Probability in the Engineering and Informational Sciences (02699648)
We consider a nonnegative random variable T representing the lifetime of a system. We discuss the residual lifetime, where X denotes the random age of the system. We also discuss the mean residual life (MRL) of T at the random time X. It is shown that the MRL at random age (MRLR) is closely related to some well-known variability measures. In particular, we show that the MRLR can be considered a generalization of Gini's mean difference (GMD). Under the proportional hazards model, we show that the MRLR gives the extended GMD and the extended cumulative residual entropy as special cases. Then, we provide a decomposition result indicating that the MRLR has a covariance representation. Some comparison results are also established for the MRLs of two systems at random ages. © The Author(s), 2025.
European Journal of Operational Research (03772217)318(3)pp. 825-835
Numerous maintenance policies have been proposed in the reliability mathematics and engineering literature. Nevertheless, little has been reported on their practical applications in industries. This gap is largely due to restrictive assumptions of the maintenance policies. Two of the main assumptions are that maintenance is conducted on typical components and that the reliability of an item under maintenance is known (where the item can be a component or a system composed of multiple components). These assumptions do not often hold in the real world: maintenance is often performed on a collection of components such as an integrated circuit plate and the reliability of each individual component may not be known. To reduce these gaps, this paper develops a new maintenance policy for a collection of components and an approximate method to estimate the reliability of this collection based on the failure data collected from the field. The maintenance policy considers that a system is composed of three subsystems with different levels of maintenance effectiveness (i.e, minimal, imperfect, and perfect). The approximate estimate of the reliability of each subsystem is derived based on the failure data that are time between failures of the system but not those of the components that cause the system to fail. An algorithm for simulating the superposition of generalised renewal processes is then proposed. Numerical examples are used to illustrate the proposed approximation method. © 2024
Naval Research Logistics (15206750)71(5)pp. 728-738
The geometric process has been widely studied in various disciplines and applied in reliability, maintenance and warranty cost analysis, among others. In its applications in maintenance policy optimisation, the geometric process assumes constant repair effectiveness by its process rate. Nevertheless, in practice, maintenance effectiveness may differ from time to time and can therefore be better depicted by a random variable. Motivated by this argument, this paper proposes a new variant of the geometric process, which is referred to as the rate randomized geometric process (RRGP). The probabilistic properties of the RRGP are then investigated. The maximum likelihood method is utilised to estimate the parameters of the RRGP. Numerical examples are given to show its applicability in both maintenance policy optimization and fitting real-world failure datasets. © 2024 Wiley Periodicals LLC.
Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability (17480078)
This paper aims to develop a general framework to use degradation models in the reliability assessments of coherent systems. In the first part of the paper, we prove that if the components of a system are subject to a degradation model, then the same holds for the system. This fact is used to study the preservation results of some aging classes and stochastic orders in systems under degradation models. In the second part, we give a survival signature-based representation of the system’s reliability under the condition that the components of the system deteriorate under a linear degradation model. We then use this representation of the system’s reliability to propose optimal preventive maintenance for coherent systems whose components are under a linear degradation model. © IMechE 2024.
Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability (17480078)238(2)pp. 291-303
In this paper, we investigate optimal age-based preventive maintenance (PM) policies for an (n-k + 1)-out-of-n system whose components are exposed to fatal shocks that arrive from various sources. We consider two different scenarios for the system failure. In the first one, it is assumed that the shock process is of the type of Marshall-Olkin where each shock affects one component of the system and puts it down, and one shock affects all components and destroys all of them. In the second scenario, it is assumed that the system is subject to an extended type of Marshall-Olkin shock process where the shocks arriving at random times may cause the breakdown of 1, 2, …, or n components. Under each scenario for the components failure, we investigate an optimal age-based PM model for the system by imposing the related cost function. Then, in each case, we explore the optimal PM time that minimizes the mean cost per unit of time. Some numerical results are presented to illustrate the applications of the proposed models. © IMechE 2023.
Quality Technology and Quantitative Management (16843703)21(5)pp. 656-673
This paper studies the reliability function of a weighted (Formula presented.) -out-of- (Formula presented.) system with multiple types of components. A weighted system with (Formula presented.) different types of components is considered and two real-life scenarios for the operation of the system are defined. In the first one, it is assumed that the system operates if the accumulated weight of all active components is more than a predetermined threshold (Formula presented.), otherwise the system fails. In the second scenario, the system is assumed to operate if the total weight of the operating components of type (Formula presented.) is more than or equal to the corresponding threshold (Formula presented.), for at least (Formula presented.) types of components, (Formula presented.). Our approaches in assessing the reliability function of the system lifetime rely on the notion of survival signature corresponding to multi-type weighted (Formula presented.) -out-of- (Formula presented.) system. The formulas for the proposed survival signature are presented and the corresponding computational algorithms are also given. To examine the theoretical results, the reliability function and other aging characteristics of a wind farm (system) consisting of three plants located in three different regions are assessed numerically and graphically. Finally, an allocation problem to determine the best choice for the distribution of each type in the weighted (Formula presented.) -out-of- (Formula presented.) systems is studied. © 2023 International Chinese Association of Quantitative Management.
Communications in Statistics - Theory and Methods (1532415X)53(11)pp. 4062-4084
The α-mixture model, as a flexible family of distributions, is an effective tool for modeling heterogeneity inS population. This article investigates the hazard rate of α-mixture in terms of hazard rates of mixed baseline distributions. In particular, when the baseline hazard rate follows either additive or multiplicative models an inverse problem to obtain the baseline hazard is solved. We, also, study the α-mixture hazard rate ordering for the ordered mixing distributions in the sense of likelihood ratio order. Sufficient conditions to order two finite α-mixtures in the sense of dispersive ordering are provided. Finally, it is shown that the hazard rate of the finite α-mixture in the multiplicative model tends to the hazard rate of the strongest (weakest) population as (Formula presented.) ((Formula presented.)). Several examples are presented to illustrate theoretical findings. © 2023 Taylor & Francis Group, LLC.
Journal of Computational and Applied Mathematics (03770427)435
An effective method for improving the reliability of a technical system is to build redundancy into it. A commonly used redundancy structure in reliability engineering is the k-out-of-n structure. A k-out-of-n structure, consisting of n components, operates if and only if at least k components operate. This paper studies a high-dimension feature of k-out-of-n systems consisting of independent and identical components. This phenomenon, commonly known as a phase transition in high-dimensional probability, shows that in a system consisting of a large number of components, if the components operate independently with common reliability, then the system reliability undergoes a ‘sharp transition’. That is, in a narrow interval, the value of the system reliability rapidly declines. We first consider the systems whose components have common static reliability p∈(0,1). Then we focus on k-out-of-n systems with time-dependent reliability. As an application of the results, we employ the phase transition property to propose optimal age-based maintenance for the system. © 2023 Elsevier B.V.
Test (11330686)33(3)pp. 717-730
In this short communication, we discuss the remaining lifetime and the mean remaining lifetime (MRL) of an item with a random age. We show that the MRL at random age is closely related to some well-known variability measures. First, we provide a decomposition result showing that the MRL at random age, similar to other variability measures, has a covariance representation. Under the proportional hazards (PH) model, we show that the MRL, depending on the parameter of proportionality, subsumes the Gini’s mean difference and the cumulative residual entropy as special cases. It is also shown that, under the PH model, the MRL can be expressed via the equilibrium distribution and the mean number of events in the generalized Pólya process. © The Author(s) under exclusive licence to Sociedad de Estadística e Investigación Operativa 2024.
Applied Stochastic Models in Business and Industry (15264025)40(4)pp. 875-894
One primary objective of reliability engineering is to achieve optimal maintenance of technical systems, which ensures they remain in good operating condition. This paper proposes an age-based preventive optimal maintenance policy for (Formula presented.) -component coherent systems. Under this proposed strategy, the system begins operating at (Formula presented.) and undergoes preventative maintenance (PM) at a time (Formula presented.). If the system fails before (Formula presented.), it will be replaced with a new one. If the system is still functioning at time (Formula presented.), an assessment is made based on the number of failed components to determine whether the system should be replaced or allowed to continue operating. If the number of failures at (Formula presented.) is below a predetermined threshold (Formula presented.), the PM time (Formula presented.) is postponed and a new PM time (Formula presented.) will be scheduled, and the system continues operating in the interval (Formula presented.). Otherwise, the entire system is preventively replaced at (Formula presented.) with a new one. In this scenario, we use a cost function to determine the optimal values of decision variables (Formula presented.), (Formula presented.), and (Formula presented.). To examine the effectiveness of our proposed model, we analyze some examples of coherent systems using graphical and numerical methods. © 2024 John Wiley & Sons Ltd.
Computational Statistics (09434062)39(5)pp. 2721-2742
The area under a receiver operating characteristic (ROC) curve is frequently used in medical studies to evaluate the effectiveness of a continuous diagnostic biomarker, with values closer to one indicating better classification. Unfortunately, the standard statistical procedures based on simple random sampling (SRS) and ranked set sampling (RSS) techniques tend to be less efficient when the values of the area under a ROC curve (AUC) get closer to one. Thus, developing some statistical procedures for efficiently estimating the AUC when it is close to one is very important. In this paper, some estimators are developed using nomination sampling to assess AUC. The proposed AUC estimators are compared with their counterparts in SRS and RSS using Monte Carlo simulation. The results show that some of the estimators developed in this study considerably improve the efficiency of the AUC estimation when it is close to one. This substantially reduces the cost and time for the sample size needed to obtain the desired precision. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.
Ricerche di Matematica (18273491)72(2)pp. 1-36
In this paper, we propose a flexible growth model that constitutes a suitable generalization of the well-known Gompertz model. We perform an analysis of various features of interest, including a sensitivity analysis of the initial value and the three parameters of the model. We show that the considered model provides a good fit to some real datasets concerning the growth of the number of individuals infected during the COVID-19 outbreak, and software failure data. The goodness of fit is established on the ground of the ISRP metric and the d2 -distance. We also analyze two time-inhomogeneous stochastic processes, namely a birth-death process and a birth process, whose means are equal to the proposed growth curve. In the first case we obtain the probability of ultimate extinction, being 0 an absorbing endpoint. We also deal with a threshold crossing problem both for the proposed growth curve and the corresponding birth process. A simulation procedure for the latter process is also exploited. © 2020, The Author(s).
Statistical Papers (09325026)64(1)pp. 161-177
The mean past lifetime (MPL) is an important tool in reliability and survival analysis for measuring the average time elapsed since the occurrence of an event, under the condition that the event has occurred before a specific time t> 0. This article develops a nonparametric estimator for MPL based on observations collected according to ranked set sampling (RSS) design. It is shown that the proposed estimator is a strongly uniform consistent estimator of MPL. It is also proved that the introduced estimator tends to a Gaussian process under some mild conditions. A Monte Carlo simulation study is employed to evaluate the performance of the proposed estimator with its competitor in simple random sampling (SRS). Our findings show the introduced estimator is more efficient than its counterpart estimator in SRS as long as the quality of ranking is better than random. Finally, an illustrative example is provided to describe the potential application of the developed estimator in assessing the average time between the infection and diagnosis in HIV patients. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
Hashemi, Marzieh,
Asadi, M.,
Asadi, M.,
Hashemi, M.,
Balakrishnan n., N. Applied Stochastic Models in Business and Industry (15264025)39(1)pp. 4-53
In reliability engineering literature, a large number of research papers on optimal preventive maintenance (PM) of technical systems (networks) have appeared based on preliminary many different approaches. According to the existing literature on PM strategies, the authors have considered two scenarios for the component failures of the system. The first scenario assumes that the components of the system fail due to aging, while the second scenario assumes the system fails according to the fatal shocks arriving at the system from external or internal sources. This article reviews different approaches on the optimal strategies proposed in the literature on the optimal maintenance of multi-component coherent systems. The emphasis of the article is on PM models given in the literature whose optimization criteria (cost function and stationary availability) are developed by using the signature-based (survival signature-based) reliability of the system lifetime. The notions of signature and survival signature, defined for systems consisting of one type or multiple types of components, respectively, are powerful tools assessing the reliability and stochastic properties of coherent systems. After giving an overview of the research works on age-based PM models of one-unit systems and k-out-of-n systems, we provide a more detailed review of recent results on the signature-based and survival signature-based PM models of complex systems. In order to illustrate the theoretical results on different proposed PM models, we examine two real examples of coherent systems both numerically and graphically.
Reliability Engineering and System Safety (18790836)234
In reliability engineering and other disciplines, there are many situations, in which one deals with the number of events in a time period that happen as an outcome of a stochastic process. Examples are the number of repairs of a system in an age-based block replacement and the number of repairs for a product sold under warranty in a specific period. In these circumstances, the mean number of failures (repair) plays a crucial role in assessing reliability characteristics and the optimal time of maintenance of the system. In this paper, we study a parametric function for the mean number of repairs of the system that covers a wide range of repair degrees, such as better than minimal repair, minimal repair and worse than minimal repair. Using the suggested model, we propose an approximation for the renewal function in the case that the repairs are complete. We also give applications of the proposed function in age-based block replacement and optimal warranty policy. © 2023 Elsevier Ltd
Communications in Statistics - Theory and Methods (1532415X)52(21)pp. 7564-7575
Mixture of survival and hazard functions have been widely applied to the analysis of data from heterogeneous populations. From the Bayesian point of view, two different predictive mixture hazard rates can be considered: prior and posterior predictive hazard rates. In this paper, we consider a heterogeneous population consists of two sub-populations with different hazard rates where each one follows a Cox proportional hazards model. A prior predictive mixture hazard model is proposed to estimate the hazard rate of the population through the assessment of some potential regression covariates. Under right-censoring, the estimating equations based on martingale are developed to estimate the parameters of the assumed mixture model. The large sample properties of the proposed estimators are established. The finite sample behavior of the resulting estimators is evaluated through simulation studies, and the approach is also applied to a kidney cancer data set collected from a clinical trial. © 2022 Taylor & Francis Group, LLC.
Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability (17480078)237(6)pp. 1100-1113
In the literature of system reliability and other fields related to the time to occurrence of an event, shock models play an important role. In this paper, we assume that a system is subject to shocks that occur according to a counting process describing the number of shocks that arrive during a specified time interval. As the magnitude of the damage imposed to the system by each shock is a crucial parameter to the system’s survival, we investigate some important random variables related to this parameter. A random variable of interest associated with this process is the first time, after a pre-specified time (Formula presented.), at which the amount of a shock damage to the system gets greater than the maximum of damages imposed to the system until time (Formula presented.). We obtain the reliability function of this random variable and investigate various properties of it and some other related random variables. In order to explore further the results, we examine two commonly used processes in the literature, that is, the non-homogeneous Poisson process and the Pólya process. © IMechE 2022.
Hashemi, Marzieh,
Behboudi z., ,
Mohtashami borzadaran g.r., ,
Asadi, M. Applied Stochastic Models in Business and Industry (15264025)39(3)pp. 333-351
This article investigates an optimal preventive standby activation policy for an m-out-of-n redundant system. We assume that for such a system, which starts operating at time t = 0, a standby component will be activated at either the failure time of the system or at a predetermined time tau, whichever occurs first. We first obtain the system reliability function under this switching policy as a function of tau. Then, we investigate the optimal switching time tau so that the mean time to failure of the system is maximized. The results indicate that the existence of an optimal value of tau ,depends on the lifetime of the standby redundancy and its virtual age in the standby state. Some illustrative examples are presented to examine the theoretical outcomes.
Applied Stochastic Models in Business and Industry (15264025)39(1)pp. 85-87
Asadi, M.,
Devarajan, K.,
Ebrahimi, N.,
Soofi, E.S.,
Spirko-burns, L. International Statistical Review (03067734)90(3)pp. 499-524
Various statistical methodologies embed a probability distribution in a more flexible family of distributions. The latter is called elaboration model, which is constructed by choice or a formal procedure and evaluated by asymmetric measures such as the likelihood ratio and Kullback–Leibler information. The use of asymmetric measures can be problematic for this purpose. This paper introduces two formal procedures, referred to as link functions, that embed any baseline distribution with a continuous density on the real line into model elaborations. Conditions are given for the link functions to render symmetric Kullback–Leibler divergence, Rényi divergence and phi-divergence family. The first link function elaborates quantiles of the baseline probability distribution. This approach produces continuous counterparts of the binary probability models. Examples include the Cauchy, probit, logit, Laplace and Student's (Formula presented.) links. The second link function elaborates the baseline survival function. Examples include the proportional odds and change point links. The logistic distribution is characterised as the one that satisfies the conditions for both links. An application demonstrates advantages of symmetric divergence measures for assessing the efficacy of covariates. © 2022 International Statistical Institute.
Methodology and Computing in Applied Probability (13875841)24(3)pp. 1485-1502
In this paper, we consider two coherent systems having shared components. We assume that in the two systems there are three different types of components; components of type one that just belong to the first system, components of type two that lie only in the second system and components of type three that are shared by the two systems. We use the concept of joint survival signature to assess the joint reliability function of the two systems. Using this concept, some representations for the joint reliability function of the system lifetimes are obtained under two different scenarios of component failures. In the first scenario, we assume that the components of the systems fail according to different counting processes such as non-homogeneous Poisson processes. In the second scenario, it is assumed that the component lifetimes of each type are exchangeable while the three types of component lifetimes can be independent or dependent. To illustrate the theoretical results, two systems with shared components are studied numerically and graphically. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Applied Stochastic Models in Business and Industry (15264025)38(4)pp. 590-608
This paper investigates some maintenance strategies for repairable systems that are exposed to two distinct types of failure: internal failures due to aging over time and fatal external shocks arriving according to a nonhomogeneous Poisson process. When the system fails, it will be repaired, and it is assumed that the repair does not bring the system to an as-good-as-new state. Two maintenance models are developed to prevent the system from failure under the mentioned scenario. By imposing the related cost functions, we investigate the corresponding optimal policy for each model such that the expected cost rate is minimized. Some numerical examples and simulation studies are presented to demonstrate the proposed maintenance models. © 2022 John Wiley & Sons, Ltd.
Reliability Engineering and System Safety (18790836)217
We investigate optimal maintenance models for warranted coherent systems consisting of n components (n≥1). The system is under a warranty policy with two consecutive phases under which the manufacturer's commitment goes in two different forms. In Phase I, upon the system failure, the failed components are replaced with new ones and a corrective maintenance is conducted on the whole system. If the system failure occurs during Phase II, only minimal repair is performed on the components. Following expiration of the warranty time, the customer is solely responsible for maintaining the system who, for a fixed length of the time period, replaces the failed components and preventively maintains the system at the end of such period. During the maintenance period, the same maintenance as that of in Phase I of the warranty period is conducted. Such a maintenance model can be considered as a generalization of the age-based maintenance model for a system under warranty model. We investigate, from the manufacturer's and customer's perspectives, the optimal length of each phase in the warranty period, and the optimal planned time of preventive maintenance based on some imposed cost functions, respectively. Numerical examples are provided to illustrate the proposed optimal maintenance models. © 2021
Computers and Industrial Engineering (03608352)163
In this paper, we study some preventive maintenance policies for coherent systems with n,n⩾1, components. We consider a coherent system consisting of K,K⩾1, different types of independent components. The system may fail due to the aging of its components, or it may fail due to fatal shocks arriving from external sources. Under this mechanism of system failure, we propose optimal age-based and block preventive maintenance models by considering the costs of preventive maintenance, corrective maintenance, and minimal repairs. We provide some formulas for the average long-run cost rate of the proposed strategies. The existence of optimal values under which the imposed cost functions are minimized has been discussed. In addition, Monte Carlo simulations are carried out to evaluate the cost functions of the proposed maintenance models. Our derivations rely on the concept of survival signature. Real examples of coherent systems are also presented to examine and illustrate the theoretical results. © 2021 Elsevier Ltd
Journal of Applied Probability (00219002)59(4)pp. 1144-1177
This paper is concerned with the optimal number of redundant allocation to n-component coherent systems consisting of heterogeneous dependent components. We assume that the system is built up of L groups of different components, L ≥ 1, where there are ni components in group i, and ΣLi=1 ni = n. The problem of interest is to allocate vi active redundant components to each component of type i, i = 1, . . ., L. To get the optimal values of vi we propose two cost-based criteria. One of them is introduced based on the costs of renewing the failed components and the costs of refreshing the alive ones at the system failure time. The other criterion is proposed based on the costs of replacing the system at its failure time or at a predetermined time τ, whichever occurs first. The expressions for the proposed functions are derived using the mixture representation of the system reliability function based on the notion of survival signature. We assume that a given copula function models the dependency structure between the components. In the particular case that the system is a series-parallel structure, we provide the formulas for the proposed cost-based functions. The results are discussed numerically for some specific coherent systems. © The Author(s), 2022.
Probability in the Engineering and Informational Sciences (02699648)36(4)pp. 1055-1079
Most of the real-life populations are heterogeneous and homogeneity is often just a simplifying assumption for the relevant statistical analysis. Mixtures of lifetime distributions that correspond to homogeneous subpopulations were intensively studied in the literature. Various distributional and stochastic properties of finite and continuous mixtures were discussed. In this paper, following recent publications, we develop further a mixture concept in the form of the generalized α-mixtures that include all mixture models that are widely explored in the literature. We study some main stochastic properties of the suggested mixture model, that is, aging and appropriate stochastic comparisons. Some relevant examples and counterexamples are given to illustrate our findings. Copyright © The Author(s), 2021. Published by Cambridge University Press.
Metrika (1435926X)84(8)pp. 1213-1240
The α-mixtures is a new, flexible family of distributions that includes many mixture models as special cases. This paper is mainly focused on relevant stochastic comparisons and ageing properties of α-mixtures of survival functions. In particular, we prove that ageing properties of α-mixtures for additive and multiplicative hazards models depend on the properties of the baseline failure rate functions and the corresponding conditional moments of mixing distributions. Partial orderings of the finite α-mixtures in the sense of the usual stochastic order and the hazard rate order are discussed. Finally, we extend some results on the shape of the mixture failure rate obtained in the literature for usual mixtures to the case of α-mixtures. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability (17480078)235(5)pp. 909-922
This article is a study on the reliability characteristics of a system under a failure model called the generalized mixed (Formula presented.) -shock model. We assume that the system is subject to shocks according to a stochastic process. Each shock may cause some damage to the system. The system fails either the magnitude of the damage caused by a shock exceeds a threshold (Formula presented.) or the time between two consecutive shocks is less than a pre-specified threshold (Formula presented.) and simultaneously magnitude of the damage is bigger than a pre-specified critical threshold (Formula presented.) ((Formula presented.)). The survival function and other characteristics of the system lifetime are investigated. By imposing a cost function, we arrive at an optimal replacement policy for the system based on the proposed failure model. Several examples are provided under which we illustrate the theoretical results numerically and graphically. © IMechE 2021.
Reliability Engineering and System Safety (18790836)205
Weighted k-out-of-n systems form an important class of redundant systems with a wide range of applications in reliability engineering. In this paper, we introduce novel optimal preventive maintenance models for the class of weighted k-out-of-n systems based on average cost and availability criteria. Depending on the average number of failed components, we set up a cost (availability) function corresponding to the time that the total weights of the working components become less than a predetermined threshold m, or the age of the system reaches TPM. The form of the cost (and availability) function relies on the mixture representation of the system reliability based on the notion of the signature of the system. Some examples of weighted k-out-of-n systems are presented to demonstrate the proposed models numerically and graphically. © 2020 Elsevier Ltd
Applied Stochastic Models in Business and Industry (15264025)37(6)pp. 1017-1041
Nowadays, networks (systems) appear in many areas of science and technology. One of the most important strategies to reduce the likelihood of the failure of an operating network is preventive maintenance (PM). In this article, we propose some optimal PM models for a network consisting of n, n ≥ 1, links (components). The criteria of interest are the “cost function” of renewing the network and “stationary availability” of the network during its mission. In the first part of the article, we consider the case that the network is formed of identical components while in the second part, we deal with the case that the network is built up from several nonidentical groups of components. In both parts, we utilize the PM under some partial information about the status of the network at a time t. The results of the article are developed using the notions of signature and survival signature. To interpret the proposed models, the results are illustrated numerically and graphically for two networks. © 2021 John Wiley & Sons, Ltd.
IISE TRANSACTIONS (24725854)53(11)pp. 1266-1280
We propose optimal preventive maintenance strategies for n-component coherent systems. We assume that in the early time of the system operation all failed components are repaired, such that the state of a failed component gets back to a working state, worse than that of prior to failure. To model this repair action, we utilize a counting process on the interval (0, tau], known as the generalized Polya process (which subsumes the non-homogeneous Poisson process as a special case). Two generalized Polya process-based repair strategies are proposed. The criteria to be optimized are the cost function formulated based on the repair costs of the components/system, and the system availability, to obtain the optimal time of preventive maintenance of the system. To illustrate the theoretical results, two coherent systems are studied for which the optimal preventive maintenance times are explored under different conditions.
REVSTAT-Statistical Journal (21830371)19(1)pp. 111-130
In this paper, we propose some estimators for the parameters of a statistical model based on Kullback–Leibler divergence of the survival function in continuous setting and apply it to type I censored data. We prove that the proposed estimators are subclass of “generalized estimating equations” estimators. The asymptotic properties of the estimators such as consistency and asymptotic normality are investigated. Some illustrative examples are also provided. In particular, in estimating the shape parameter of generalized Pareto distribution, we show that our procedure dominates some existing methods in the sense of bias and mean squared error. © 2021, National Statistical Institute. All rights reserved.
Behboudi z., ,
Mohtashami borzadaran g.r., ,
Asadi, M. Applied Mathematical Modelling (0307904X)92pp. 176-195
In the present paper, we use the concept of virtual age to propose a periodic switching policy for assessing the reliability of a two-unit cold standby system. This approach is suitable for units that recover with discontinuous activity and get younger. To restore the units and bring back the state of the system to a desirable condition, the active unit and cold standby unit are periodically replaced with each other at some given time interval. The survival function of the system under the periodic switching policy is derived in terms of the baseline time-to-failure distribution. The results indicate that, when the units have an increasing failure rate, the new switching policy improves the performance of the units which, in turn, increases the reliability of the system. Since the switching is not always failure-free, we derive also the survival function of the system in the case of imperfect switching. By considering the cost of switching and the system profit, we investigate the optimal length of the time interval for switching the active unit with the standby unit such that the mean profit of the system is maximized. To examine the theoretical results, we present some illustrative examples. Finally, we use a Monte-Carlo simulation to evaluate the system reliability. © 2020 Elsevier Inc.
Ardakani, O.M.,
Asadi, M.,
Ebrahimi, N.,
Soofi, E.S. Journal Of The Iranian Statistical Society (17264057)20(1)pp. 27-59
In recent years, we have studied information properties of various types of mixtures of probability distributions and introduced a new type, which includes previously known mixtures as special cases. These studies are disseminated in different fields: reliability engineering, econometrics, operations research, probability, the information theory, and data mining. This paper presents a holistic view of these studies and provides further insights and examples. We note that the insightful probabilistic formulation of the mixing parameters stipulated by Behboodian (1972) is required for a representation of the well-known information measure of the arithmetic mixture. Applications of this information measure presented in this paper include lifetime modeling, system reliability, measuring uncertainty and disagreement of forecasters, probability modeling with partial information, and information loss of kernel estimation. Probabilistic formulations of the mixing weights for various types of mixtures provide the Bayes-Fisher information and the Bayes risk of the mean residual function. © 2021,Journal of the Iranian Statistical Society. All rights reserved.
Metrika (1435926X)83(6)pp. 657-676
Measuring the correlation between two random variables is an important goal in various statistical applications. The standardized covariance is a widely used criterion for measuring the linear association. In this paper, first, we propose a covariance-based unified measure of variability for a continuous random variable X and show that several measures of variability and uncertainty, such as variance, Gini mean difference and cumulative residual entropy arise as special cases. Then, we propose a unified measure of correlation between two continuous random variables X and Y, with distribution functions (DFs) F and G. Assuming that H is a continuous DF, the proposed measure is defined based on the covariance between X and the transformed random variable H- 1G(Y) (known as the Q-transformation of H on G). We show that our proposed measure of association subsumes some of the existing measures of correlation. Under some mild condition on H, it is shown that the suggested index ranges in [- 1 , 1] where the extremes of the range, i.e., - 1 and 1, are attainable by the Fréchet bivariate minimal and maximal DFs, respectively. A special case of the proposed correlation measure leads to a variant of the Pearson correlation coefficient which has absolute values greater than or equal to Pearson correlation. The results are examined numerically for some well known bivariate DFs. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
Communications in Statistics - Theory and Methods (1532415X)49(1)pp. 44-60
In order to discriminate between two probability distributions extensions of Kullback–Leibler (KL) information have been proposed in the literature. In recent years, an extension called cumulative Kullback–Leibler (CKL) information is considered by authors which is closely related to equilibrium distributions. In this paper, we propose an adjusted version of CKL based on equilibrium distributions. Some properties of the proposed measure of divergence are investigated. A test of exponentiality based on the adjusted measure, is proposed. The empirical power of the presented test is calculated and compared with some existing standard tests of exponentiality. The results show that our proposed test, for some important alternative distributions, has better performance than some of the existing tests. © 2018, © 2018 Taylor & Francis Group, LLC.
Statistics (02331888)54(6)pp. 1311-1328
A time-dependent divergence measure is proposed to compare the survival functions of two lifetime random variables. It is shown that the proposed measure ranges between (Formula presented.) and for the proportional hazards case has the metric properties. Several properties of the divergence measure are investigated, among others, it is shown that the divergence between two survival functions does not depend on time if and only if they follow the proportional hazards model. The measure is also examined for various other well-known survival models such as the proportional odds model. The estimation of the suggested divergence measure for survival data is also discussed, and the asymptotic normal distribution of the resulting estimator is established. The proposed estimation is evaluated via simulation and further employed to compare the effects of two treatment groups on the overall survival times of kidney cancer patients. © 2020 Informa UK Limited, trading as Taylor & Francis Group.
Journal of Statistical Computation and Simulation (15635163)90(17)pp. 3232-3249
We consider a coherent system consists of n components with independent and identically distributed lifetimes. We use the well-known signature-based representation of the system reliability to estimate the reliability of components of the system when the data is the system failure times collected according to a progressively censored scheme. Different estimators for the reliability of components are proposed and their properties are examined. A Monte Carlo simulation study have been carried out to compare the performance of the estimation methods. © 2020 Informa UK Limited, trading as Taylor & Francis Group.
Ardakani, O.M.,
Asadi, M.,
Ebrahimi, N.,
Soofi, E.S. Statistical Analysis and Data Mining (19321864)13(4)pp. 405-418
Big data enables reliable estimation of continuous probability density, cumulative distribution, survival, hazard rate, and mean residual functions (MRFs). We illustrate that plot of the MRF provides the best resolution for distinguishing between distributions. At each point, the MRF gives the mean excess of the data beyond the threshold. Graph of the empirical MRF, called here the MR plot, provides an effective visualization tool. A variety of theoretical and data driven examples illustrate that MR plots of big data preserve the shape of the MRF and complex models require bigger data. The MRF is an optimal predictor of the excess of the random variable. With a suitable prior, the expected MRF gives the Bayes risk in the form of the entropy functional of the survival function, called here the survival entropy. We show that the survival entropy is dominated by the standard deviation (SD) and the equality between the two measures characterizes the exponential distribution. The empirical survival entropy provides a data concentration statistic which is strongly consistent, easy to compute, and less sensitive than the SD to heavy tailed data. An application uses the New York City Taxi database with millions of trip times to illustrate the MR plot as a powerful tool for distinguishing distributions. © 2020 Wiley Periodicals LLC.
Mathematical Methods of Statistics (19348045)29(3)pp. 135-148
Abstract: Recently, a time-dependent measure of divergence has been introduced by Mansourvar and Asadi (2020) to assess the discrepancy between the survival functions of two residual lifetime random variables. In this paper, we derive various time-dependent results on the proposed divergence measure in connection to other well-known measures in reliability engineering. The proposed criterion is also examined in mixture models and a general class of survival transformation models which results in some well-known models in the lifetime studies and survival analysis. In addition, the time-dependent measure is employed to evaluate the divergence between the lifetime distributions of k-out-of-n systems and also to assess the discrepancy between the distribution functions of the epoch times of a non-homogeneous Poisson process. © 2020, Allerton Press, Inc.
Advances in Applied Probability (00018678)52(4)pp. 1197-1223
Providing optimal strategies for maintaining technical systems in good working condition is an important goal in reliability engineering. The main aim of this paper is to propose some optimal maintenance policies for coherent systems based on some partial information about the status of components in the system. For this purpose, in the first part of the paper, we propose two criteria under which we compute the probability of the number of failed components in a coherent system with independent and identically distributed components. The first proposed criterion utilizes partial information about the status of the components with a single inspection of the system, and the second one uses partial information about the status of component failure under double monitoring of the system. In the computation of both criteria, we use the notion of the signature vector associated with the system. Some stochastic comparisons between two coherent systems have been made based on the proposed concepts. Then, by imposing some cost functions, we introduce new approaches to the optimal corrective and preventive maintenance of coherent systems. To illustrate the results, some examples are examined numerically and graphically. © 2020 Applied Probability Trust.
Reliability Engineering and System Safety (18790836)195
We propose two new optimal maintenance strategies for preservation of a complex n-component coherent system. The criteria that will be optimized are some given imposed cost functions formulated based on the costs of the repairs of components (system) to get the optimal time of preventive maintenance of the system. The first strategy involves minimal repair on the failed components in an early life period of the system and then corrective maintenance on the entire system upon its failure or preventive/corrective maintenance on each component when the age of the system reaches t. In the second strategy, we assume that the system is inspected at times t,2t,…,mt, m=1,2⋯. At each inspection time, depending on the system conditions, a maintenance action is performed on the components of the system and then the imposed cost function is minimized to get the optimal preventive maintenance time. To interpret the theoretical results, a system constructed from 12 components of three types is examined for which the optimal preventive maintenance times are explored for both strategies under given cost functions. © 2019 Elsevier Ltd
Reliability Engineering and System Safety (18790836)185pp. 124-132
This article is a study on the reliability and preventive maintenance of the coherent systems whose components are subject to failure according to multiple external shocks. We consider an n-component coherent system in which the components are categorized to L different batches, 2 ≤ L ≤ n. It will assume that the components of the batches are subject to failure according to independent external shocks arriving based on independent counting processes. Under this model of components failure, we obtain the survival signature based reliability function of the system lifetime. Then, we investigate the optimal time of preventive maintenance of the system by imposing some cost functions and some criteria on the stationary availability of the system. In order to illustrate the results, some examples have presented in which the failure of components in different batches occur due to the external shocks which arrive according to independent nonhomogeneous Poisson processes with different mean value functions. © 2018 Elsevier Ltd
Computational Statistics and Data Analysis (01679473)135pp. 35-55
The mean residual life (MRL) of a nonnegative random variable X plays an important role in various disciplines such as reliability, survival analysis, and extreme value theory. This paper deals with the problem of estimating the MRL in ranked set sampling (RSS) design. An RSS-based estimator for MRL is proposed and its properties are investigated. For finite sample sizes, a Monte Carlo simulation study is carried out to show that the resulting estimator is more efficient than its counterpart in simple random sampling (SRS) design. It is proved that the proposed estimator asymptotically follows a Gaussian process and its asymptotic variance is no larger than its counterpart in the SRS design, regardless of the quality of ranking. Different methods of constructing a confidence interval for MRL in the RSS and SRS designs are then discussed. It is observed that while both the RSS and SRS-based confidence intervals do not control the nominal confidence level equally well, the RSS-based confidence intervals have generally shorter lengths than those in the SRS scheme. Finally, a potential application in the context of medical studies is presented for illustration purpose. © 2019 Elsevier B.V.
Asadi, M.,
Ebrahimi, N.,
Kharazmi, O.,
Soofi, E.S. IEEE Transactions on Information Theory (00189448)65(4)pp. 2316-2321
This paper presents the Bayes Fisher information measures, defined by the expected Fisher information under a distribution for the parameter, for the arithmetic, geometric, and generalized mixtures of two probability density functions. The Fisher information of the arithmetic mixture about the mixing parameter is related to chi-square divergence, Shannon entropy, and the Jensen-Shannon divergence. The Bayes Fisher measures of the three mixture models are related to the Kullback-Leibler, Jeffreys, Jensen-Shannon, Rényi, and Tsallis divergences. These measures indicate that the farther away are the components from each other, the more informative are data about the mixing parameter. We also unify three different relative entropy derivations of the geometric mixture scattered in statistics and physics literatures. Extensions of two of the formulations to the minimization of Tsallis divergence give the generalized mixture as the solution. © 2018 IEEE.
Metrika (1435926X)82(5)pp. 589-605
The weighted k-out-of-n (briefly denoted as weighted k / n) systems are among the most important kind of redundancy structures. We consider a weighted k / n system with dependent components where the system is built up from two classes CX and CY of components that are categorized according to their weights and reliability functions. It is assumed that a random number M, M= 0 , 1 , ⋯ , m, of the components are chosen from set CX whose components are distributed as FX and the remaining n- M components selected from the set CY whose components have distribution function FY. We further assume that the structure of dependency of the components can be modeled by a copula function. The reliability of the system, at any time t, is expressed as a mixture of reliability of weighted k / n systems with fixed number of the components of types CX and CY in terms of the probability mass function M. Some stochastic orderings are made between two different weighted k / n systems. It is shown that when the random mechanism of the chosen components for two systems are ordered in usual stochastic (st) order then, under some conditions, the lifetimes of the two systems are also ordered in st order. We also compare the lifetimes of two different systems in the sense of stochastic precedence concept. The results are examined by several illustrative examples under different conditions. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
International Journal of Biostatistics (15574679)15(1)
The mean past lifetime provides the expected time elapsed since the failure of a subject given that he/she has failed before the time of observation. In this paper, we propose the proportional mean past lifetime model to study the association between the mean past lifetime function and potential regression covariates. In the presence of left censoring, martingale estimating equations are developed to estimate the model parameters, and the asymptotic properties of the resulting estimators are studied. To assess the adequacy of the model, a goodness of fit test is also investigated. The proposed method is evaluated via simulation studies and further applied to a data set. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.
Probability in the Engineering and Informational Sciences (02699648)33(3)pp. 438-459
The signature matrix of an n-component three-state network (system), which depends only on the network structure, is a useful tool for comparing the reliability and stochastic properties of networks. In this paper, we consider a three-state network with states up, partial performance, and down. We assume that the network remains in state up, for a random time T1 and then moves to state partial performance until it fails at time T>T1. The signature-based expressions for the conditional entropy of T given T1, the joint entropy, Kullback-Leibler (K-L) information, and mutual information of the lifetimes T and T1 are presented. It is shown that the K-L information, and mutual information between T1 and T depend only on the network structure (i.e., depend only to the signature matrix of the network). Some signature-based stochastic comparisons are also made to compare the K-L of the state lifetimes in two different three-state networks. Upper and lower bounds for the K-L divergence and mutual information between T1 and T are investigated. Finally the results are extended to n-component multi-state networks. Several examples are examined graphically and numerically. Copyright © Cambridge University Press 2018.
Statistical Papers (09325026)60(3)pp. 455-471
In the study of reliability and stochastic properties of technical systems a realistic assumption is to consider the dependency between the components of the system. We investigate the reliability and stochastic properties of a coherent system where the component lifetimes of the system are identically distributed and the structural dependency of the components is expressed using a copula. We use the notion of distortion function to explore several ageing and stochastic properties of the residual and inactivity time of coherent systems and order statistics. Some illustrative examples are also provided. © 2016, Springer-Verlag Berlin Heidelberg.
Journal of Applied Probability (00219002)56(4)pp. 1151-1167
This paper presents a flexible family which we call the -mixture of survival functions. This family includes the survival mixture, failure rate mixture, models that are stochastically closer to each of these conventional mixtures, and many other models. The -mixture is endowed by the stochastic order and uniquely possesses a mathematical property known in economics as the constant elasticity of substitution, which provides an interpretation for α. We study failure rate properties of this family and establish closures under monotone failure rates of the mixture's components. Examples include potential applications for comparing systems. © Applied Probability Trust 2019.
Metron (00261424)76(1)pp. 115-131
In the present paper, we define a new measure of divergence between two probability distribution functions F1 and F2 based on Jensen inequality and Gini mean difference. The proposed measure, which we call it Jensen–Gini measure of divergence (JG), is symmetric and its square root is a metric. We show that the JG can be represented as a mixture of Cramér’s distance (CD) between the two distributions F1 and F2. A generalization of JG for measuring the overall difference between several probability distributions is also proposed. The proposed JG measure of divergence is applied to estimate the unknown parameters of a probability distribution. We consider a statistical model F(x; θ) , where the parameter θ∈ Θ is assumed to be unknown. Based on a random sample drawn from the distribution, we consider the JG between the distribution F(x; θ) and the empirical estimator of the distribution. Then, we estimate the parameter θ as a value in the parameter space Θ which minimizes the JG between the distribution F(x; θ) and its empirical estimator. We call this estimator as minimum Jensen–Gini estimator (MJGE) of the parameter. Several properties of MJGE are investigated. It is shown that the MJGE is in the class of generalized estimating equations. Asymptotic properties of MJGE such as consistency and normality are explored. Some simulation studies are performed to evaluate the performance of MJGE. © 2017, Sapienza Università di Roma.
Mathematics (22277390)6(10)
In this paper, we investigate the reliability and stochastic properties of an n-component network under the assumption that the components of the network fail according to a counting process called a geometric counting process (GCP). The paper has two parts. In the first part, we consider a two-state network (with states up and down) and we assume that its components are subjected to failure based on a GCP. Some mixture representations for the network reliability are obtained in terms of signature of the network and the reliability function of the arrival times of the GCP. Several aging and stochastic properties of the network are investigated. The reliabilities of two different networks subjected to the same or different GCPs are compared based on the stochastic order between their signature vectors. The residual lifetime of the network is also assessed where the components fail based on a GCP. The second part of the paper is concerned with three-state networks. We consider a network made up of n components which starts operating at time t = 0. It is assumed that, at any time t > 0, the network can be in one of three states up, partial performance or down. The components of the network are subjected to failure on the basis of a GCP, which leads to change of network states. Under these scenarios, we obtain several stochastic and dependency characteristics of the network lifetime. Some illustrative examples and plots are also provided throughout the article. © 2018 by the author.
Brazilian Journal of Probability and Statistics (01030752)32(4)pp. 795-814
Fisher information is a very important and fundamental criterion in statistical inference especially in optimal and large sample studies in estimation theory. It also plays a key role in physics, thermodynamic, information theory and other applications. In the literature there have been defined two forms of Fisher information: one for the parameters of a distribution function and one for the density function of a distribution. In this paper, we consider a nonnegative continuous random (lifetime) variable X and define a time-dependent Fisher information for density function of the residual random variable associated to X. We also propose a time-dependent version of Fisher information distance (relative Fisher information) between the densities of two nonnegative random variables. Several properties of the proposed measures and their relations to other statistical measures are investigated. To illustrate the results various examples are also provided. © Brazilian Statistical Association, 2018.
European Journal of Operational Research (03772217)270(2)pp. 723-733
The proportional hazards (PH), mixture hazards (MH), proportional reversed hazards (PRH), and mixture reversed hazards (MRH) models are widely used in various applications in many fields, but their optimality properties remain unknown. We represent these important reliability models as general escort models and derive them as solutions to several information theoretic formulations. Thus far, the escort models are defined in physics by the normalized powers of one or product of two probability mass or density functions and have been derived in statistics and physics as the solutions to different information formulations. The general escort models introduced in this paper include the escorts of densities, as well as the escorts of survival functions and cumulative distribution functions which represent the hazards models. Moreover, we show that the MH and MRH models are also optimal according to formulations in terms of the mean variation distance. Additional results explore reliability properties of the escort of two densities. A notable property is that the escort of two densities with non-constant hazard rates can be a constant hazard rate model. Another result characterizes the PH model in terms of the survival function of the escort of two densities. Comparisons of the MH, escort of two densities, and the mixture of two distributions are illustrated. © 2018 Elsevier B.V.
Journal of Applied Probability (00219002)55(3)pp. 845-861
In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples. Copyright © Applied Probability Trust 2018.
Metrika (1435926X)80(6-8)pp. 649-661
We propose a new measure of association between two continuous random variables X and Y based on the covariance between X and the log-odds rate associated to Y. The proposed index of correlation lies in the range [- 1 , 1]. We show that the extremes of the range, i.e., - 1 and 1, are attainable by the Fre ´ chet bivariate minimal and maximal distributions, respectively. It is also shown that if X and Y have bivariate normal distribution, the resulting measure of correlation equals the Pearson correlation coefficient ρ. Some interpretations and relationships to other variability measures are presented. Among others, it is shown that for non-negative random variables the proposed association measure can be represented in terms of the mean residual and mean inactivity functions. Some illustrative examples are also provided. © 2017, Springer-Verlag GmbH Germany.
Metrika (1435926X)80(2)pp. 227-241
The study on the inactivity times is useful in evaluating the aging and reliability properties of coherent systems in reliability engineering. In the present paper, we investigate the inactivity time of a coherent system consisting of n i.i.d. components. We drive some mixture representations for the reliability function of conditional inactivity times of coherent systems under two specific conditions on the status of the system components. Some ageing and stochastic properties of the proposed conditional inactivity times are also explored. © 2016, Springer-Verlag Berlin Heidelberg.
Journal of Applied Probability (00219002)54(4)pp. 1027-1050
The proportional hazards (PH) model and its associated distributions provide suitable media for exploring connections between the Gini coefficient, Fisher information, and Shannon entropy. The connecting threads are Bayes risks of the mean excess of a random variable with the PH distribution and Bayes risks of the Fisher information of the equilibrium distribution of the PH model. Under various priors, these Bayes risks are generalized entropy functionals of the survival functions of the baseline and PH models and the expected asymptotic age of the renewal process with the PH renewal time distribution. Bounds for a Bayes risk of the mean excess and the Gini's coefficient are given. The Shannon entropy integral of the equilibrium distribution of the PH model is represented in derivative forms. Several examples illustrate implementation of the results and provide insights for potential applications. Copyright © Applied Probability Trust 2017.
Journal of Applied Probability (00219002)54(4)pp. 1051-1070
In this paper we investigate the stochastic properties of the number of failed components of a three-state network. We consider a network made up of n components which is designed for a specific purpose according to the performance of its components. The network starts operating at time t = 0 and it is assumed that, at any time t > 0, it can be in one of states up, partial performance, or down. We further suppose that the state of the network is inspected at two time instants t 1 and t 2 (t 1 < t 2). Using the notion of the two-dimensional signature, the probability of the number of failed components of the network is calculated, at t 1 and t 2, under several scenarios about the states of the network. Stochastic and ageing properties of the proposed failure probabilities are studied under different conditions. We present some optimal age replacement policies to show applications of the proposed criteria. Several illustrative examples are also provided. Copyright © Applied Probability Trust 2017.
IEEE Transactions on Reliability (15581721)65(2)pp. 992-1000
We consider a network consisting of n components and assume that the network has two states up and down. We further suppose that the network is subject to shocks that appear according to a counting process and that each shock may lead to the component failures. Under some assumptions on the shock occurrences, we present a new variant of the notion of signature which we call t-signature. Then, t-signature-based mixture representations for the reliability function of the network are obtained. Several stochastic properties of the network lifetime are investigated. In particular, under the assumption that the number of failures at each shock follows a binomial distribution and the process of shocks is nonhomogeneous Poisson process, explicit form of the network reliability is derived and its aging properties are explored. Several examples are also provided. © 2015 IEEE.
Journal of Applied Statistical Science (10675817)22(1-2)pp. 53-74
The consecutive k-out-of-n systems are important types of coherent structures and have applications in various areas. There are consecutive k-out-of-n systems with the property that some of the components of the system remain unfailed at any time t. There are also consecutive k-out-of-n systems with the property that, at the time of the failure of the system, some components fail before the failure of the system. In the first part of the paper, we study the stochastic properties of residual lifetime of the live components of the system under the condition that the system is working at time t. In the second part, under the condition that the system is not working at time t, the inactivity time of the failed components of the system is investigated. © Nova Science Publishers, Inc.
Asadi, M.,
Ebrahimi, N.,
Soofi, E.S.,
Zohrevand, Y. Reliability Engineering and System Safety (18790836)156pp. 244-255
The signature of a coherent system with n components is an n-dimensional vector whose ith element is the probability that the ith failure of the components is fatal to the system. The signature depends only on the system design and provides useful tools for comparison of systems. We propose the Jensen–Shannon information (JS) criteria for comparison of systems, which is a scalar function of the signature and ranks systems based on their designs. The JS of a system is interpreted in terms of the remaining uncertainty about the system lifetime, the utility of dependence between the lifetime and the number of failures of components fatal to the system, and the Bayesian decision theory. The JS is non-negative and its minimum is attained by k-out-of-n systems, which are the least complex systems. This property offers JS as a measure of complexity of a system. Effects of expansion of a system on JS are studied. Application examples include comparisons of various sets of new systems and used but still working systems discussed in the literature. We also give an upper bound for the JS at the general level and compare it with a known upper bound. © 2016 Elsevier Ltd
Statistics (02331888)50(6)pp. 1421-1433
In survival analysis and reliability theory, a fundamental problem is the study of lifetime properties of a live organism or system. In this regard, there have been considered and studied several models based on different concepts of ageing such as hazard rate and mean residual life. In this paper, we consider an additive-multiplicative hazard model (AMHM) and study some reliability and ageing properties of the proposed model. We then specify the bivariate models whose conditionals satisfy AMHM. Several properties of the proposed bivariate model are investigated and adequacy of the model is evaluated based on a real data set. © 2016 Informa UK Limited, trading as Taylor & Francis Group.
Navarro j., J.,
Esna-ashari, M.,
Asadi, M.,
Sarabia j.m., Metrika (1435926X)78(6)pp. 691-709
New bivariate models are obtained with conditional distributions (in two different senses) satisfying the proportional generalized odds rate (PGOR) model. The PGORsemi-parametric model includes as particular cases theCox proportional hazard rate (PHR) model and the proportional odds rate (POR) model. Thus the new bivariate models are very flexible and include, as particular cases, the bivariate extensions of PHR and POR models. Moreover, some well known parametric bivariate models are also included in these general models. The basic theoretical properties of the new models are obtained. An application to fit a real data set is also provided. © Springer-Verlag Berlin Heidelberg 2015.
IEEE Transactions on Reliability (15581721)64(4)pp. 1276-1286
We consider a coherent system consisting of n components where the component lifetimes are assumed to be random variables following a probabilistically exchangeable joint distribution function, where the probability distribution does not change with any permutation of the components. We study stochastic properties of the residual lifetime of live components of the system, under different conditions. The number of failed components of the system are explored, under the assumption that the system is operating at time t. We also study the probability that the ith component failure in the system causes the system failure, given that the system is operating at time t. The results of the paper extend some of the existing results in the literature. © 1963-2012 IEEE.
Journal of Applied Probability (00219002)52(1)pp. 305-305
Metrika (1435926X)78(3)pp. 261-281
Suppose that a system has three states up, partial performance and down. We assume that for a random time T1 the system is in state up, then it moves to state partial performance for time T2 and then the system fails and goes to state down. We also denote the lifetime of the system by T, which is clearly T = T1+T2. In this paper, several stochastic comparisons are made between T, T1 and T2 and their reliability properties are also investigated. We prove, among other results, that different concepts of dependence between the elements of the signatures (which are structural properties of the system) are preserved by the lifetimes of the states of the system (which are aging properties of the system). Various illustrative examples are provided. © 2014, Springer-Verlag Berlin Heidelberg.
Sankhya B (09768394)77(1)pp. 141-164
Most of the research, on the study of the reliability properties of technical systems, assume that the components of the systems operate independently. However, in real life situation, it is more reasonable to assume that there is dependency among the components of the system. In this paper, we consider a (n−k+1)-out-of-n structure in which the component lifetimes are dependent random variables. We investigate stochastic properties of the inactivity time of the failed components of the system, extending some existing results in the literature where the components of the system are assumed to be independent and identically distributed. The results are then extended to the case where the system has an arbitrary coherent structure with exchangeable components. © 2014, Indian Statistical Institute.
Zarezadeh s., S.,
Asadi, M.,
Balakrishnan n., N. European Journal of Operational Research (03772217)232(3)pp. 561-571
In this paper, we consider a two-state (up and down) network consisting of n links. We study the D-spectrum based dynamic reliability of the network under the assumption that the links are subject to failure according to a nonhomogeneous Poisson process. Several mixture representations are provided for the reliability function of residual lifetime of used networks, under different conditions on the status of the network or its links. These representations enable us to explore the residual reliability of operating networks in terms of the reliability functions of residual lifetimes of upper record values. The distribution function of inactivity time of a network is examined under the condition that the network has failed by inspection time t. Stochastic ordering properties of the residual lifetimes of networks under conditional D-spectra are investigated. Several examples and graphs are also provided to illustrate the established results. © 2013 Elsevier B.V. All rights reserved.
Journal of Applied Probability (00219002)51(4)pp. 999-1020
This paper is an investigation into the reliability and stochastic properties of three-state networks. We consider a single-step network consisting of n links and we assume that the links are subject to failure. We assume that the network can be in three states, up (K = 2), partial performance (K = 1), and down (K = 0). Using the concept of the two-dimensional signature, we study the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network. In the process of doing so, we define variants of the concept of the dynamic signature in a bivariate setting. Then, we obtain signature based mixture representations of the reliability of the residual lifetimes of the network states under the condition that the network is in state K = 2 (or K = 1) and exactly k links in the network have failed. We prove preservation theorems showing that stochastic orderings and dependence between the elements of the dynamic signatures (which relies on the network structure) are preserved by the residual lifetimes of the states of the network (which relies on the network ageing). Various illustrative examples are also provided. © Applied Probability Trust 2014.
Asadi, M.,
Ebrahimi, N.,
Soofi, E.S.,
Zarezadeh s., S. Naval Research Logistics (15206750)61(6)pp. 427-434
This article introduces two new maximum entropy (ME) methods for modeling the distribution of time to an event. One method is within the classical ME framework and provides characterizations of change point models such as the piecewise exponential distribution. The second method uses the entropy of the equilibrium distribution (ED) for the objective function and provides new characterizations of the exponential, Weibull, Pareto, and uniform distributions. With the same moment constraints, the classical ME and the maximum ED entropy algorithms generate different models for the interarrival time. © 2014 Wiley Periodicals, Inc.
Communications in Statistics - Theory and Methods (1532415X)43(10-12)pp. 2468-2475
In analyzing the lifetime properties of a coherent system, the concept of "signature" is a useful tool. Let T be the lifetime of a coherent system having n iid components. The signature of the system is a probability vector s=(s1, s2,., sn), such that s i=P(T=Xi:n), where, Xi:n, i=1, 2,., n denote the ordered lifetimes of the components. In this note, we assume that the system is working at time t>0. We consider the conditional signature of the system as a vector in which the ith element is defined as pi(t)=P(T=Xi: n|T>t) and investigate its properties as a function of time. © 2014 Taylor & Francis Group, LLC.
Studia Scientiarum Mathematicarum Hungarica (15882896)51(2)pp. 243-270
In recent years, slash and skew slash distributions have been employed, as flexible models, in various fields. In this paper, we study several properties of these distributions in both univariate and multivariate cases. Some recurrence relations for the probability density functions are derived and the behavior of reliability measures, such as hazard rate and mean residual life, associated to these distributions are investigated.
Methodology and Computing in Applied Probability (13875841)16(3)pp. 675-691
Some coherent systems are such that failure of the system does not mean that all components fail. This paper investigates the stochastic behavior and reliability properties of the residual lifetime of live components in coherent systems under the assumption that the system fails at time t. We also investigate the stochastic properties of inactivity time of failed components in coherent systems where failure of some components does not cause the failure of the complete system. © 2013 Springer Science+Business Media New York.
IEEE Transactions on Reliability (15581721)62(4)pp. 917-929
The concept of D-spectrum is a useful tool to investigate the reliability and stochastic properties of networks. In this paper, we consider a network consisting of n components (links or nodes), and assume that the network has two states: up, and down. We study the D-spectrum-based reliability of the network under the assumption that the components are subject to failure according to a counting process. Mixture representation of the reliability of the network lifetime is given in terms of the reliability functions of arrival times. It is shown that, when the D-spectra of two networks are stochastically ordered, then under some conditions the lifetimes of two networks are also stochastically ordered. Under the special case when the process of the failure of the components is a nonhomogeneous Poisson process, we arrive at a mixture representation for the reliability function of the network lifetime, and explore several stochastic and aging properties of the network lifetime under different scenarios. The failure rate of the network lifetime and its asymptotic behavior is investigated. D-spectrum based representation theorems are also given on the basis of the stochastic precedence concept. © 1963-2012 IEEE.
Metrika (1435926X)76(7)pp. 979-996
In this note, we consider a coherent system with the property that, upon failure of the system, some of its components remain unfailed in the system. Under this condition, we study the residual lifetime of the live components of the system. Signature based mixture representation of the joint and marginal reliability functions of the live components are obtained. Various stochastic and aging properties of the residual lifetime of such components are investigated. Some characterization results on exponential distributions are also provided. © 2012 Springer-Verlag Berlin Heidelberg.
IEEE Transactions on Reliability (15581721)61(1)pp. 41-49
The concept of the signature of a coherent system is useful to study the stochastic and aging properties of the system. Let X 1:n,X 2:n,⋯,X n:n denote the ordered lifetimes of the components of a coherent system consisting of n i.i.d components. If T denotes the lifetime of the system, then the signature vector of the system is defined to be a probability vector s = (s 1,s 2,⋯,s n) such that s i = P(T = X i:n), i=1,2,⋯,n. Here we consider a coherent system with signature of the form s=(s 1,s 2,⋯s i,0⋯,0), where s k>0, k=1,2,⋯,i. Under the condition that the system is working at time t , we propose a time dependent measure to calculate the probability of residual life of live components of the system, i.e., X k:n, k=i+1,⋯,n. Several stochastic and aging properties of the proposed measure are explored. © 2006 IEEE.
Communications in Statistics - Theory and Methods (1532415X)41(6)pp. 983-999
This article considers the properties of a nonparametric estimator developed for a reliability function which is used in many reliability problems. Properties such as asymptotic unbiasedness and consistency are proven for the estimator and using U-statistics, weak convergence of the estimator to a normal distribution is shown. Finally, numerical examples based on an extensive simulation study are presented to illustrate the theory and compare the estimator developed in this article with another based directly on the ratio of two empirical distributions studied in Zardasht and Asadi (2010). Copyright © Taylor & Francis Group, LLC.
Test (11330686)21(1)pp. 93-115
In recent years, consecutive systems were shown to have many applications in various branches of science such as engineering. This paper is a study on the stochastic and aging properties of residual lifetime of consecutive k-out-of-n systems under the condition that n - r + 1, r ≤ n, components of the system are working at time t. We consider the linear and circular consecutive k-out-of-n systems and propose a mean residual lifetime (MRL) for such systems. Several properties of the proposed MRL is investigated. The mixture representation of the MRL of the systems with respect to the vector of signatures of the system is also studied. © 2011 Sociedad de Estadística e Investigación Operativa.
Statistics and Probability Letters (01677152)82(12)pp. 2156-2163
In this paper, we investigate the number of failed components in a coherent system. We consider an (n- m+ 1)-out-of- n system and we compute the probability that there are exactly i failures, i=0, ... , m-1, in the system under the condition that it is operating at time t. Several properties of the proposed time dependent probabilities are studied. The results are then extended to coherent systems consisting of n components with an arbitrary signature vector. © 2012 Elsevier B.V..
Goliforushani s., S.,
Asadi, M.,
Balakrishnan n., N. Journal of Applied Probability (00219002)49(2)pp. 385-404
In the study of the reliability of technical systems in reliability engineering, coherent systems play a key role. In this paper we consider a coherent system consisting of n components with independent and identically distributed components and propose two time-dependent criteria. The first criterion is a measure of the residual lifetime of live components of a coherent system having some of the components alive when the system fails at time t . The second criterion is a time-dependent measure which enables us to investigate the inactivity times of the failed components of a coherent system still functioning though some of its components have failed. Several ageing and stochastic properties of the proposed measures are then established. © Applied Probability Trust 2012.
Statistics and Probability Letters (01677152)82(3)pp. 574-585
To study the ageing and stochastic properties of lifetime random variables, several classes of lifetime distributions have been defined in reliability literature. In this context, an interesting problem is then to investigate the closure properties of such classes under different operators. In this paper, we study the preservation properties of classes of increasing mean inactivity time (IMIT), increasing variance inactivity time (IVIT), and some other recently introduced classes, such as NRBU, NRBUE, and NBRUrh, under a Poisson process stopped at a random time. We also analyze the preservation properties of Poisson shock models for the above-mentioned ageing classes. © 2011 Elsevier B.V.
Statistics (02331888)46(3)pp. 405-417
Let T be a lifetime random variable. In order to study the properties of T in reliability theory and survival analysis, several measures are proposed in the literature. Among these measures, hazard rate, mean residual lifetime, reversed hazard rate and the mean past lifetime (MPL) play important roles. In the present paper, we focus mainly on the MPL. We investigate its properties in connection with other reliability measures. Some results on partial ordering and characterization are also given. Finally, we deal with its statistical estimation. © 2012 Copyright Taylor and Francis Group, LLC.
Metrika (1435926X)75(4)pp. 439-454
We consider a (n - k + 1)-out-of-n system with independent and nonidentical components. Under the condition that at time t the system has failed we study the past lifetime of the components of the system. The mean past lifetime of the components is defined and some of its properties are investigated. Stochastic comparisons are also made between the past lifetime of different systems. © 2010 Springer-Verlag.
Metrika (1435926X)75(7)pp. 997-1007
In the present study we extend and unify some existing results in the literature on characterization of the generalized Pareto distributions based on generalized order statistics. © 2011 Springer-Verlag.
Probability in the Engineering and Informational Sciences (02699648)25(2)pp. 187-204
The generalized order statistics (GOS) model is a unified model that contains the well-known ordered random data such as order statistics and record values. In the present article, we investigate some stochastic ordering results and aging properties of the conditional GOS. The results of the article subsume some of the existing results, which recently are obtained in the literature, on conditional GOS. In particular, our results hold for the model of progressively type II right censored order statistics without any restriction on the censoring scheme. © 2011 Cambridge University Press.
Journal of Statistical Planning and Inference (03783758)141(8)pp. 2920-2932
In recent years, the study of reliability properties of consecutive k-out-of-n systems has attracted a great deal of attention from both theoretical and practical perspectives. In this paper we consider linear and circular consecutive k-out-of-n systems. It is assumed that lifetimes of components of the systems are independent but their probability distributions are non-identical. We study the reliability properties of the residual lifetimes of such systems under the condition that at least (n-r+1), r≤n, components of the system are operating. We also investigate the probability that a specific number of components of the above-mentioned system operate at time t, t>0, under the condition that the system is alive at time t. © 2011 Elsevier B.V.
Statistics (02331888)45(3)pp. 237-255
In this paper, we propose a measure for obtaining the expectation of time between two lower k-records under the condition that the greater one is given. Several properties of the proposed measure are derived. Some characterization results and stochastic comparisons based on the new measure are also provided. © 2011 Taylor & Francis.
Sankhya: The Indian Journal of Statistics (09727671)73(2)pp. 241-262
The concept of "signature" is a useful tool to study the reliability properties of a coherent system. In this paper, we consider a coherent system consisting of n components and assume that the system is not working at time t. Mixture representations of the inactivity times (IT) of the system and IT of the components of the system are obtained under different scenarios on the signatures of the system. Some stochastic comparisons are made on IT of the coherent systems with same type and different type of components and some aging properties of the IT of the system and its components are investigated. It is proved, under some conditions on the vector of signatures of the system, that when the components of the system have decreasing reversed hazard rate, the mean of the IT (MIT) of the system and the MIT of the components of the system are increasing in time. Several examples and illustrative graphs are also provided. © Indian Statistical Institute 2012.
IEEE Transactions on Reliability (15581721)60(4)pp. 817-822
The concept of signature is a useful tool to study the stochastic and aging properties of coherent systems. We consider a coherent system, and assume that there is some partial information about the failure status of the system lifetime. We study various properties of the conditional signature. © 2011 IEEE.
Metrika (1435926X)72(1)pp. 59-73
In the present paper, we consider a (n - k + 1)-out-of-n system with identical components where it is assumed that the lifetimes of the components are independent and have a common distribution function F. We assume that the system fails at time t or sometime before t, t > 0. Under these conditions, we are interested in the study of the mean time elapsed since the failure of the components. We call this as the mean past lifetime (MPL) of the components at the system level. Several properties of the MPL are studied. It is proved that the relation between the proposed MPL and the underlying distribution is one-to-one. We have shown that when the components of the system have decreasing reversed hazard then the MPL of the system is increasing with respect to time. Some examples are also provided. © Springer-Verlag 2009.
Statistica Neerlandica (00390402)64(4)pp. 460-481
Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of them are Xt=X-t | X>t and Yt=Y-t | Y>t, respectively. In this paper, we introduce, by taking into account the age of systems, a time-dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t):=P(Xt>Yt) which enables one to obtain, at time t, the probability that the residual lifetime Xt is greater than the residual lifetime Yt. It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability, 22, 1973) that the probability of type R(t) is important for designing as long-lived a product as possible. Several properties of R(t) and its connection with well-known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed. © 2010 The Authors. Statistica Neerlandica © 2010 VVS.
Probability in the Engineering and Informational Sciences (02699648)24(4)pp. 561-584
This article develops information optimal models for the joint distribution based on partial information about the survival function or hazard gradient in terms of inequalities. In the class of all distributions that satisfy the partial information, the optimal model is characterized by well-known information criteria. General results relate these information criteria with the upper orthant and the hazard gradient orderings. Applications include information characterizations of the bivariate Farlie-Gumbel-Morgenstern, bivariate Gumbel, and bivariate generalized Gumbel, for which no other information characterization are available. The generalized bivariate Gumbel model is obtained from partial information about the survival function and hazard gradient in terms of marginal hazard rates. Other examples include dynamic information characterizations of the bivariate Lomax and generalized bivariate Gumbel models having marginals that are transformations of exponential such as Pareto, Weibull, and extreme value. Mixtures of bivariate Gumbel and generalized Gumbel are obtained from partial information given in terms of mixtures of the marginal hazard rates. Copyright © 2010 Cambridge University Press.
Information Sciences (00200255)180(21)pp. 4195-4206
This paper explores properties of the residual Rényi entropy of some ordered random variables. The residual Rényi entropy of the kth order statistic from a continuous distribution function is represented in terms of the residual Rényi entropy of the kth order statistic from uniform distribution. The monotone behavior of the residual Rényi entropy of order statistic under various conditions is discussed. Analogues results for the residual Rényi entropy of record values are also given. © 2010 Published by Elsevier Inc.
Bulletin Of The Iranian Mathematical Society (1017060X)36(1)pp. 257-272
It is well-known that most of the characterization results on exponential distribution are based on the solution of Cauchy functional equation and integrated Cauchy functional equation. Here, we consider the functional equation F(x) = F(xy) + F(xQ(y)), x, xQ(y) ∈ [0, θ), y ∈ [0, 1], where F and Q satisfy certain conditions, to give some new characterization results on exponential, power and Pareto distributions using the concepts of conditional random variables and order statistics. © 2010 Iranian Mathematical Society.
Journal of Statistical Planning and Inference (03783758)140(1)pp. 310-322
The residual entropy function is a relevant dynamic measure of uncertainty in reliability and survival studies. Recently, Rao et al. [2004. Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50, 1220-1228] and Asadi and Zohrevand [2007. On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137, 1931-1941] define the cumulative residual entropy and the dynamic cumulative residual entropy, respectively, as some new measures of uncertainty. They study some properties and applications of these measures showing how the cumulative residual entropy and the dynamic cumulative residual entropy are connected with the mean residual life function. In this paper, we obtain some new results on these functions. We also define and study the dynamic cumulative past entropy function. Some results are given connecting these measures of a lifetime distribution and that of the associated weighted distribution. © 2009 Elsevier B.V. All rights reserved.
Statistics and Probability Letters (01677152)80(9-10)pp. 848-859
In the present paper we propose a new concept of the residual lifetime of progressively Type-II right censored order statistics (PCOS) and study some of its properties. Some new stochastic comparisons are made on the residual lifetimes of PCOS. The case when the behavior of the data generator F, in terms of different aging concepts, is given the behavior of the residual lifetimes of PCOS is investigated. We also concentrate on the mean of the residual lifetime of the PCOS. Some characterizations of the generalized Pareto distribution based on the mean of the residual lifetime of PCOS are also presented. © 2010 Elsevier B.V. All rights reserved.
Statistics (02331888)44(5)pp. 493-504
Often, in reliability theory, risk analysis, renewal processes and actuarial studies, mean residual life function or life expectancy plays an important role in studying the conditional tail measure of lifetime data. In this paper, we introduce the notion of the mean residual waiting time of records and present some monotonic and aging properties. Sharp bounds for the mean residual waiting time of records are also investigated. © 2010 Taylor & Francis.
Metrika (1435926X)72(2)pp. 251-264
In the study of reliability of the technical systems, records model plays an important role. Suppose that a technical system is subject to shocks, e. g. peaks of voltages or stresses. The successive large shocks may be viewed as realizations of records from a sequence of identically independent voltages. Assume that the lower limit value of the mth record voltage (stress) is v > 0. Under these conditions, we propose a mean residual of records (MRR's) for the future records. We study several properties of MRR. We show that the proposed MRR uniquely characterizes the distribution function that generated the sequence of records. It is proved that when the model under study has an increasing hazard rate the corresponding MRR is decreasing. We also compare between two record systems based on their MRR's when these systems are ordered in terms of their hazard rates. © 2009 Springer-Verlag.
ANZIAM Journal (14461811)51(SUPPL.)
We consider systems having many independent components connected in a parallel or series configuration with non-identical failure distributions. A time dependent measure is proposed which evaluate the probability that a failure of the system occurred at a specified component under the condition that the system has failed by some time. We proved several properties of this measure for parallel and serial systems. © Austral. Mathematical Soc. 2010.
Communications in Statistics - Theory and Methods (1532415X)37(9)pp. 1347-1352
In recent years, several attempts have been made to characterize the generalized Pareto distribution (GPD) based on the properties of order statistics and record values. In the present article, we give a characterization result on GPD based on the spacing of generalized order statistics.
Metrika (1435926X)68(2)pp. 209-217
Let T denote a positive discrete survival time and n a non-negative integer number. Properties of the mean past lifetime E(n - T|T < n) are provided. © 2007 Springer-Verlag.
IEEE Transactions on Reliability (15581721)57(4)pp. 574-580
We consider a coherent structure consisting of n components having the property that if it is known that at most r components (r[removed]
Journal of Statistical Planning and Inference (03783758)138(12)pp. 3660-3666
Suppose that a technical system is subject to shocks, e.g. peaks of voltages from a sequence of identically independent voltages having a lower limit value v > 0. We propose a new definition for the mean residual life of the records of the sequence and study its various properties. © 2008 Elsevier B.V. All rights reserved.
Journal of Statistical Planning and Inference (03783758)137(6)pp. 1931-1941
Recently, Rao et al. [(2004) Cumulative residual entropy: a new measure of information. IEEE Trans. Inform. Theory 50(6), 1220-1228] have proposed a new measure of uncertainty, called cumulative residual entropy (CRE), in a distribution function F and obtained some properties and applications of that. In the present paper, we propose a dynamic form of CRE and obtain some of its properties. We show how CRE (and its dynamic version) is connected with well-known reliability measures such as the mean residual life time. © 2006 Elsevier B.V. All rights reserved.
Journal of Statistical Planning and Inference (03783758)136(4)pp. 1197-1206
In the study of reliability of the technical systems and subsystems, parallel systems play a very important role. In the present paper, we consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F. It is assumed that at time t the system has failed. Under these conditions, we obtain the mean past lifetime (MPL) of the components of the system. Some properties of MPL are studied. It is shown that the underlying distribution function F can be recovered from the proposed MPL. Also, a comparison between two parallel systems are made based on their MPLs in the case where the components of the system are ordered in terms of reversed hazard rate. Finally a characterization of the uniform distribution is given based on MPL. © 2004 Elsevier B.V. All rights reserved.
IEEE Transactions on Reliability (15581721)55(2)pp. 314-318
In the study of the reliability of technical systems, k-out-of-n systems play an important role. In the present paper, we consider k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that, at time t, all the components of the system are working, we propose a new definition for the mean residual life (MRL) function of the system, and obtain several properties of that system. © 2006 IEEE.
Communications in Statistics - Theory and Methods (1532415X)34(2)pp. 475-484
One of the most important types of system structures is the parallel structure. In the present article, we propose a definition for the mean residual life function of a parallel system and obtain some of its properties. The proposed definition measures the mean residual life function of a parallel system consisting of n identical and independent components wider the condition that n -i, i = 0, 2,..., n -1, components of the system are working and other components of the system have already failed. It is shown that, for the case where the components of the system have increasing hazard rate, the mean residual life function of the system is a nonincreasing function of time. Finally, we will obtain an upper bound for the proposed mean residual life function. Copyright © Taylor & Francis, Inc.
Asadi, M.,
Ebrahimi, N.,
Hamedani, G.,
Soofi, E.S. Journal of Applied Probability (00219002)41(2)pp. 379-390
A formal approach to produce a model for the data-generating distribution based on partial knowledge is the well-known maximum entropy method. In this approach, partial knowledge about the data-generating distribution is formulated in terms of some information constraints and the model is obtained by maximizing the Shannon entropy under these constraints. Frequently, in reliability analysis the problem of interest is the lifetime beyond an age t. In such cases, the distribution of interest for computing uncertainty and information is the residual distribution. The information functions involving a residual life distribution depend on t, and hence are dynamic. The maximum dynamic entropy (MDE) model is the distribution with the density that maximizes the dynamic entropy for all t. We provide a result that relates the orderings of dynamic entropy and the hazard function for distributions with monotone densities. Applications include dynamic entropy ordering within some parametric families of distributions, orderings of distributions of lifetimes of systems and their components connected in series and parallel, record values, and formulation of constraints for the MDE model in terms of the evolution paths of the hazard function and mean residual lifetime function. In particular, we identify classes of distributions in which some well-known distributions, including the mixture of two exponential distributions and the mixture of two Pareto distributions, are the MDE models.
Handbook of Statistics (01697161)20pp. 199-214
Statistics and Probability Letters (01677152)49(3)pp. 263-269
A direct approach to measure uncertainty in the residual life time distribution has been initiated by Ebrahimi (1996, Sankhya Ser. A 58, 48-57) and explored further by Ebrahimi and Pellerey (1995) and Ebrahimi and Kirmani (1996). In this paper, some new properties of the proposed measure in connection to order statistics and record values are derived. The generalized Pareto distribution has been widely used in the literature. We have also given several characterizations of this distribution in terms of the proposed measure.
Journal of Statistical Planning and Inference (03783758)81(2)pp. 201-207
Many characterization results of the bivariate exponential distribution and the bivariate geometric distribution have been proved in the literature. Recently Nair and Nair (1988b, Ann. Inst. Statist. Math. 40 (2), 267-271) obtained a characterization result of the Gumbel bivariate exponential distribution and a bivariate geometric distribution based on truncated moments. In this note, we extend the results of to obtain a general result, characterizing these two bivariate distributions based on the truncated expectation of a function h, satisfying some mild conditions.
Metrika (1435926X)49(2)pp. 121-126
In this paper, we characterize some multivariate distributions based on a relationship between the multivariate hazard rate, as defined by Johnson and Kotz (1975) and Marshall (1975), and the multivariate mean residual life as defined by Arnold and Zahedi (1988). The results are extensions of the results obtained earlier by Roy (1989, 1990) and Ma (1996, 1997).
Statistical Papers (09325026)39(4)pp. 347-360
Let X be a random variable and X(w) be a weighted random variable corresponding to X. In this paper, we intend to characterize the Pearson system of distributions by a relationship between reliability measures of X and X(w), for some weight function w>0.
Journal of Multivariate Analysis (0047259X)67(2)pp. 190-202
Recently attempts have been made to characterize probability distributions via truncated expectations in both univariate and multivariate cases. In this paper we will use a well known theorem of Lau and Rao (1982) to obtain some characterization results, based on the truncated expectations of a functionh, for the bivariate Gumbel distribution, a bivariate Lomax distribution, and a bivariate power distribution. The results of the paper subsume some earlier results appearing in the literature. © 1998 Academic Press.
