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Journal articles on the topic 'Mean residual lifetime'

Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Alshehri, Mashael A., and Mohamed Kayid. "Mean Residual Lifetime Frailty Models: A Weighted Perspective." Mathematical Problems in Engineering 2021 (December 13, 2021): 1–21. http://dx.doi.org/10.1155/2021/3974858.

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The mean residual life frailty model and a subsequent weighted multiplicative mean residual life model that requires weighted multiplicative mean residual lives are considered. The expression and the shape of a mean residual life for some semiparametric models and also for a multiplicative degradation model are given in separate examples. The frailty model represents the lifetime of the population in which the random parameter combines the effects of the subpopulations. We show that for some regular dependencies of the population lifetime on the random parameter, some aging properties of the subpopulations’ lifetimes are preserved for the population lifetime. We indicate that the weighted multiplicative mean residual life model generates positive dependencies of this type. The copula function associated with the model is also derived. Necessary and sufficient conditions for certain aging properties of population lifetimes in the model are determined. Preservation of stochastic orders of two random parameters for the resulting population lifetimes in the model is acquired.

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2

Tang, L. C., Y. Lu, and E. P. Chew. "Mean residual life of lifetime distributions." IEEE Transactions on Reliability 48, no. 1 (March 1999): 73–78. http://dx.doi.org/10.1109/24.765930.

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3

Jupp, P. E. "Characterization of matrix probability distributions by mean residual lifetime." Mathematical Proceedings of the Cambridge Philosophical Society 100, no. 3 (November 1986): 583–89. http://dx.doi.org/10.1017/s0305004100066305.

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The mean residual lifetime of a real-valued random variable X is the function e defined byOne of the more important properties of the mean residual lifetime function is that it determines the distribution of X. See, for example, Swartz [10]. References to related characterizations are given by Galambos and Kotz [3], pages 30–35. It was established by Jupp and Mardia[6] that this property holds also for vector-valued X. As (1·1) makes sense if X is a random symmetric matrix, it is natural to ask whether the property holds in this case also. The purpose of this note is to show that, under certain regularity conditions, the distributions of such matrices are indeed determined by their mean residual lifetimes.

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4

Kayid, M. "Characterizations of the Weak Bivariate Failure Rate Order and Bivariate IFR Aging Class." Journal of Mathematics 2022 (April 30, 2022): 1–9. http://dx.doi.org/10.1155/2022/2573667.

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In this paper, two characterizations of the weak bivariate failure rate order over the bivariate Laplace transform order of two-dimensional residual lifetimes are given. The results are applied to characterize the weak bivariate failure rate ordering of random pairs by the weak bivariate mean residual lifetime ordering of the minima of pairs with exponentially distributed random pairs with unspecified mean. Moreover, a well-known bivariate aging term, namely, the bivariate increasing failure rate, is characterized by the weaker bivariate decreasing mean residual lifetime property of a random pair of minima.

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5

Ebrahimi, Nader. "Estimation of Two Ordered Mean Residual Lifetime Functions." Biometrics 49, no. 2 (June 1993): 409. http://dx.doi.org/10.2307/2532554.

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6

Finkelstein, Maxim. "On relative ordering of mean residual lifetime functions." Statistics & Probability Letters 76, no. 9 (May 2006): 939–44. http://dx.doi.org/10.1016/j.spl.2005.10.027.

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7

Triantafyllou, Ioannis S. "On the Lifetime and Signature of Constrained (k, d)-out-of-n: F Reliability Systems." International Journal of Mathematical, Engineering and Management Sciences 6, no. 1 (October 29, 2020): 66–78. http://dx.doi.org/10.33889/ijmems.2021.6.1.006.

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In the present paper we carry out a reliability study of the constrained (k, d)-out-of-n: F systems with exchangeable components. The signature vector is computed by the aid of the proposed algorithm. In addition, explicit signature-based expressions for the corresponding mean residual lifetime and the conditional mean residual lifetime of the aforementioned reliability system are also provided. For illustration purposes, a well-known multivariate distribution for modelling the lifetimes of the components of the constrained (k, d)-out-of-n: F structure is considered.

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8

Finkelstein, M. S. "On the shape of the mean residual lifetime function." Applied Stochastic Models in Business and Industry 18, no. 2 (2002): 135–46. http://dx.doi.org/10.1002/asmb.461.

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9

Navarro, Jorge, and Serkan Eryilmaz. "Mean Residual Lifetimes of Consecutive-k-out-of-n Systems." Journal of Applied Probability 44, no. 1 (March 2007): 82–98. http://dx.doi.org/10.1239/jap/1175267165.

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In this paper we study reliability properties of consecutive-k-out-of-n systems with exchangeable components. For 2k ≦ n, we show that the reliability functions of these systems can be written as negative mixtures (i.e. mixtures with some negative weights) of two series (or parallel) systems. Some monotonicity and asymptotic properties for the mean residual lifetime function are obtained and some ordering properties between these systems are established. We prove that, under some assumptions, the mean residual lifetime function of the consecutive-k-out-of-n: G system (i.e. a system that functions if and only if at least k consecutive components function) is asymptotically equivalent to that of a series system with k components. When the components are independent and identically distributed, we show that consecutive-k-out-of-n systems are ordered in the likelihood ratio order and, hence, in the mean residual lifetime order, for 2k ≦ n. However, we show that this is not necessarily true when the components are dependent.

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10

Navarro, Jorge, and Serkan Eryilmaz. "Mean Residual Lifetimes of Consecutive-k-out-of-n Systems." Journal of Applied Probability 44, no. 01 (March 2007): 82–98. http://dx.doi.org/10.1017/s0021900200002734.

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In this paper we study reliability properties of consecutive-k-out-of-n systems with exchangeable components. For 2k ≦ n, we show that the reliability functions of these systems can be written as negative mixtures (i.e. mixtures with some negative weights) of two series (or parallel) systems. Some monotonicity and asymptotic properties for the mean residual lifetime function are obtained and some ordering properties between these systems are established. We prove that, under some assumptions, the mean residual lifetime function of the consecutive-k-out-of-n: G system (i.e. a system that functions if and only if at least k consecutive components function) is asymptotically equivalent to that of a series system with k components. When the components are independent and identically distributed, we show that consecutive-k-out-of-n systems are ordered in the likelihood ratio order and, hence, in the mean residual lifetime order, for 2k ≦ n. However, we show that this is not necessarily true when the components are dependent.

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11

Eryilmaz, Serkan. "Mean Residual and Mean Past Lifetime of Multi-State Systems With Identical Components." IEEE Transactions on Reliability 59, no. 4 (December 2010): 644–49. http://dx.doi.org/10.1109/tr.2010.2054173.

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12

Economou, Polychronis, Georgios Psarrakos, and Abdolsaeed Toomaj. "Recruitment and survival time under mean residual lifetime bias sampling." Statistics 54, no. 4 (July 3, 2020): 885–907. http://dx.doi.org/10.1080/02331888.2020.1804905.

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13

Zhou, Mai, and Jong-Hyeon Jeong. "Empirical likelihood ratio test for median and mean residual lifetime." Statistics in Medicine 30, no. 2 (November 5, 2010): 152–59. http://dx.doi.org/10.1002/sim.4110.

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14

KUMAR, VIKAS, and H. C. TANEJA. "A GENERALIZED ENTROPY-BASED RESIDUAL LIFETIME DISTRIBUTIONS." International Journal of Biomathematics 04, no. 02 (June 2011): 171–84. http://dx.doi.org/10.1142/s1793524511001416.

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The present communication considers Havrda and Charvat entropy measure to propose a generalized dynamic information measure. It is shown that the proposed measure determines the survival function uniquely. The residual lifetime distributions have been characterized. A bound for the dynamic entropy measure in terms of mean residual life function has been derived, and its monotonicity property is studied.

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15

Iscioglu, Funda. "A new approach in the mean residual lifetime evaluation of a multi-state system." Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 235, no. 4 (March 29, 2021): 700–710. http://dx.doi.org/10.1177/1748006x20969573.

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Dynamic reliability measures are important characteristics for understanding the lifetime behaviour of multi-state systems. This study aims to analyze the mean residual lifetime (MRL) function for a one-unit three-state system.The system considered has just three states such as “0, 1, 2”. State “2” and “1” signify the perfect functioning and partially working states, respectively while state “0” is the failure state. We let [Formula: see text] and [Formula: see text] be the lifetimes of the system spent at state “1” and “2,” respectively. We assess a particular performance characteristic, MRL, of this three-state system via conditional survival functions, when [Formula: see text] and [Formula: see text] are both independent and dependent. In case of independency, MRL functions are obtained and the results are discussed with reference to a case study. The effect of different distributions on the MRLs are also investigated in this case study. In case of dependency, on the other hand, the effect of dependency on MRL functions is especially underlined given that the system is at state “ j” ([Formula: see text]) for [Formula: see text]. To understand the time dependent behaviour of the related MRL functions, certain graphical illustrations are also presented.

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16

Lee, Cheol-Eung. "Simplified Method for Estimation of Mean Residual Life of Rubble-mound Breakwaters." Journal of Korean Society of Coastal and Ocean Engineers 34, no. 2 (April 28, 2022): 37–45. http://dx.doi.org/10.9765/kscoe.2022.34.2.37.

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A simplified model using the lifetime distribution has been presented to estimate the Mean Residual Life (MRL) of rubble-mound breakwaters, which is not like a stochastic process model based on time-dependent history data to the cumulative damage progress of rubble-mound breakwaters. The parameters involved in the lifetime distribution can be easily estimated by using the upper and lower limits of lifetime and their likelihood that made a judgement by several experts taking account of the initial design lifetime, the past sequences of loads, and others. The simplified model presented in this paper has been applied to the rubble-mound breakwater with TTP armor layer. Wiener Process (WP)-based stochastic model also has been applied together with Monte-Carlo Simulation (MCS) technique to the breakwater of the same condition having time-dependent cumulative damage to TTP armor layer. From the comparison of lifetime distribution obtained from each models including Mean Time To Failure (MTTF), it has found that the lifetime distributions of rubble-mound breakwater can be very satisfactorily fitted by log-normal distribution for all types of cumulative damage progresses, such as exponential, linear, and logarithmic deterioration which are feasible in the real situations. Finally, the MRL of rubble-mound breakwaters estimated by the simplified model presented in this paper have been compared with those by WP stochastic process. It can be shown that results of the presented simplified model have been identical with those of WP stochastic process until any ages in the range of MTT F regardless of the deterioration types. However, a little of differences have been seen at the ages in the neighborhood of MTTF, specially, for the linear and logarithmic deterioration of cumulative damages. For the accurate estimation of MRL of harbor structures, it may be desirable that the stochastic processes should be used to consider properly time-dependent uncertainties of damage deterioration. Nevertheless, the simplified model presented in this paper can be useful in the building of the MRL-based preventive maintenance planning for several kinds of harbor structures, because of which is not needed timedependent history data about the damage deterioration of structures as mentioned above.

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17

Mahdy, Mervat. "Probabilistic properties of discrete mean and variance reversed residual lifetime functions." Applied Mathematical Sciences 7 (2013): 6167–79. http://dx.doi.org/10.12988/ams.2013.37400.

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18

Raqab, Mohammad Z. "Evaluations of the mean residual lifetime of an -out-of- system." Statistics & Probability Letters 80, no. 5-6 (March 2010): 333–42. http://dx.doi.org/10.1016/j.spl.2009.11.007.

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19

Misagh, Farsam. "On Shift-Dependent Cumulative Entropy Measures." International Journal of Mathematics and Mathematical Sciences 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/7213285.

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Measures of cumulative residual entropy (CRE) and cumulative entropy (CE) about predictability of failure time of a system have been introduced in the studies of reliability and life testing. In this paper, cumulative distribution and survival function are used to develop weighted forms of CRE and CE. These new measures are denominated as weighted cumulative residual entropy (WCRE) and weighted cumulative entropy (WCE) and the connections of these new measures with hazard and reversed hazard rates are assessed. These information-theoretic uncertainty measures are shift-dependent and various properties of these measures are studied, including their connections with CRE, CE, mean residual lifetime, and mean inactivity time. The notions of weighted mean residual lifetime (WMRL) and weighted mean inactivity time (WMIT) are defined. The connections of weighted cumulative uncertainties with WMRL and WMIT are used to calculate the cumulative entropies of some well-known distributions. The joint versions of WCE and WCRE are defined which have the additive properties similar to those of Shannon entropy for two independent random lifetimes. The upper boundaries of newly introduced measures and the effect of linear transformations on them are considered. Finally, empirical WCRE and WCE are proposed by virtue of sample mean, sample variance, and order statistics to estimate the new measures of uncertainty. The consistency of these estimators is studied under specific choices of distributions.

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20

Badía, Francisco Germán, and María Dolores Berrade. "On the Residual Lifetime and Inactivity Time in Mixtures." Mathematics 10, no. 15 (August 6, 2022): 2795. http://dx.doi.org/10.3390/math10152795.

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In this paper we study the aging characteristics in mixtures of distributions, providing characterizations for their derivatives that explain the smooth behavior of the mixture. The classical preservation results for the reversed hazard rate, mean residual life and mean inactivity time are derived under a different approach than in previous studies. We focus on the variance of both the residual life and inactivity time in mixtures, obtaining some preservation properties. We also state conditions for weak and strong bending properties for the variance of the residual life and the inactivity time in mixtures.

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21

Hashempour, Majid. "A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations." Mathematica Slovaca 71, no. 4 (August 1, 2021): 983–1004. http://dx.doi.org/10.1515/ms-2021-0034.

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Abstract In this paper, we introduce a new two-parameter lifetime distribution which is called extended Half-Logistic (EHL) distribution. Theoretical properties of this model including the hazard function, quantile function, asymptotic, extreme value, moments, conditional moments, mean residual life, mean past lifetime, residual entropy, cumulative residual entropy and order statistics are derived and studied in details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, two real data sets are illustration the purposes.

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22

Di Crescenzo, Antonio, Barbara Martinucci, and Julio Mulero. "A QUANTILE-BASED PROBABILISTIC MEAN VALUE THEOREM." Probability in the Engineering and Informational Sciences 30, no. 2 (December 9, 2015): 261–80. http://dx.doi.org/10.1017/s0269964815000376.

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For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

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23

Rychlik, Tomasz, and Magdalena Szymkowiak. "Bounds on the Lifetime Expectations of Series Systems with IFR Component Lifetimes." Entropy 23, no. 4 (March 24, 2021): 385. http://dx.doi.org/10.3390/e23040385.

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We consider series systems built of components which have independent identically distributed (iid) lifetimes with an increasing failure rate (IFR). We determine sharp upper bounds for the expectations of the system lifetimes expressed in terms of the mean, and various scale units based on absolute central moments of component lifetimes. We further establish analogous bounds under a more stringent assumption that the component lifetimes have an increasing density (ID) function. We also indicate the relationship between the IFR property of the components and the generalized cumulative residual entropy of the series system lifetime.

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24

Rykov, Vladimir, Nika Ivanova, Dmitry Kozyrev, and Tatyana Milovanova. "On Reliability Function of a k-Out-Of-n System with Decreasing Residual Lifetime of Surviving Components after their Failures." Mathematics 10, no. 22 (November 13, 2022): 4243. http://dx.doi.org/10.3390/math10224243.

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We consider the reliability function of a k-out-of-n system under conditions that failures of its components lead to an increase in the load on the remaining ones and, consequently, to a change in their residual lifetimes. Development of models able to take into account that failures of a system’s components lead to a decrease in the residual lifetime of the surviving ones is of crucial significance in the system reliability enhancement tasks. This paper proposes a novel approach based on the application of order statistics of the system’s components lifetime to model this situation. An algorithm for calculation of the system reliability function and two moments of its uptime has been developed. Numerical study includes sensitivity analysis for special cases of the considered model based on two real-world systems. The results obtained show the sensitivity of system’s reliability characteristics to the shape of lifetime distribution, as well as to the value of its coefficient of variation at a fixed mean.

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25

Salehi, E. T., M. Asadi, and S. Eryılmaz. "On the mean residual lifetime of consecutive k-out-of-n systems." TEST 21, no. 1 (March 3, 2011): 93–115. http://dx.doi.org/10.1007/s11749-011-0237-3.

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26

Trukhanov, V. M., and M. P. Kukhtik. "METHODS OF PROLONGATION OF OPERATING LIFETIMES FOR COMPLEX TECHNICAL SYSTEMS LIKE MOVABLE MOUNTINGS OF SPECIAL PURPOSE." Kontrol'. Diagnostika, no. 253 (July 2019): 46–51. http://dx.doi.org/10.14489/td.2019.07.pp.046-051.

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The article is devoted to the relevant issue of prolongation of operating lifetimes for complex technical systems like movable mountings of special purpose. The issues of mean and residual lives have been considered, mathematical models of mean and residual lives have been represented. Methods of prolongation of operating lifetimes for expensive objects of special purpose like movable mountings have been described. In particular, mathematical model of prolongation of operating lifetimes by the calculation and analytical method, which is based on of normal distribution law of actual load and durability, has been represented. The experimental method, which is based on analysis of actual technical state of element base and material, has been described. The probabilistic method of calculation of reliability function during prolonged operating lifetime taking into account actual technical state of element base has been represented.

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27

Khorashadizadeh, M., A. H. Rezaei Roknabadi, and G. R. Mohtashami Borzadaran. "Characterizations of Lifetime Distributions Based on Doubly Truncated Mean Residual Life and Mean Past to Failure." Communications in Statistics - Theory and Methods 41, no. 6 (March 15, 2012): 1105–15. http://dx.doi.org/10.1080/03610926.2010.535626.

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28

Tavangar, Mahdi. "Some representation theorems in terms of mean residual lifetime and mean inactivity time of coherent systems." Communications in Statistics - Theory and Methods 45, no. 2 (February 18, 2015): 226–35. http://dx.doi.org/10.1080/03610926.2013.830745.

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29

Anwar, Masood, and Jawaria Zahoor. "The Half-Logistic Lomax Distribution for Lifetime Modeling." Journal of Probability and Statistics 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/3152807.

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We introduce a new two-parameter lifetime distribution called the half-logistic Lomax (HLL) distribution. The proposed distribution is obtained by compounding half-logistic and Lomax distributions. We derive some mathematical properties of the proposed distribution such as the survival and hazard rate function, quantile function, mode, median, moments and moment generating functions, mean deviations from mean and median, mean residual life function, order statistics, and entropies. The estimation of parameters is performed by maximum likelihood and the formulas for the elements of the Fisher information matrix are provided. A simulation study is run to assess the performance of maximum-likelihood estimators (MLEs). The flexibility and potentiality of the proposed model are illustrated by means of real and simulated data sets.

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30

Skoczylas, Agnieszka, Kazimierz Zaleski, Jakub Matuszak, Krzysztof Ciecieląg, Radosław Zaleski, and Marek Gorgol. "Influence of Slide Burnishing Parameters on the Surface Layer Properties of Stainless Steel and Mean Positron Lifetime." Materials 15, no. 22 (November 16, 2022): 8131. http://dx.doi.org/10.3390/ma15228131.

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This paper presents the results of an experimental study on the impact of slide burnishing on surface roughness parameters (Sa, Sz, Sp, Sv, Ssk, and Sku), topography, surface layer microhardness, residual stress, and mean positron lifetime (τmean). In the study, specimens of X6CrNiTi18 stainless steel were subjected to slide burnishing. The experimental variables were feed and slide burnishing force. The slide burnishing process led to changes in the surface structure and residual stress distribution and increased the surface layer microhardness. After slide burnishing, the analyzed roughness parameters decreased compared with their pre-treatment (grinding) values. The slide burnishing of X6CrNiTi18 steel specimens increased their degree of strengthening e from 8.77% to 42.74%, while the hardened layer thickness gh increased after the treatment from about 10 µm to 100 µm. The maximum compressive residual stress was about 450 MPa, and the maximum depth of compressive residual stresses was gσ = 1.1 mm. The positron mean lifetime τmean slightly yet systematically increased with the increase in burnishing force F, while an increase in feed led to changes of a different nature.

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31

Ahmadi, Reza. "A bivariate process-based mean residual lifetime model for maintenance and inspection planning." Computers & Industrial Engineering 163 (January 2022): 107792. http://dx.doi.org/10.1016/j.cie.2021.107792.

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32

Yong, Zhou, Ma Qiuxia, and Pan Sheng. "Composite estimator of mean residual lifetime with length-biased and right-censored data." SCIENTIA SINICA Mathematica 49, no. 5 (April 25, 2019): 781. http://dx.doi.org/10.1360/n012018-00114.

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33

Oluyede, Broderick O. "ON BOUNDS AND APPROXIMATING WEIGHTED DISTRIBUTIONS BY EXPONENTIAL DISTRIBUTIONS." Probability in the Engineering and Informational Sciences 20, no. 3 (June 1, 2006): 517–28. http://dx.doi.org/10.1017/s0269964806060311.

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In this article, we obtain error bounds for exponential approximations to the classes of weighted residual and equilibrium lifetime distributions with monotone weight functions. These bounds are obtained for the class of distributions with increasing (decreasing) hazard rate and mean residual life functions.

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34

Willmot, Gordon E., and Jun Cai. "ON CLASSES OF LIFETIME DISTRIBUTIONS WITH UNKNOWN AGE." Probability in the Engineering and Informational Sciences 14, no. 4 (October 2000): 473–84. http://dx.doi.org/10.1017/s0269964800144067.

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Some class properties of the used better (worse) than aged [UBA (UWA)] and the used better (worse) than aged in expectation [UBAE (UWAE)] classes of lifetime distributions are considered. Relationships with the decreasing (increasing) mean residual lifetime [DMRL (IMRL)] class and the decreasing (increasing) variance residual lifetime [DVRL (IVRL)] class are established. Discrete UBA and UWA distributions are introduced and studied. Characterizations of UBA and UWA distributions are derived by using discrete aging properties of mixed Poisson distributions. Applications of these results to queueing theory and ruin are then considered. In particular, preservation of UBA (UWA) and UBAE (UWAE) under a transform of life distributions is given.

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35

Alotaibi, Refah, Mervat Khalifa, Lamya A. Baharith, Sanku Dey, and H. Rezk. "The Mixture of the Marshall–Olkin Extended Weibull Distribution under Type-II Censoring and Different Loss Functions." Mathematical Problems in Engineering 2021 (May 18, 2021): 1–15. http://dx.doi.org/10.1155/2021/6654101.

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To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing as compared to simple models. This paper considers a mixture of the Marshall–Olkin extended Weibull distribution for efficient modeling of failure, survival, and COVID-19 data under classical and Bayesian perspectives based on type-II censored data. We derive several properties of the new distribution such as moments, incomplete moments, mean deviation, average lifetime, mean residual lifetime, Rényi entropy, Shannon entropy, and order statistics of the proposed distribution. Maximum likelihood and Bayes procedure are used to derive both point and interval estimates of the parameters involved in the model. Bayes estimators of the unknown parameters of the model are obtained under symmetric (squared error) and asymmetric (linear exponential (LINEX)) loss functions using gamma priors for both the shape and the scale parameters. Furthermore, approximate confidence intervals and Bayes credible intervals (CIs) are also obtained. Monte Carlo simulation study is carried out to assess the performance of the maximum likelihood estimators and Bayes estimators with respect to their estimated risk. The flexibility and importance of the proposed distribution are illustrated by means of four real datasets.

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36

Mansourvar, Zahra, and Torben Martinussen. "Estimation of average causal effect using the restricted mean residual lifetime as effect measure." Lifetime Data Analysis 23, no. 3 (April 1, 2016): 426–38. http://dx.doi.org/10.1007/s10985-016-9366-z.

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37

Fagiuoli, Enrico, and Franco Pellerey. "Moment inequalities for sums of DMRL random variables." Journal of Applied Probability 34, no. 2 (June 1997): 525–35. http://dx.doi.org/10.2307/3215391.

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Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.

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38

Fagiuoli, Enrico, and Franco Pellerey. "Moment inequalities for sums of DMRL random variables." Journal of Applied Probability 34, no. 02 (June 1997): 525–35. http://dx.doi.org/10.1017/s0021900200101159.

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Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.

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39

Li, Xiaohu, and Maochao Xu. "SOME RESULTS ABOUT MIT ORDER AND IMIT CLASS OF LIFE DISTRIBUTIONS." Probability in the Engineering and Informational Sciences 20, no. 3 (June 1, 2006): 481–96. http://dx.doi.org/10.1017/s0269964806060293.

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We investigate some new properties of mean inactivity time (MIT) order and increasing MIT (IMIT) class of life distributions. The preservation property of MIT order under increasing and concave transformations, reversed preservation properties of MIT order, and IMIT class of life distributions under the taking of maximum are developed. Based on the residual life at a random time and the excess lifetime in a renewal process, stochastic comparisons of both IMIT and decreasing mean residual life distributions are conducted as well.

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40

Goldoust, Mehdi, and Adel Mohammadpour. "Generalized extended Marshall-Olkin family of lifetime distributions." Statistics in Transition New Series 23, no. 1 (March 1, 2022): 55–74. http://dx.doi.org/10.2478/stattrans-2022-0004.

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Abstract We introduce a new generalized family of nonnegative continuous distributions by adding two extra parameters to a lifetime distribution, called the baseline distribution, by twice compounding a power series distribution. The new family, called the lifetime power series-power series family, has a serial arrangement of parallel structures, which extends the Marshall and Olkin structure. Four special models are discussed. A mathematical treatment of the new distributions is provided, including ordinary and incomplete moments, quantile, moment generating and mean residual functions. The maximum likelihood estimation technique is used to estimate the model parameters and a simulation study is conducted to investigate the performance of the maximum likelihood estimates. Its applicability is also illustrated by means of two real data sets.

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Asadi, Majid, Nader Ebrahimi, G. G. Hamedani, and Ehsan S. Soofi. "Maximum dynamic entropy models." Journal of Applied Probability 41, no. 2 (June 2004): 379–90. http://dx.doi.org/10.1239/jap/1082999073.

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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.

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Asadi, Majid, Nader Ebrahimi, G. G. Hamedani, and Ehsan S. Soofi. "Maximum dynamic entropy models." Journal of Applied Probability 41, no. 02 (June 2004): 379–90. http://dx.doi.org/10.1017/s0021900200014376.

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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.

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43

Poynor, Valerie, and Athanasios Kottas. "Nonparametric Bayesian inference for mean residual life functions in survival analysis." Biostatistics 20, no. 2 (January 19, 2018): 240–55. http://dx.doi.org/10.1093/biostatistics/kxx075.

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SUMMARY Modeling and inference for survival analysis problems typically revolves around different functions related to the survival distribution. Here, we focus on the mean residual life (MRL) function, which provides the expected remaining lifetime given that a subject has survived (i.e. is event-free) up to a particular time. This function is of direct interest in reliability, medical, and actuarial fields. In addition to its practical interpretation, the MRL function characterizes the survival distribution. We develop general Bayesian nonparametric inference for MRL functions built from a Dirichlet process mixture model for the associated survival distribution. The resulting model for the MRL function admits a representation as a mixture of the kernel MRL functions with time-dependent mixture weights. This model structure allows for a wide range of shapes for the MRL function. Particular emphasis is placed on the selection of the mixture kernel, taken to be a gamma distribution, to obtain desirable properties for the MRL function arising from the mixture model. The inference method is illustrated with a data set of two experimental groups and a data set involving right censoring. The supplementary material available at Biostatistics online provides further results on empirical performance of the model, using simulated data examples.

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Alomani, Ghadah, and Mohamed Kayid. "Stochastic Properties of Fractional Generalized Cumulative Residual Entropy and Its Extensions." Entropy 24, no. 8 (July 28, 2022): 1041. http://dx.doi.org/10.3390/e24081041.

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The fractional generalized cumulative residual entropy (FGCRE) has been introduced recently as a novel uncertainty measure which can be compared with the fractional Shannon entropy. Various properties of the FGCRE have been studied in the literature. In this paper, further results for this measure are obtained. The results include new representations of the FGCRE and a derivation of some bounds for it. We conduct a number of stochastic comparisons using this measure and detect the connections it has with some well-known stochastic orders and other reliability measures. We also show that the FGCRE is the Bayesian risk of a mean residual lifetime (MRL) under a suitable prior distribution function. A normalized version of the FGCRE is considered and its properties and connections with the Lorenz curve ordering are studied. The dynamic version of the measure is considered in the context of the residual lifetime and appropriate aging paths.

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Wu, Hongping, and Yihui Luan. "An Efficient Estimation of the Mean Residual Life Function with Length-Biased Right-Censored Data." Mathematical Problems in Engineering 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/937397.

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The mean residual life (MRL) function for a lifetime random variableT0is one of the basic parameters of interest in survival analysis. In this paper, we propose a new estimator of the MRL function with length-biased right-censored data and evaluate its performance through a small Monte Carlo simulation study. The results of the simulations show that the proposed estimator outperforms the existing one referred to in Data and Model Setup Section in terms of Monte Carlo bias and mean square error, especially when the censoring rate is heavy. We also show that the proposed estimator converges in distribution under some conditions.

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Ocloo, Selasi Kwaku, Lewis Brew, Suleman Nasiru, and Benjamin Odoi. "Harmonic Mixture Fréchet Distribution: Properties and Applications to Lifetime Data." International Journal of Mathematics and Mathematical Sciences 2022 (September 28, 2022): 1–20. http://dx.doi.org/10.1155/2022/6460362.

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In this study, we propose a four-parameter probability distribution called the harmonic mixture Fréchet. Some useful expansions and statistical properties such as moments, incomplete moments, quantile functions, entropy, mean deviation, median deviation, mean residual life, moment-generating function, and stress-strength reliability are presented. Estimators for the parameters of the harmonic mixture Fréchet distribution are derived using the estimation techniques such as the maximum-likelihood estimation, the ordinary least-squares estimation, the weighted least-squares estimation, the Cramér–von Mises estimation, and the Anderson–Darling estimation. A simulation study was conducted to assess the biases and mean square errors of the estimators. The new distribution was applied to three-lifetime datasets and compared with the classical Fréchet distribution and eight (8) other extensions of the Fréchet distribution.

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Muhammad, Mustapha, Huda M. Alshanbari, Ayed R. A. Alanzi, Lixia Liu, Waqas Sami, Christophe Chesneau, and Farrukh Jamal. "A New Generator of Probability Models: The Exponentiated Sine-G Family for Lifetime Studies." Entropy 23, no. 11 (October 24, 2021): 1394. http://dx.doi.org/10.3390/e23111394.

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In this article, we propose the exponentiated sine-generated family of distributions. Some important properties are demonstrated, such as the series representation of the probability density function, quantile function, moments, stress-strength reliability, and Rényi entropy. A particular member, called the exponentiated sine Weibull distribution, is highlighted; we analyze its skewness and kurtosis, moments, quantile function, residual mean and reversed mean residual life functions, order statistics, and extreme value distributions. Maximum likelihood estimation and Bayes estimation under the square error loss function are considered. Simulation studies are used to assess the techniques, and their performance gives satisfactory results as discussed by the mean square error, confidence intervals, and coverage probabilities of the estimates. The stress-strength reliability parameter of the exponentiated sine Weibull model is derived and estimated by the maximum likelihood estimation method. Also, nonparametric bootstrap techniques are used to approximate the confidence interval of the reliability parameter. A simulation is conducted to examine the mean square error, standard deviations, confidence intervals, and coverage probabilities of the reliability parameter. Finally, three real applications of the exponentiated sine Weibull model are provided. One of them considers stress-strength data.

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48

Louzada, Francisco, Vitor Marchi, and James Carpenter. "The Complementary Exponentiated Exponential Geometric Lifetime Distribution." Journal of Probability and Statistics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/502159.

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We proposed a new family of lifetime distributions, namely, complementary exponentiated exponential geometric distribution. This new family arises on a latent competing risk scenario, where the lifetime associated with a particular risk is not observable but only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments,rth moment of theith order statistic, mean residual lifetime, and modal value. Inference is implemented via a straightforwardly maximum likelihood procedure. The practical importance of the new distribution was demonstrated in three applications where our distribution outperforms several former lifetime distributions, such as the exponential, the exponential-geometric, the Weibull, the modified Weibull, and the generalized exponential-Poisson distribution.

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Salehi, E., and S. S. Hashemi-Bosra. "The mean general residual lifetime of (n − k + 1)-out-of-n systems with exchangeable components." Communications in Statistics - Theory and Methods 46, no. 13 (June 21, 2016): 6382–400. http://dx.doi.org/10.1080/03610926.2015.1129413.

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Eliwa, M. S., Medhat EL-Damcese, A. H. El-Bassiouny, Abhishek Tyag, and M. El-Morshedy. "The Weibull Distribution: Reliability Characterization Based on Linear and Circular Consecutive Systems." Statistics, Optimization & Information Computing 9, no. 4 (September 24, 2021): 974–83. http://dx.doi.org/10.19139/soic-2310-5070-1132.

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Linear and circular consecutive models play a vital role to study the mechanical systems emerging in various fields including survival analysis, reliability theory, biological disciplines, and other lifetime sciences. As a result, analysis of reliability properties of consecutive k − out − of − n : F systems has gained a lot of attention in recent years from a theoretical and practical point of view. In the present article, we have studied some important stochastic and aging properties of residual lifetime of consecutive k − out − of − n : F systems under the condition n − k + 1, k ≤ n and all components of the system are working at time t. The mean residual lifetime (MRL) and its hazard rate function are proposed for the linear consecutive k − out − of − n : F (lin/con/k/n:F) and circular consecutive k − out − of − n : F (cir/con/k/n:F) systems. Furthermore, several mathematical properties of the proposed MRL are examined. Finally, the Weibull distribution with two parameters is used as an example to explain the theoretical results.

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