scholarly journals Hazard rate ordering of order statistics and systems

2006 ◽  
Vol 43 (2) ◽  
pp. 391-408 ◽  
Author(s):  
Jorge Navarro ◽  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Random Vector ◽  
Hazard Rate ◽  
Rate Function ◽  
Reliability Theory ◽  
Hazard Rate Function ◽  
Coherent Systems ◽  
Hazard Rate Order ◽  
Limiting Behaviour ◽  
Hazard Rate Ordering

Let X = (X1, X2, …, Xn) be an exchangeable random vector, and write X(1:i) = min{X1, X2, …, Xi}, 1 ≤ i ≤ n. In this paper we obtain conditions under which X(1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

scholarly journals Hazard rate ordering of order statistics and systems

2006 ◽  
Vol 43 (02) ◽  
pp. 391-408 ◽  
Author(s):  
Jorge Navarro ◽  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Random Vector ◽  
Hazard Rate ◽  
Rate Function ◽  
Reliability Theory ◽  
Hazard Rate Function ◽  
Coherent Systems ◽  
Hazard Rate Order ◽  
Limiting Behaviour ◽  
Hazard Rate Ordering

LetX= (X1,X2, …,Xn) be an exchangeable random vector, and writeX(1:i)= min{X1,X2, …,Xi}, 1 ≤i≤n. In this paper we obtain conditions under whichX(1:i)decreases iniin the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

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.

Hazard rate ordering of the largest order statistics from geometric random variables

2018 ◽  
Vol 55 (2) ◽  
pp. 652-658
Author(s):  
Bara Kim ◽  
Jeongsim Kim
Keyword(s):  
Order Statistics ◽  
Open Problem ◽  
Hazard Rate ◽  
Random Variables ◽  
Hazard Rate Order ◽  
Two Samples ◽  
Hazard Rate Ordering

Abstract Mao and Hu (2010) left an open problem about the hazard rate order between the largest order statistics from two samples of n geometric random variables. Du et al. (2012) solved this open problem when n = 2, and Wang (2015) solved for 2 ≤ n ≤ 9. In this paper we completely solve this problem for any value of n.

Discrete Distribution for the Stochastic Range of a Wiener Process and Its Properties

2019 ◽  
Vol 18 (04) ◽  
pp. 1950024 ◽  
Author(s):  
Mohamed Abd Allah El-Hadidy
Keyword(s):  
Order Statistics ◽  
Wiener Process ◽  
Hazard Rate ◽  
Discrete Distribution ◽  
Rate Function ◽  
Strength Parameter ◽  
Hazard Rate Function ◽  
Data Set ◽  
Basic Properties ◽  
Distributional Properties

We introduce the discrete distribution of a Wiener process range (DDWPR). Rather than finding some basic distributional properties including hazard rate function, moments, stress-strength parameter and order statistics of this distribution, this paper studies some basic properties of the truncated version of this distribution. The effectiveness of this distribution is established using a data set.

scholarly journals Dynamic Multivariate Quantile Residual Life in Reliability Theory

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
M. Shafaei Noughabi ◽  
M. Kayid ◽  
A. M. Abouammoh
Keyword(s):  
Hazard Rate ◽  
Residual Life ◽  
Rate Function ◽  
Reliability Theory ◽  
Dynamic Feature ◽  
Hazard Rate Function ◽  
Residual Life Function ◽  
Quantile Residual Life ◽  
Multivariate Quantile ◽  
Life Function

We extend the univariate α-quantile residual life function to multivariate setting preserving its dynamic feature. Principal attributes of this function are derived and their relationship to the dynamic multivariate hazard rate function is discussed. A corresponding ordering, namely, α-quantile residual life order, for random vectors of lifetimes is introduced and studied. Based on the proposed ordering, a notion of positive dependency is presented. Finally, a discussion about conditions characterizing the class of decreasing multivariate α-quantile residual life functions is pointed out.

The Reversed Hazard Rate Function

1998 ◽  
Vol 12 (1) ◽  
pp. 69-90 ◽  
Author(s):  
Henry W. Block ◽  
Thomas H. Savits ◽  
Harshinder Singh
Keyword(s):  
Time Scale ◽  
Hazard Rate ◽  
Rate Function ◽  
Hazard Rates ◽  
Hazard Rate Function ◽  
Reversed Hazard Rate ◽  
Rate Functions ◽  
Hazard Rate Ordering ◽  
Series Systems ◽  
Reversed Hazard Rate Function

In this paper we discuss some properties of the reversed hazard rate function. This function has been shown to be useful in the analysis of data in the presence of left censored observations. It is also natural in discussing lifetimes with reversed time scale. In fact, ordinary hazard rate functions are most useful for lifetimes, and reverse hazard rates are natural if the time scale is reversed. Mixing up these concepts can often, although not always, lead to anomalies. For example, one result gives that if the reversed hazard rate function is increasing, its interval of support must be (—∞, b) where b is finite. Consequently nonnegative random variables cannot have increasing reversed hazard rates. Because of this result some existing results in the literature on the reversed hazard rate ordering require modification.Reversed hazard rates are also important in the study of systems. Hazard rates have an affinity to series systems; reversed hazard rates seem more appropriate for studying parallel systems. Several results are given that demonstrate this. In studying systems, one problem is to relate derivatives of hazard rate functions and reversed hazard rate functions of systems to similar quantities for components. We give some results that address this. Finally, we carry out comparisons for k-out-of-n systems with respect to the reversed hazard rate ordering.

Initial and final behaviour of failure rate functions for mixtures and systems

2003 ◽  
Vol 40 (03) ◽  
pp. 721-740 ◽  
Author(s):  
Henry W. Block ◽  
Yulin Li ◽  
Thomas H. Savits
Keyword(s):  
Asymptotic Behaviour ◽  
Failure Rate ◽  
Rate Function ◽  
Coherent Systems ◽  
Failure Rate Function ◽  
Rate Functions ◽  
Limiting Behaviour

In this paper we consider the initial and asymptotic behaviour of the failure rate function resulting from mixtures of subpopulations and formation of coherent systems. In particular, it is shown that the failure rate of a mixture has the same limiting behaviour as the failure rate of the strongest subpopulation. A similar result holds for systems except the role of strongest subpopulation is replaced by strongest min path set.

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