Mean Residual Life Research Articles

A study on utilization of two cold standby components to increase reliability of a coherent system

. This article focuses on a specific type of coherent system in which, when the coherent system fails, either no active component remains or the remaining active components are connected in parallel. Roy and Gupta (2023) also considered the same type of coherent systems. We observe the restart of such a coherent system using two cold standby components. In our study, standbys are not placed into action simultaneously. They are put into action one by one. We compute the reliability function of the considered coherent system, which is equipped with two cold standby components. In addition, we show that when we use a stronger standby component (in the sense of failure rate order) at first, we get a longer system lifetime. Also, we compute three different mean residual life functions of the system. We have presented a few numerical results to illustrate the importance of our analysis.

An exponentiated XLindley distribution with properties, inference and applications

In this paper, we propose exponentiated XLindley (EXL) distribution. The novel model is adaptable due to the mixt shapes of its density and failure rate functions. The following key statistical properties of EXL distribution are derived: quantile function, moments, hazard function, mean residual life, and Rényi entropy. The parameters are estimated using the maximum likelihood, Anderson Darling, Cramer von Misses, maximum product spacing, ordinary and weighted least square estimation procedures. To examine the behavior of the estimate, Monte Carlo simulation is used. Further Bayesian technique is also utilized to estimate the EXL parameters. The traceplot and Geweke diagnostics are used to track the convergence of simulated processes. The applicability of the EXL distribution is demonstrated by three datasets from different domains such as mortality rate due to COVID-19, precipitation in inches, and failure time for repairable items. The proposed distribution provides efficient results as compared to renowned competitive distributions.

Secant Kumaraswamy Family of Distributions: Properties, Regression Model, and Applications

In this study, Secant Kumaraswamy family of distributions is proposed and studied. This is motivated by the fact that no one distribution can model all types of data from different fields. Therefore, there is the need to develop distributions with desirable properties and flexible enough for modelling data exhibiting different characteristics. Some properties of the new family of distributions, including the quantile function, moments, moment generating function, and mean residual life function, are derived. Five special cases of the family of distributions are presented, and their flexibility is shown by the varying degrees of skewness and kurtosis and nonmonotonic hazard rates. The maximum likelihood estimation method is used to obtain estimators of the family of distributions. Two location-scale regression models are developed for the Secant Kumaraswamy Weibull distribution, which is a special case of the family of distributions. Six different real datasets are used to demonstrate the usefulness of the family of distributions and the regression models. The results show that the family of distributions can be used to model real datasets.

Weighted extropy measures in general Morgenstern family under k-record values with application to medical data

In this paper, we study the marginal distribution of concomitants of k-record (KR) values from generalized Farlie–Gumbel–Morgenstern (GFGM) of bivariate distributions. In addition, the joint distribution of concomitants of KR for this family is obtained. Furthermore, some useful recurrence relations between moments of concomitants are derived. In addition, the hazard rate, the reversed hazard rate, and mean residual life functions of concomitants for this family are obtained. Some recent new measures of information, such as weighted extropy, weighted cumulative past extropy, and weighted cumulative residual extropy, are investigated for the concomitant of KR under the GFGM family. A non-parametric estimator of the proposed measure is provided by combining the empirical method with the concurrent use of KR in the GFGM family. Finally, we analyzed real-world data to examine our findings.

Regression analysis of partially linear transformed mean residual life models

Regression analysis of partially linear transformed mean residual life models

Estimation of the Xrama Distribution Parameter Under Complete and Progressive Type-II Censored Schemes

In this paper, a two-component mixture of Exponential and Rama distribution named Xrama distribution is introduced and discussed. The distributional properties such as moment and the related measures including other characteristics namely quantile function, mean residual life function, Bonferroni curve, Lorenz curve, order statistics, stochastic ordering, and entropy were derived. Simulation study for both complete and progressive type-2 censored schemes is used to illustrate and analyze the behavior of the model. The proposed model’s performance is assessed through a comparison with various other lifetime distributions using negative log-likelihood, Akaike Information Criterion, Corrected Akaike Information Criterion, Bayesian Information Criterion, and Hannan–Quinn information criterion on an actual dataset. The results reveal that, when modeling real lifetime data, the suggested model exhibits greater flexibility than the competing models examined in this study.

The Marshall–Olkin Pareto Type-I Distribution: Properties, Inference under Complete and Censored Samples with Application to Breast Cancer Data

In this paper, we introduce the Marshall–Olkin Pareto type-I (MOPTI) distribution. Structural properties of the MOPTI distribution including the quantile function, mean residual life, and a new theorem for strength-stress measure are introduced. Five methods of estimation for the MOPTI parameters based on complete samples are presented. Furthermore, we explore the estimation of the MOPTI parameters under type-I and type-II censoring. Two Monte Carlo simulation studies are conducted to evaluate the performance of the estimation methods under complete and censored samples. A real-life data set is used to validate the proposed methods.

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.

The Type II Topp-Leone Inverse Power Lomax distribution with Simulation and Applications

In this work, we present a four-parameter lifetime model that can be used to model reliability issues, fatigue life studies, and survival data called the Type II Topp-Leone Inverse Power Lomax distribution. It has the Type II inverse Lomax, Inverse Power Lomax, and Inverse Lomax distributions as sub-models. Some of its statistical properties, including complete and incomplete moments, generating functions, characteristics functions, mean residual life, mean inactivity time, Renyi entropy, Tsallis entropy, order statistics, stress-strength reliability, and weighted probability moment, have formal formulas that we have developed. The model's parameters are estimated using the maximum likelihood estimation technique. The effectiveness of maximum likelihood estimators is evaluated in terms of absolute bias and simulation study standard error. Two lifetime data sets are used to demonstrate how the new model can be applied. Using the same comparative criteria, the proposed distribution offers a better fit than a few well-known distributions.

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.

Preservation of mean inactivity time ordering for coherent systems

AbstractPreservation of stochastic orders through the system signature has captured the attention of researchers in recent years. Signature-based comparisons have been made for the usual stochastic order, hazard rate order, and likelihood ratio orders. However, for the mean residual life (MRL) order, it has recently been proved that the preservation result does not hold true in general, but rather holds for a particular class of distributions. In this paper, we study whether or not a similar preservation result holds for the mean inactivity time (MIT) order. We prove that the MIT order is not preserved from signatures to system lifetimes with independent and identically distributed (i.i.d.) components, but holds for special classes of distributions. The relationship between these classes and the order statistics is also highlighted. Furthermore, the distribution-free comparison of the performance of coherent systems with dependent and identically distributed (d.i.d.) components is studied under the MIT ordering, using diagonal-dependent copulas and distorted distributions.

Comparison of Extreme Wind and Waves Using Different Statistical Methods in 40 Offshore Wind Energy Lease Areas Worldwide

With the ongoing global drive towards renewable energy, several potential offshore wind energy lease areas worldwide have come into focus. This study aims to estimate the extreme wind and wave conditions across several newly designated offshore wind lease sites spanning six continents that are crucial for risk assessment and the design of offshore wind turbines. Firstly, the raw data of wind speeds and wave heights prevailing in these different lease areas were obtained. Following this, an in-depth extreme value analysis was performed over different return periods. Two principal methodologies were applied for this comparative study: the block-maxima and the peaks-over-threshold (POT) approaches. Various statistical techniques, including the Gumbel method of moments, Gumbel maximum likelihood, Gumbel least-squares, and the three-parameter GEV, were employed under the block-maxima approach to obtain the distribution parameters. The threshold for the POT approach was defined using the mean residual life method, and the distribution parameters were obtained using the maximum likelihood method. The Gumbel least-squares method emerged as the most conservative estimator of extreme values in the majority of cases, while the POT approach generally yielded lower extreme values compared to the block-maxima approach. However, the results from the POT approach showed large variations based on the selected threshold. This comprehensive study’s findings will provide valuable input for the efficient planning, design, and construction of future offshore wind farms.

An Improved XShanker Distribution with Applications to Rainfall and Vinyl Chloride Data

This article is an improvement on the XShanker distribution having a single parameter which is the scale parameter and also in the class of Lindley distributions. It is named Double XShanker following the approach that generated it. The distributional properties which include the non-central moment with the associated statistics, the moment generating function, characteristic function, mean residual life function, stress-strength reliability function, Bonferroni and Lorenz curve functions, odd function, stochastic ordering, distribution of order statistics, and Reny entropy. The parameter was estimated using the method of maximum likelihood. Some visualizations based on theoretical values were presented. Some statistics were computed from theoretical values with convergence behavior recorded. A simulation study was conducted with 1000 samples of different sizes and the behavior observed is that as sample size increases the estimates decrease indicating precision. Data on rainfall and Vinyl chloride were used to validate the usefulness of the suggested distribution.

A quasi Rama distribution

A two-parameter quasi Rama distribution which contains the Rama distribution as particular case has been proposed. Its statistical properties based on moments have been discussed. The hazard rate function, mean residual life function, mean deviations and stochastic ordering of the distribution have been derived and studied. The estimation of parameters using method of moments and maximum likelihood methods has been discussed. A simulation study has been presented to know the performance of maximum likelihood estimates. The goodness of fit of the proposed distribution on two datasets relating to failure times has been presented.

Stochastic Comparisons of Largest-Order Statistics and Ranges from Marshall–Olkin Bivariate Exponential and Independent Exponential Variables

Sample range and the associated functions such as survival function and mean residual life function have found many important applications in the reliability field. In this work, we establish some results that are in two different directions. In the first part, we establish some conditions for comparing the largest-order statistics (in the sense of mean residual life order) arising from bivariate Marshall–Olkin exponential distribution. Then, in the second part, we present some sufficient conditions for comparing sample ranges (in the sense of usual stochastic order and reversed hazard rate order) arising from independent exponential random variables.

Odd Chen Exponential Distribution: Properties and Applications

In this study, a new statistical distribution with three parameters called the Odd-Chen Exponential has been proposed. The statistical properties of the proposed distribution, such as the quantile, moments, incomplete moments, moment-generating function, and mean residual life, were developed. The density shows different shapes, making it more flexible for analyzing different forms of data. The hazard function also exhibits different shapes, including the well-known bathtub shape, which means that the distribution is flexible with real-life data. To estimate the distribution parameters, ordinary least squares estimators, Cramér-von Mises estimators, and maximum likelihood estimators were derived. The results were compared using a Monte Carlo simulation. Two-time datasets; one from the mining field and the other from survival analysis, were used to check the applicability of the proposed distribution. The results revealed that the OCE distribution performed better than the Odd Chen Weibull, Odd Chen Rayleigh, Rayleigh, Cauchy, Generalised Inverse Weibull and the Modified Extended Chen distributions.

A weighted‐Weibull distribution: Properties and applications

AbstractThe paper describes a two parameter model and its relationship to the widely used Weibull model. Mathematical properties of the distribution like survival and hazard functions, moments, harmonic and geometric means, Shannon entropy and mean residual life are derived. Different methods of estimation are discussed and a simulation study is performed to verify the efficiency of estimation methods. Applications of our distribution in different scenarios observed in real life are illustrated.

A study on utilization of a cold standby component to enhance the mean residual life function of a coherent system

In this article, we consider a particular type of coherent systems, in which, when the coherent system fails, either no active component remains or the remaining active components are connected in parallel. This work studies two coherent systems of such type with two components, four coherent systems with three components, and seven coherent systems with four components as examples. Lifetime as well as system signature of each of these coherent systems is given. We investigate the restart of such a coherent system using a single cold standby component. The reliability of a system is studied through the (reliability) characteristics such as reliability function, mean time to failure, mean residual life functions of the system. We compute signature-based exact expression for the reliability function of the considered coherent system, which is equipped with a cold standby component. Also, we obtain three different mean residual life functions of the system. So far, no attempt has been made to study such coherent systems in cold standby sparing; therefore, we have provided some new results. We illustrate the theoretical results by considering three particular coherent systems as examples.

DYNAMIC PREDICTION OF RESIDUAL LIFE WITH LONGITUDINAL COVARIATES USING LONG SHORT-TERM MEMORY NETWORKS.

Sepsis, a complex medical condition that involves severe infections with life-threatening organ dysfunction, is a leading cause of death worldwide. Treatment of sepsis is highly challenging. When making treatment decisions, clinicians and patients desire accurate predictions of mean residual life (MRL) that leverage all available patient information, including longitudinal biomarker data. Biomarkers are biological, clinical, and other variables reflecting disease progression that are often measured repeatedly on patients in the clinical setting. Dynamic prediction methods leverage accruing biomarker measurements to improve performance, providing updated predictions as new measurements become available. We introduce two methods for dynamic prediction of MRL using longitudinal biomarkers. in both methods, we begin by using long short-term memory networks (LSTMs) to construct encoded representations of the biomarker trajectories, referred to as "context vectors." In our first method, the LSTM-GLM, we dynamically predict MRL via a transformed MRL model that includes the context vectors as covariates. In our second method, the LSTM-NN, we dynamically predict MRL from the context vectors using a feed-forward neural network. We demonstrate the improved performance of both proposed methods relative to competing methods in simulation studies. We apply the proposed methods to dynamically predict the restricted mean residual life (RMRL) of septic patients in the intensive care unit using electronic medical record data. We demonstrate that the LSTM-GLM and the LSTM-NN are useful tools for producing individualized, real-time predictions of RMRL that can help inform the treatment decisions of septic patients.

Power Lambert uniform distribution: Statistical properties, actuarial measures, regression analysis, and applications

Here, we present a new bounded distribution known as the power Lambert uniform distribution, and we deduce some of its statistical properties such as quantile function, moments, incomplete moments, mean residual life and mean inactivity time, Lorenz, Bonferroni, and Zenga curves, and order statistics. We presented different shapes of the probability density function and the hazard function of the proposed model. Eleven traditional methods are used to estimate its parameters. The behavior of these estimators is investigated using simulation results. Some actuarial measures are derived mathematically for our proposed model. Some numerical computations for these actuarial measures are given for some choices of parameters and significance levels. A new quantile regression model is constructed based on the new unit distribution. The maximum likelihood estimation method is used to estimate the unknown parameters of the regression model. Furthermore, the usability of the new distribution and regression models is demonstrated with the COVID-19 and educational datasets, respectively.