scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (01) ◽  
pp. 102-116
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.

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (01) ◽  
pp. 102-116 ◽  
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.

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (1) ◽  
pp. 102-116 ◽  
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 Xλ1, Xλ2, …, Xλn be independent Weibull random variables with Xλi ∼ W(α, λi), where λi > 0 for i = 1, …, n. Let Xn:nλ denote the lifetime of the parallel system formed from Xλ1, Xλ2, …, Xλn. We investigate the effect of the changes in the scale parameters (λ1, …, λn) on the magnitude of Xn:nλ according to reverse hazard rate and likelihood ratio orderings.

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.

ON PARALLEL SYSTEMS WITH HETEROGENEOUS GAMMA COMPONENTS

2011 ◽  
Vol 25 (3) ◽  
pp. 369-391 ◽  
Author(s):  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Random Variables ◽  
Parallel Systems ◽  
Likelihood Ratio Order ◽  
Maximum Order ◽  
Hazard Rate Order

In this article, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous gamma components in terms of the likelihood ratio order and the hazard rate order. LetX1andX2be two independent gamma random variables withXihaving shape parameterr>0 and scale parameter λi,i=1, 2, and letX*1andX*2be another set of independent gamma random variables withX*ihaving shape parameterrand scale parameter λ*i,i=1, 2. Denote byX2:2andX*2:2the corresponding maximum order statistics, respectively. It is proved that, among others, if (λ1, λ2) weakly majorize (λ*1, λ*2), thenX2:2is stochastically greater thanX*2:2in the sense of likelihood ratio order. We also establish, among others, that if 0<r≤1 and (λ1, λ2) isp-larger than (λ*1, λ*2), thenX2:2is stochastically greater thanX*2:2in the sense of hazard rate order. The results derived here strengthen and generalize some of the results known in the literature.

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 On the Convolution of Heterogeneous Bernoulli Random Variables

2011 ◽  
Vol 48 (3) ◽  
pp. 877-884 ◽  
Author(s):  
Maochao Xu ◽  
N. Balakrishnan
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Random Variables ◽  
Likelihood Ratio Order ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Bernoulli Random Variables

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

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

2012 ◽  
Vol 26 (2) ◽  
pp. 159-182 ◽  
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.

scholarly journals Dependency in multisensory integration: a copula-based analysis

2019 ◽  
Vol 377 (2157) ◽  
pp. 20180364
Author(s):  
Hans Colonius ◽  
Adele Diederich
Keyword(s):  
Multisensory Integration ◽  
Stochastic Order ◽  
Reaction Times ◽  
Theme Issue ◽  
One Dimensional ◽  
Usual Stochastic Order ◽  
Order Relations ◽  
Specific Choice ◽  
Sensory Modalities ◽  
Stochastic Order Relations

The notion of copula has attracted attention from the field of contextuality and probability. A copula is a function that joins a multivariate distribution to its one-dimensional marginal distributions. Thereby, it allows characterizing the multivariate dependency separately from the specific choice of margins. Here, we demonstrate the use of copulas by investigating the structure of dependency between processing stages in a stochastic model of multisensory integration, which describes the effect of stimulation by several sensory modalities on human reaction times. We derive explicit terms for the covariance and Kendall's tau between the processing stages and point out the specific role played by two stochastic order relations, the usual stochastic order and the likelihood ratio order, in determining the sign of dependency. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

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

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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