Mean Residual Lifetime Research Articles

Extreme Values and Reliability Measures of Continuous Erlang Mixtures

This study delves into Extreme Value Theory, focusing on the limiting distributions of continuous Erlang mixtures and their mixing distributions. It categorizes Erlang mixtures based on their extreme value types, demonstrating that the limiting distributions of the Erlang mixtures can be either Type I (Gumbel) or Type II (Frechet), contingent on the mixing distribution. Further, this study derives the mean residual lifetime and equilibrium distributions of these mixed distributions.

Surveillance of the recurrence time of the effectiveness of national and region-level quarantines of COVID-19 pandemic in Chile.

The World Health Organization has recommended a range of social and health measures to mitigate the spread of coronavirus disease 2019 (COVID-19), including strategies such as quarantines, border closures, social distancing, and mask usage, among others. Specifically, the Chilean authorities implemented the "step-by-step" plan, built on the concept of dynamic quarantine. Numerous studies have examined the effectiveness of these quarantines in Chile during the pandemic, utilizing data published by the Chilean Ministry of Health. This study's primary aim was to enhance our understanding of quarantine effectiveness in Chile during the pandemic. We accomplished this by analyzing the distributional behavior of the time until the COVID-19 pandemic was deemed under control or not. In our study, we defined an event with two potential outcomes related to the instantaneous reproductive number (Rt), which signifies the time until a change in the event outcome occurs. Importantly, we did not predefine a specific temporal observation unit. These findings allowed us to complement the concept of effective quarantine by considering the dynamics generated by the protocols, such as phase 1 of the quarantine, in achieving natural herd immunity in response to the number of COVID-19 cases and Rt. We assessed the behavior of the mean and median residual lifetime until the initiation of controlled/uncontrolled episodes of the COVID-19 pandemic based on Rt in all regions of Chile. Despite variations in the distribution of residual times for controlled/uncontrolled episodes in different regions, there was a similar observation during the period considered (between March 2020 and March 2021): the mitigation measures did not produce a clear positive effect for controlling the epidemic. The residual times of episodes with Rr,t > 1 were not different from those episodes with Rr,t ≤ 1.

Residual life modeling and maintenance planning for repairable systems

Abstract Accepted by: Phil Scarf This paper formulates a state-dependent mean residual lifetime model for a repairable system operating in a dynamic environment. The problem is addressed by means of a two-state damage process reflecting the effect of operating environment on the system and a repair process associated with the damage process. As the damage process shifts to a higher state, to maintain a minimum level of performance, the decision maker repairs the system at times that arise according to a point process with a constant intensity. We demonstrate the generality of the proposed model and show how existing models emerge as specific cases. Our approach stimulates further research on the determination of two types of maintenance policies: maintenance policy based on the number of imperfect repairs (Model I) and time-based maintenance policy (Model II). In both cases, using the renewal reward theorem argument, we aim at minimizing the long-run average maintenance cost per unit time by determining optimal replacement policies and the optimal level of imperfect repairs. We illustrate the proposed models and carry out a comparative analysis of maintenance policies through numerical examples. The main conclusions drawn are that repair and maintenance policies depend on the failure mechanism, repair frequency and the level of costs involved. Also, numerical comparison shows that the maintenance modelling based on the number of imperfect repairs (model I) outperforms the time-based replacement model (model II) and two baseline models ignoring the effect of operating environment or whose attention is restricted to perfect repair.

Medical application with an extended Ailamujia inverted Weibull model: Properties, estimation and simulation

In this article, we propose a new lifetime model called Kavya-Manoharan Ailanujia inverted Weibull distribution, which is obtained from the Kavya-Manoharan (KM) transformation family and Ailanujia inverted Weibull distribution. Various statistical properties are obtained, involving moments, incomplete moments, conditional moments, mean deviations, mean residual lifetime and mean past lifetime and entropy. Moreover, the newly proposed distribution, characterized by two parameters, proves to be remarkably versatile, as demonstrated in the application results. Several statistical properties of the KMAIW distribution are deduced, and the estimation of its parameters is accomplished through three distinct techniques. To assess the effectiveness of these approaches, a comprehensive simulation study is conducted. The significance and applicability of the proposed model are underscored through the analysis of a real-world dataset. In comparative evaluations against well-known distributions featuring two, three, and four parameters, the suggested model consistently outperforms them all. This highlights the superior fitting capability and adaptability of the proposed model in comparison to established distributions with varying parameter complexities.

Fitting various medical and engineering applications using the new lifetime distribution with a mathematical framework

Sustainability considerations play a crucial role in informing the modeling and fitting of medical and engineering data, ensuring the development of robust and environmentally conscious solutions. This paper delves into the investigation of a novel continuous distribution, aiming to provide a thorough understanding of its various fundamental mathematical and statistical properties. The analysis encompasses an exploration of survival functions, hazard rate functions, quantile, skewness, kurtosis, moments, mean time to failure, mean time to repair, insurance pricing principles, availability, and mean residual (past) lifetime functions. The proposed model demonstrates versatility in modeling both asymmetric and symmetric data across various kurtosis shapes. It can effectively handle outlier observations and accommodate different shapes of failure rates, including unimodal, bathtub, increasing, or decreasing patterns. This makes the proposed model suitable for modeling data in diverse fields. The maximum likelihood approach is employed to estimate model parameters using complete and upper recorded values. A simulation study is conducted to evaluate the performance of the estimators under different sample sizes for both complete and upper recorded values. To further demonstrate the flexibility and effectiveness of the new model, two datasets from medical and engineering domains are utilized for validation and testing purposes.

Nonparametric estimation of mean residual lifetime in ranked set sampling with a concomitant variable

The mean residual lifetime (MRL) of a unit is its expected additional lifetime provided that it has survived until time t. The MRL estimation problem has been frequently addressed in the literature since it has wide applications in statistics, reliability and survival analysis. In this paper, we consider the problem of estimating the MRL in ranked set sampling when actual quantifications of a concomitant variable are available. To exploit the additional information of the concomitant variable, we introduce several MRL estimators based on some regression techniques. We then compare them with the standard MRL estimator in simple random sampling using Monte Carlo simulation and a real dataset from the Surveillance, Epidemiology, and End Results Program. Our results indicate the superiority of the procedures that we have developed when the quality of ranking is fairly good.

Evaluating the Discrete Generalized Rayleigh Distribution: Statistical Inferences and Applications to Real Data Analysis

Various discrete lifetime distributions have been observed in real data analysis. Numerous discrete models have been derived from a continuous distribution using the survival discretization method, owing to its simplicity and appealing formulation. This study focuses on the discrete analog of the newly generalized Rayleigh distribution. Both classical and Bayesian statistical inferences are performed to evaluate the efficacy of the new discrete model, particularly in terms of relative bias, mean square error, and coverage probability. Additionally, the study explores different important submodels and limiting behavior for the new discrete distribution. Various statistical functions have been examined, including moments, stress–strength, mean residual lifetime, mean past time, and order statistics. Finally, two real data examples are employed to evaluate the new discrete model. Simulations and numerical analyses play a pivotal role in facilitating statistical estimation and data modeling. The study concludes that the discrete generalized Rayleigh distribution presents a notably appealing alternative to other competing discrete distributions.

Failure rate, vitality, and residual lifetime measures: Characterizations based on stress-strength bivariate model with application to an automated life test data

In this article, we introduce some reliability concepts for the bivariate Pareto Type II distribution including joint hazard rate function, CDF for parallel and series systems, joint mean residual lifetime, and joint vitality function. The maximum likelihood and Bayesian estimation methods are utilized to estimate the model parameters. Simulation is carried out to assess the performance of the maximum likelihood and Bayesian estimators, and it is found that the two approaches work quite well in estimation process. Finally, a real lifetime data is analyzed to show the flexibility and the importance of the introduced bivariate mode.

Condition-based maintenance optimisation for multi-component systems using mean residual life

This paper aims to propose a Novel Condition-based maintenance (CBM) decision aid model for optimising the maintenance of complex multi-component systems. As the degradation level of each component is assumed to be independent and stochastic, it follows a specific probability distribution determined from historical data of experimental observations and inspection. The main objective is to optimise the total cost for providing maintenance actions and reducing the excess of spare parts usage. The decision support model consists of determining measurements on components with the aim of estimating the instant of time of removing predictively one or a group of components before they fail. The measurement model includes the mean residual lifetime (MRL) and some extensions developed for this purpose. For demonstrating the pertinency of the proposed model, we use a preventive maintenance strategy for one-component systems and a grouping/opportunistic maintenance for multi-component systems. Besides, a numerical comparative study performing these measurements is carried out using several examples and a case study from Electric energy distribution systems. The solution is illustrated as a decision-making optimal model for optimising the maintenance operations’ costs and the total number of spare parts. The numerical results and the comparison show the efficiency of the proposed approach.

Kernel estimators for mean residual lifetime in length-biased sampling

In this article, we propose three non parametric kernel estimators for the mean residual life function when the data are selected proportionally to their length. We evaluate the mean squared error of the three estimators and investigate the consistency for all three of them.

Estimation in optimal treatment regimes based on mean residual lifetimes with right-censored data.

An optimal individualized treatment regime (ITR) is a decision rule in allocating the best treatment to each patient and, hence, maximizing overall benefits. In this paper, we propose a novel framework based on nonparametric inverse probability weighting (IPW) and augmented inverse probability weighting (AIPW) estimators of the value function when the data are subject to right censoring. In contrast to most existing approaches that are designed to maximize the expected survival time under a binary treatment framework, the proposed method targets maximizing the mean residual lifetime of patients. Specifically, the proposed IPW method searches the optimal ITR by maximizing an estimator for the overall population outcome directly, without specifying the regression model for the conditional mean residual lifetime, whereas the AIPW method integrates the model information of the mean residual lifetime to improve the robustness. Furthermore, to overcome the computational difficulty in a nonsmooth value estimator, smoothed IPW and AIPW estimators are constructed. In theory, we establish the asymptotic properties of the proposed method under suitable regularity conditions. The empirical performances of the proposed IPW and AIPW estimators are evaluated using simulation studies and are further illustrated with an application to the real-world data set from the Acquired Immunodeficiency Syndrome Clinical Trial Group Protocol 175 (ACTG175).

Reliability analysis of the triple modular redundancy system under step-partially accelerated life tests using Lomax distribution

Triple modular redundancy (TMR) is a robust technique utilized in safety-critical applications to enhance fault-tolerance and reliability. This article focuses on estimating the distribution parameters of a TMR system under step-stress partially accelerated life tests, where each component included in the system follows a Lomax distribution. The study aims to analyze the system’s reliability and mean residual lifetime based on the estimated parameters. Various estimation techniques, including maximum likelihood, percentile, least squares, and maximum product of spacings, are explored. Additionally, the optimal stress change time is determined using two criteria. An illustrative example supported by two actual data sets is presented to showcase the methodology’s application. By conducting Monte Carlo simulations, the assessment of the estimation methods’ effectiveness reveals that the maximum likelihood method outperforms the other three methods in terms of both accuracy and performance, as indicated by the numerical outcomes. This research contributes to the understanding and practical implementation of TMR systems in safety-critical industries, potentially saving lives and preventing catastrophic events.

A new extension of the modified Weibull distribution with applications for engineering data

In this study, a novel extension of the three-parameter modified Weibull distribution called the NMW3 distribution is proposed. The NMW3 distribution exhibits increasing or bathtub-shaped hazard rates, which can be beneficial in the context of reliability. The statistical characteristics of the NMW3 distribution are examined, including the difference between the change points of the hazard rate and the mean residual lifetime functions. Additionally, the sensitivities of the parameters and the flatness of the bathtub curve of the NMW3 distribution are discussed. The Weibull probability paper is utilized to characterize plots related to the NMW3 distribution. The parameters of the NMW3 distribution are estimated using both the graphical approach and the maximum likelihood method. Through the analysis of three real data sets, it is demonstrated that the NMW3 distribution outperforms existing extensions of the modified Weibull distributions, highlighting its practical importance. Finally, a reduced distribution of the NMW3 distribution is proposed, and it shows that the reduced distribution has almost the same advantageous characteristics as the NMW3 distribution.

Combined m-Consecutive-k-Out-of-n: F and Consecutive kc-Out-of-n: F Structures with Cold Standby Redundancy

In the present work, we study the combined m-consecutive-k-out-of-n: F and kc-out-of-n: F reliability systems, which consist of independent and identically distributed components. Two different redundancy policies are considered, and their general frameworks are described and illustrated. The main objective of the paper refers to the investigation of the effect of adding cold standby redundancy to the system at the the system level and the component level. Exact formulae for determining the crucial characteristics of the enhanced structure, such as its survival function, the mean time to failure and the mean residual lifetime, are provided. All formulae proved in the present manuscript are explicit expressions which are based on the signature vector of the resulting reliability schemes. An extensive numerical investigation is carried out to shed light on the performance of the combined m-consecutive-k-out-of-n: F and consecutive kc-out-of-n: F reliability systems with cold standby redundancy. Some concluding remarks and comments are provided upon the determination of the optimal design parameters.

Weighted negative cumulative extropy with application in testing uniformity

In this paper, we propose a new information measure, related to cumulative extropy, called weighted negative cumulative extropy measure. Various properties of the proposed measure are obtained and it is shown that weighted negative cumulative extropy measure has relationship with both weighted mean residual and mean past lifetimes of the associated random variable. Non-parametric estimator of the proposed measure is provided and, using this estimator, a goodness-of-fit test for uniform distribution is developed. The proposed test performs reasonably well.

Mean residual lifetime assessment approach for a multi-state standby system

In this paper, a new MRL assessment approach for a multi-state standby system is considered. The three-state system is backed up with a binary cold standby unit. Given that the system is at a specific state at time t, obtaining the MRL is worth considering in conducting the maintenance and repair plans of the system. For different degradation rates and time points, MRL results are examined. An HCTMP is considered for the degradation. Therefore, when the system is observed to be at its perfect state, the MRL decrease with an increase in all the failure rates of the system. However, when the system is observed to be at its partial state, the MRL is not affected by the increase in the failure rate pertained to the perfect state. The MRL when the system has known to be failed before time t and backed up with the standby unit increases with the time increase whereas the MRL when the system is at its perfect(or partial) state is constant when time increases. Moreover, cost evaluation of the system is analyzed. The results are supported with numerical examples and graphical representations.

A general class of shock models with dependent inter-arrival times

We introduce and study a general class of shock models with dependent inter-arrival times of shocks that occur according to the homogeneous Poisson generalized gamma process. A lifetime of a system affected by a shock process from this class is represented by the convolution of inter-arrival times of shocks. This class contains many popular shock models, namely the extreme shock model, the generalized extreme shock model, the run shock model, the generalized run shock model, specific mixed shock models, etc. For systems operating under shocks, we derive and discuss the main reliability characteristics (namely the survival function, the failure rate function, the mean residual lifetime function and the mean lifetime) and study relevant stochastic comparisons. Finally, we provide some numerical examples and illustrate our findings by the application that considers an optimal mission duration policy.

Reliability modeling and maintenance planning for a parallel system with respect to the state-dependent mean residual time

Parallel configuration is one of the most important structures for complex systems. Due to its role in reliability analysis, there has been a growing interest in using the mean residual lifetime as index of the failure prognosis and also maintenance planning. When inspections of the system are involved, mean residual life can be improved through incorporating the information gathered in the inspection times. In this article, maintenance actions are performed based on a mean residual time index and the information about the number of failed components gathered in the inspection times. The main novelty of this article is that the mean residual time is computed as the expected remaining time to reach a defective state instead of the expected remaining time to reach the system failure. Different properties of the expected remaining time to reach a defective state are shown in this article. Based on the expected remaining time to reach a defective state, two maintenance models (Model I and Model) are developed. Compared to Model I, Model II allows the inclusion of minimal repairs. Different numerical examples are computed to highlight distinctions between the two models.

The test of exponentiality based on the mean residual life function revisited

We revisit the family of goodness-of-fit tests for exponentiality based on the mean residual life time proposed by Baringhaus and Henze [(2008), ‘A New Weighted Integral Goodness-of-Fit Statistic for Exponentiality’, Statistics & Probability Letters, 78(8), 1006–1016]. We motivate the test statistic by a characterisation of Shanbhag [(1970), ‘The Characterizations for Exponential and Geometric Distributions’, Journal of the American Statistical Association, 65(331), 1256–1259] and provide an alternative representation, which leads to simple and short proofs for the known theory and an easy to access covariance structure of the limiting Gaussian process under the null hypothesis. Explicit formulas for the eigenvalues and eigenfunctions of the operator associated with the limit covariance are given using results on weighted Brownian bridges. In addition, we derive further asymptotic theory under fixed alternatives as well as approximate Bahadur efficiencies, which provide an insight into the choice of the tuning parameter with regard to the power performance of the tests.

A New Formulation of Generalized Gamma: Some Results and Applications

We extend the 2-parameter Weibull to the generalized gamma distribution by adding a new partial parameter. The new shape parameter can be used to easily generate generalized gamma distributions with different shapes of the density function, hazard rate, and mean residual lifetimes that are useful in simulating various business processes such as manufacturing processes, and reliability systems. We derived some theoretical results and created visual presentations to show the influence (or effect) of this new shape parameter as well. A new Monte Carlo simulation based on the new parameter was proposed to assess the discrepancy between the generalized gamma and its subfamilies. The power analysis of the proposed test was evaluated via simulation studies. We also present some numerical examples.