Some new results on stochastic comparisons of parallel systems

2000 ◽  
Vol 37 (04) ◽  
pp. 1123-1128 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Lower Bound ◽  
Hazard Rate ◽  
Geometric Mean ◽  
Random Variables ◽  
Parallel Systems ◽  
Rate Function ◽  
Arithmetic Mean ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Exponential Random Variables

Let X 1,…,X n be independent exponential random variables with X i having hazard rate . Let Y 1,…,Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏ i=1 n λ i )1/n , the geometric mean of the λis. Let X n:n = max{X 1,…,X n }. It is shown that X n:n is greater than Y n:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X n:n and an upper bound on the hazard rate function of X n:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference 65, 203–211), which are in terms of the arithmetic mean of the λ i s. Furthermore, let X 1 *,…,X n ∗ be another set of independent exponential random variables with X i ∗ having hazard rate λ i ∗, i = 1,…,n. It is proved that if (logλ1,…,logλ n ) weakly majorizes (logλ1 ∗,…,logλ n ∗, then X n:n is stochastically greater than X n:n ∗.

Some new results on stochastic comparisons of parallel systems

2000 ◽  
Vol 37 (4) ◽  
pp. 1123-1128 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Lower Bound ◽  
Hazard Rate ◽  
Geometric Mean ◽  
Random Variables ◽  
Parallel Systems ◽  
Rate Function ◽  
Arithmetic Mean ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Exponential Random Variables

Let X1,…,Xn be independent exponential random variables with Xi having hazard rate . Let Y1,…,Yn be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏i=1nλi)1/n, the geometric mean of the λis. Let Xn:n = max{X1,…,Xn}. It is shown that Xn:n is greater than Yn:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of Xn:n and an upper bound on the hazard rate function of Xn:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference65, 203–211), which are in terms of the arithmetic mean of the λis. Furthermore, let X1*,…,Xn∗ be another set of independent exponential random variables with Xi∗ having hazard rate λi∗, i = 1,…,n. It is proved that if (logλ1,…,logλn) weakly majorizes (logλ1∗,…,logλn∗, then Xn:n is stochastically greater than Xn:n∗.

COPULA REPRESENTATIONS FOR THE SUM OF DEPENDENT RISKS: MODELS AND COMPARISONS

2020 ◽  
pp. 1-21
Author(s):  
Jorge Navarro ◽  
José María Sarabia
Keyword(s):  
Hazard Rate ◽  
Marginal Distribution ◽  
Rate Function ◽  
Distribution Functions ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Limiting Behavior ◽  
Dependent Risks ◽  
Analytical Expressions ◽  
Theoretical Results

The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

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.

scholarly journals ON STOCHASTIC COMPARISONS OF ORDER STATISTICS FROM HETEROGENEOUS EXPONENTIAL SAMPLES

2019 ◽  
pp. 1-6
Author(s):  
Yaming Yu
Keyword(s):  
Order Statistics ◽  
Order Statistic ◽  
Open Problem ◽  
Hazard Rate ◽  
Random Variables ◽  
Stochastic Comparisons ◽  
Heterogeneous Sample ◽  
Exponential Random Variables ◽  
Hazard Rate Ordering

Abstract We show that the kth order statistic from a heterogeneous sample of n ≥ k exponential random variables is larger than that from a homogeneous exponential sample in the sense of star ordering, as conjectured by Xu and Balakrishnan [14]. As a consequence, we establish hazard rate ordering for order statistics between heterogeneous and homogeneous exponential samples, resolving an open problem of Pǎltǎnea [11]. Extensions to general spacings are also presented.

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.

Reliability Concepts

2018 ◽  
pp. 1-22
Keyword(s):  
Hazard Rate ◽  
Failure Process ◽  
Reliability Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Hazard Rate Function ◽  
Reliability Engineering ◽  
Probability Models ◽  
Engineering Design Process ◽  
Basic Concepts

The first chapter introduces basic concepts of Reliability and their relationships. Four probability functions—reliability function, cumulative distribution function, probability density function, and hazard rate function—that completely characterize the failure process are defined. Three failure rates—MTBF, MTTF, MTTR—that play important role in reliability engineering design process are explained here. The three patterns of failures, DFR, CFR, and IFR, are discussed with reference to the bathtub curve. Two probability models, Exponential and Weibull, are presented. Series and parallel systems and application areas of reliability are also presented.

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