Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications

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.

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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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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 largest-order statistics for proportional reversed hazard rate model and applications

Journal of Applied Probability ◽

10.1017/jpr.2020.40 ◽

2020 ◽

Vol 57 (3) ◽

pp. 832-852

Author(s):

Lu Li ◽

Qinyu Wu ◽

Tiantian Mao

Keyword(s):

Order Statistics ◽

Likelihood Ratio ◽

Hazard Rate ◽

Parallel Systems ◽

Rate Model ◽

Stochastic Comparisons ◽

Reversed Hazard Rate ◽

Generalized Gamma ◽

Pareto Distributions ◽

Hazard Rate Model

AbstractWe investigate stochastic comparisons of parallel systems (corresponding to the largest-order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate (PRHR) model. As applications of the main results, we obtain the equivalent characterizations of stochastic comparisons with respect to the reversed hazard rate and likelihood rate orders for the exponentiated generalized gamma and exponentiated Pareto distributions. Our results recover and strengthen some recent results in the literature.

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COMPARISONS OF SERIES AND PARALLEL SYSTEMS WITH HETEROGENEOUS COMPONENTS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964813000314 ◽

2013 ◽

Vol 28 (1) ◽

pp. 39-53 ◽

Cited By ~ 2

Author(s):

Weiyong Ding ◽

Gaofeng Da ◽

Xiaohu Li

Keyword(s):

Likelihood Ratio ◽

Hazard Rate ◽

Parallel Systems ◽

Likelihood Ratio Order ◽

Stochastic Comparisons ◽

Hazard Rate Order ◽

Reversed Hazard Rate ◽

Series Systems

This paper carries out stochastic comparisons of series and parallel systems with independent and heterogeneous components in the sense of the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. The main results extend and strengthen the corresponding ones by Misra and Misra [18] and by Ding, Zhang, and Zhao [8]. Meanwhile, the results on the hazard rate order of parallel systems and the reversed hazard order of series systems serve as nice supplements to Theorem 16.B.1 of Boland and Proschan [4] and Theorem 3.2 of Nanda and Shaked [20], respectively.

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Comparisons of Parallel Systems with Components Having Proportional Reversed Hazard Rates and Starting Devices

Mathematics ◽

10.3390/math9080856 ◽

2021 ◽

Vol 9 (8) ◽

pp. 856

Author(s):

Narayanaswamy Balakrishnan ◽

Ghobad Barmalzan ◽

Sajad Kosari

Keyword(s):

Hazard Rate ◽

Stochastic Order ◽

Parallel Systems ◽

Hazard Rates ◽

Stochastic Comparisons ◽

Reversed Hazard Rate ◽

Usual Stochastic Order ◽

Distributed Components

In this paper, we consider stochastic comparisons of parallel systems with proportional reversed hazard rate (PRHR) distributed components equipped with starting devices. By considering parallel systems with two components that PRHR and starting devices, we prove the hazard rate and reversed hazard rate orders. These results are then generalized for such parallel systems with n components in terms of usual stochastic order. The establish results are illustrated with some examples.

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Further Results on Dynamic Additive Hazard Rate Model

Mathematical Problems in Engineering ◽

10.1155/2014/752831 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5 ◽

Cited By ~ 1

Author(s):

Zhengcheng Zhang ◽

Limin Zhang

Keyword(s):

Hazard Rate ◽

Stochastic Orders ◽

Rate Model ◽

Stochastic Comparisons ◽

Reversed Hazard Rate ◽

The Past ◽

Hazard Rate Model ◽

Aging Properties ◽

Hazard Rate Models

In the past, the proportional and additive hazard rate models have been investigated in the works. Nanda and Das (2011) introduced and studied the dynamic proportional (reversed) hazard rate model. In this paper we study the dynamic additive hazard rate model, and investigate its aging properties for different aging classes. The closure of the model under some stochastic orders has also been investigated. Some examples are also given to illustrate different aging properties and stochastic comparisons of the model.

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Stochastic comparisons of coherent systems under different random environments

Journal of Applied Probability ◽

10.1017/jpr.2018.30 ◽

2018 ◽

Vol 55 (2) ◽

pp. 459-472 ◽

Cited By ~ 2

Author(s):

Ebrahim Amini-Seresht ◽

Yiying Zhang ◽

Narayanaswamy Balakrishnan

Keyword(s):

Hazard Rate ◽

Sufficient Conditions ◽

Random Environments ◽

Reliability Engineering ◽

Coherent Systems ◽

Stochastic Comparisons ◽

Numerical Examples ◽

Reversed Hazard Rate ◽

Collaborative Environment ◽

Theoretical Results

Abstract For many practical situations in reliability engineering, components in the system are usually dependent since they generally work in a collaborative environment. In this paper we build sufficient conditions for comparing two coherent systems under different random environments in the sense of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orders. Applications and numerical examples are provided to illustrate all the theoretical results established here.

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