COMPARISONS OF SERIES AND PARALLEL SYSTEMS WITH HETEROGENEOUS COMPONENTS

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

International Journal of Quality & Reliability Management ◽

10.1108/ijqrm-02-2020-0040 ◽

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.

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On the Convolution of Heterogeneous Bernoulli Random Variables

Journal of Applied Probability ◽

10.1239/jap/1316796922 ◽

2011 ◽

Vol 48 (3) ◽

pp. 877-884 ◽

Cited By ~ 8

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).

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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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ON PARALLEL SYSTEMS WITH HETEROGENEOUS GAMMA COMPONENTS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964811000064 ◽

2011 ◽

Vol 25 (3) ◽

pp. 369-391 ◽

Cited By ~ 10

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.

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

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964811000234 ◽

2011 ◽

Vol 26 (1) ◽

pp. 61-75 ◽

Cited By ~ 1

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].

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On the Convolution of Heterogeneous Bernoulli Random Variables

Journal of Applied Probability ◽

10.1017/s0021900200008391 ◽

2011 ◽

Vol 48 (03) ◽

pp. 877-884 ◽

Cited By ~ 1

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).

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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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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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