochar

Keyword(s):

Likelihood Ratio ◽

Hazard Rate ◽

Stochastic Order ◽

Random Variables ◽

Parallel Systems ◽

Parallel System ◽

Weibull Distributions ◽

Scale Parameters ◽

Order Relations ◽

Stochastic Order Relations

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

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COMPARISONS OF SAMPLE RANGES ARISING FROM MULTIPLE-OUTLIER MODELS: IN MEMORY OF MOSHE SHAKED

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964817000468 ◽

2017 ◽

Vol 33 (1) ◽

pp. 28-49

Author(s):

Narayanaswamy Balakrishnan ◽

Jianbin Chen ◽

Yiying Zhang ◽

Peng Zhao

Keyword(s):

Hazard Rate ◽

Stochastic Order ◽

Sufficient Conditions ◽

Hazard Rates ◽

Proportional Hazard ◽

Stochastic Comparisons ◽

Hazard Rate Order ◽

Numerical Examples ◽

Reversed Hazard Rate ◽

Usual Stochastic Order

In this paper, we discuss the ordering properties of sample ranges arising from multiple-outlier exponential and proportional hazard rate (PHR) models. The purpose of this paper is twofold. First, sufficient conditions on the parameter vectors are provided for the reversed hazard rate order and the usual stochastic order between the sample ranges arising from multiple-outlier exponential models with common sample size. Next, stochastic comparisons are separately carried out for sample ranges arising from multiple-outlier exponential and PHR models with different sample sizes as well as different hazard rates. Some numerical examples are also presented to illustrate the results established here.

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Stochastic Comparisons of Residual Lifetimes and Inactivity Times of Coherent Systems

Journal of Applied Probability ◽

10.1239/jap/1378401240 ◽

2013 ◽

Vol 50 (3) ◽

pp. 848-860 ◽

Cited By ~ 5

Author(s):

Nitin Gupta

Keyword(s):

Likelihood Ratio ◽

Sufficient Conditions ◽

Structure Functions ◽

Coherent System ◽

Stochastic Comparison ◽

Coherent Systems ◽

Stochastic Comparisons ◽

Likelihood Ratio Ordering ◽

Necessary And Sufficient ◽

Residual Lifetimes

Under the assumption of independent and identically distributed (i.i.d.) components, the problem of the stochastic comparison of a coherent system having used components and a used coherent system has been considered. Necessary and sufficient conditions on structure functions have been provided for the stochastic comparison of a coherent system having used/inactive i.i.d. components and a used/inactive coherent system. As a consequence, for r-out-of-n systems, it has been shown that systems having used i.i.d. components stochastically dominate used systems in the likelihood ratio ordering.

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On a new stochastic model for cascading failures

Journal of Applied Probability ◽

10.1017/jpr.2020.68 ◽

2020 ◽

Vol 57 (4) ◽

pp. 1150-1161

Author(s):

Hyunju Lee

Keyword(s):

Stochastic Order ◽

Accelerated Life Test ◽

Cascading Failures ◽

Test Model ◽

Failure Model ◽

Life Test ◽

Usual Stochastic Order ◽

Accelerated Life ◽

Failure Models ◽

Residual Lifetimes

AbstractIn this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.

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