Mean Residual Lifetime Function Research Articles

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.

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.

Some new bounds for the mean value function of the residual lifetime process

In this study, mean residual lifetime (MRL) function of the residual lifetime process is investigated. Various two-sided bounds are obtained for this function by using the monotone convergence of the sequence of the functions. Since many important features of any stochastic process depend on classifications of distributions, improved bounds are found for MRL function by considering the taxonomy of the distributions. Moreover, some new bounds are achieved by using the asymptotic expression of MRL function and its boundary relation with the MRL of a single component at age t. Three examples are given to illustrate the results proposed.

Further theories on application of new generalized probability density function and its applications

AbstractThe exponential‐based probability density functions (PDFs) such as Gaussian, exponential, and Rayleigh are widely used in lifetime modeling, survival analysis, and engineering. Recently, the generalized probability density functions consisting a well‐known exponential PDFs were introduced and successfully applied in electrical and electronic engineering. After encouraging results of generalized PDF, further theories and applications are considered within this paper. The parameters of the new (GE) PDF were successfully derived by using the maximum likelihood estimation method. The applications in the paper show that the new generalized probability density function based on maximum likelihood parameter estimators provides more effective probabilistic representation. In this study, statistical properties of new GE distribution is discussed. Some reliability characteristics such as hazard function, mean residual lifetime (MRL) function, and mean past lifetime (MPL) functions are obtained based on special functions. Also, th moment of the new GE random variable is obtained in analytical form. Maximum likelihood estimates (MLEs) of parameters are also discussed. Lastly, some illustrative examples in engineering and survival analysis are provided based on different real data sets.

A new approach in the mean residual lifetime evaluation of a multi-state system

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.

Reliability analysis of a multi-state system with identical units having two dependent components

The performance evaluation of a system having n identical units, each of which consists of two components has been successfully discussed in binary-state reliability analysis. In this paper, we study the performance evaluation of a multi-state system based on bivariate order statistics. The multi-state system consists of n independent and identical units, each having two components. The components of each unit are assumed to be s-dependent. However, the units work s-independently with each other. The system and each component of each unit having three performance levels “0 (failure), 1 (partially working) and 2 (completely working)” are considered. The degradation of the components follows Markov Process and also Farlie-Gumbel-Morgenstern distribution is used to model the s-dependence of the components. The reliability analysis of a multi-state k-out-of- n system are evaluated under the assumptions. Some dynamic performance measures for the system such as the mean residual and mean past lifetime functions based on bivariate order statistics are also evaluated. The performance of the system is especially examined for different values of s-dependence parameter, the degradation rates and different number of units for the system. The results are supported with some numerical examples and graphical representations.

A New Generalized Family of Lifetime Distributions Motivated by Parallel and Series Structures

Any given system can be represented as a parallel arrangement of series structures. Motivated by this fact, a general family of distributions is introduced, by adding two extra parameters to a distribution (called baseline distribution),twice compounding with power series distribution. The new family can allow various hazard rate curves that compete well with other alternatives in fitting real data. We derive formal expressions for its moments, generating function, mean residual lifetime and other reliability functions. Certain characterizations of the new family are presented in terms of the ratio of two truncated moments as well as based on the hazard rate function. 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. Finally, two applications of the model with real data sets are presented to illustrate the usefulness of the proposed distribution.

Comparisons of aggregate claim numbers and amounts: a study of heterogeneity

ABSTRACTIn this paper, we investigate how the heterogeneity among occurrence probabilities and claim severities affects the aggregate claim numbers and aggregate claim amount for an insurance portfolio. We show that higher heterogeneity (and dependence) among occurrence probabilities results in both smaller aggregate claim numbers and aggregate claim amount in the sense of the mean residual lifetime order. We also prove that as the heterogeneity among the claims increases, the aggregate claim amount increases in the sense of the usual stochastic order when the vector of occurrence probabilities is left tail weakly stochastic arrangement increasing. These theoretical findings are applied to (i) study ordering properties of convolutions of binomial random variables, (ii) provide upper bounds for the mean residual lifetime functions of the aggregate claim numbers and amount, and (iii) compare stop-loss premiums and risk capital of different insurance portfolios.

A Study on the Multi-state (r, s)-out-of- n Systems with Dependent Components

In the study of technical systems in reliability engineering, multi-state systems play a useful role. A multi-state system is a system consisting n components that system and its components may have several level performance. In the present paper, we introduce the multi-state (r, s)-out-of-n system consisting n elements having the property that each element consists two dependent components and each component of the elements and the system can be in one of m+1 possible states: 0, 1, 2,..., m. We investigate an efficient method to compute the exact reliability by using the distribution of bivariate order statistics. Depending on the number of active components of the multi-state (r, s)-out-of-n systems at time t, the mean residual lifetime function of the system is studied. Also, an example and illustrative graph is provided.

Bootstrapping the Kaplan–Meier estimator on the whole line

This article is concerned with proving the consistency of Efron’s bootstrap for the Kaplan–Meier estimator on the whole support of a survival function. While previous works address the asymptotic Gaussianity of the Kaplan–Meier estimator without restricting time, we enable the construction of bootstrap-based time-simultaneous confidence bands for the whole survival function. Other practical applications include bootstrap-based confidence bands for the mean residual lifetime function or the Lorenz curve as well as confidence intervals for the Gini index. Theoretical results are complemented with a simulation study and a real data example which result in statistical recommendations.

On length and area-biased Maxwell distributions

ABSTRACTThis article addresses the various properties and different methods of estimation of the unknown parameter of length and area-biased Maxwell distributions. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of length and area-biased Maxwell distributions (such as moments, moment-generating function (mgf), hazard rate function, mean residual lifetime function, residual lifetime function, reversed residual life function, conditional moments and conditional mgf, stochastic ordering, and measures of uncertainty) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimator, moments estimator, least-square and weighted least-square estimators, maximum product of spacings estimator and compare them using extensive numerical simulations. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using inverted gamma prior for the scale parameter. Furthermore, Bayes estimators and their respective posterior risks are computed and compared using Markov chain Monte Carlo (MCMC) algorithm. Also, bootstrap confidence intervals using frequentist approaches are provided to compare with Bayes credible intervals. Finally, a real dataset has been analyzed for illustrative purposes.

Reliability Analysis of the CNC System Based on Field Failure Data in Operating Environments

Reliability is a measure of how well a product will perform under a certain set of conditions for a specified amount of time especially in the field environments. In this paper, a reliability study of a computer numerical control (CNC) system is described. For this analysis, field failure data from a shop manufacturing factory collected over the course of a year on approximately 20 CNC machine tools during their operating period were analyzed. Based on the field failure data, the two-parameter exponential distribution was found to be applicable to describe the time between failures of the CNC system from among many distributions including Weibull, gamma, two-parameter exponential, normal, and logistic using the chi-squared test. In this paper, we discuss the reliability estimation of the CNC system based on the collected field failure data from a manufacturing factory using the maximum likelihood estimate (MLE) and uniform minimum variance estimate (UMVUE) methods. We also discuss the confidence intervals of the mean residual lifetime and reliability function. The result shows that the UMVUE method can provide much better and more accurate results in estimating the reliability of the CNC system than the MLE. This finding, on the one hand, seems to be obvious because the UMVUE is not only an unbiased estimator but also sufficient statistic with the smallest variance; on the other hand, it is not straightforward to obtain the UMVUE of any complex function, which is the reliability function in this case. This is a very important finding and is very encouraging because it indicates that the reliability analysis of the CNC system based on the UMVUE can be more than compensated by the ability of the complexity of parameter estimation method to better evaluate and predict the reliability of the CNC system. Hence, we believe that it is worth the effort to derive those parameter functions using UMVUE method. Copyright © 2015 John Wiley & Sons, Ltd.

Entropy for Past Residual Life Time Distributions

As we are familiar that existence of life is uncertain. In the context of reliability and lifetime distributions, there are some measures such as the hazard rate function or the mean residual lifetime function that have been used to characterize or compare the aging process of a component. This definition deals with random variable truncated above some t, i.e. the support of the random variable is taken to be (0, t). We outline some common methods for past residual lifetime distributions with the aim to provide some insights on general construction mechanisms. Some applications are given to provide the readers a possible source of ideas to draw upon. Applications of past residual lifetime distributions in reliability, survival analysis and mortality studies are briefly discussed.

Weighted Weibull Distribution: Properties and Estimation

We take a closer look at the weighted Weibull distribution. First, we study the structural properties of the probability density function, hazard rate, and mean residual lifetime functions of this distribution. We put forward the estimation for the parameters of the weighted Weibull distribution via maximum likelihood estimation technique. We also obtain expected Fisher’s information matrix, as well as discuss the existence and uniqueness of the maximum likelihood estimates. With regard to Bayesian inference of the unknown parameters, we are using importance sampling technique to calculate Bayes estimates and the corresponding highest posterior density intervals. We perform a data analysis for illustrative purposes.

Conditional quantile residual lifetime models for right censored data

Quantile residual lifetime function is a more comprehensive quantitative measure for residual lifetimes than the mean residual lifetime function. It also incorporates the median residual life function, which is less restrictive than the model based on the mean residual lifetime. In this study, we propose a semiparametric estimator of the conditional quantile residual lifetime under different covariate effects at a specified time point by the reinforcement of the auxiliary models. Two kind of test statistics are proposed to compare two quantile residual lifetimes at fixed time points. Asymptotic properties are also established and a revised bootstrap method is proposed to estimate the asymptotic variance of the estimator. Simulation studies are reported to assess the finite sample properties of the proposed estimator and the performance of test statistics in terms of type I error probabilities and powers at fixed time points. We also compare the proposed method with the method of Jung et al. (Biometrics 65:1203-1212, 2009) through simulation studies. The proposed methods are applied to HIV data and some interesting results are presented.

The Kumaraswamy Birnbaum–Saunders Distribution

Motivated by the recent work of Cordeiro and Castro (2011), we study the Kumaraswamy Birnbaum–Saunders (Kw- ) distribution. This distribution provides an enormous flexibility in modeling heavy-tailed and skewed data. We derive some mathematical properties of the Kw- including moments, quantile function, average lifetime function, mean residual lifetime function, and order statistics. In addition, we discuss maximum likelihood estimation of the model parameters.

An effective method for constructing bounds for ruin probabilities for the surplus process perturbed by diffusion

In this paper, we first study orders, valid up to a certain positive initial surplus, between a pair of ruin probabilities resulting from two individual claim size random variables for corresponding continuous time surplus processes perturbed by diffusion. The results are then applied to obtain a smooth upper (lower) bound for the underlying ruin probability; the upper (lower) bound is constructed from exponentially distributed claims, provided that the mean residual lifetime function of the underlying random variable is non-decreasing (non-increasing). Finally, numerical examples are given to illustrate the constructed upper bounds for ruin probabilities with comparisons to some existing ones.

Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems

Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of them are Xt=X−t | X>t and Yt=Y−t | Y>t, respectively. In this paper, we introduce, by taking into account the age of systems, a time-dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t ):=P(Xt>Yt) which enables one to obtain, at time t, the probability that the residual lifetime Xt is greater than the residual lifetime Yt. It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability, 22, 1973) that the probability of type R(t) is important for designing as long-lived a product as possible. Several properties of R(t) and its connection with well-known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed.

Mean Residual Lifetimes of Consecutive-k-out-of-n Systems

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.

Mean Residual Lifetimes of Consecutive-k-out-of-n Systems

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.