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 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.

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.

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.

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.

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.

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.

On unbounded hazard rates for smoothed perturbation analysis

1995 ◽  
Vol 32 (3) ◽  
pp. 659-667 ◽  
Author(s):  
Michael C. Fu ◽  
Jian-Qiang Hu
Keyword(s):  
Perturbation Analysis ◽  
Hazard Rate ◽  
Rate Function ◽  
Hazard Rates ◽  
Hazard Rate Function ◽  
Finite Support ◽  
Restrictive Assumption ◽  
Rate Functions

Many applications of smoothed perturbation analysis lead to estimators with hazard rate functions of underlying distributions. A key assumption used in proving unbiasedness of the resulting estimator is that the hazard rate function be bounded, a restrictive assumption which excludes all distributions with finite support. Here, we prove through a simple example that this assumption can in fact be removed.

Statistical and information based (physical) minimal repair for k out of n Systems

1998 ◽  
Vol 35 (3) ◽  
pp. 731-740 ◽  
Author(s):  
Philip J. Boland ◽  
Emad El-Neweihi
Keyword(s):  
Failure Time ◽  
Parallel Systems ◽  
Rate Function ◽  
Black Box ◽  
System Failure ◽  
Minimal Repair ◽  
Two Component ◽  
Component Parallel

The ‘minimal’ repair of a system can take several forms. Statistical or black box minimal repair at failure time t is equivalent to replacing the system with another functioning one of the same age, but without knowledge of precisely what went wrong with the system. Its major attribute is its mathematical tractability. In physical minimal repair, at system failure time t, we minimally repair the ‘component’ which brought the system down at time t. The work of Arjas and Norros, Finkelstein, and Natvig is reviewed. The concept of a rate function for minimal repairs of the statistical and physical types are discussed and developed. It is shown that the number of physical minimal repairs is stochastically larger than the number of statistical minimal repairs for k out of n systems with similar components. Some majorization results are given for physical minimal repair for two component parallel systems with exponential components.

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∗.

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