Mean Residual Life Research Articles

Comparison of Coherent Systems with Dependent Components Based on ‌‎the ‎Reversed Mean Residual Life Order

Comparison of Coherent Systems with Dependent Components Based on ‌‎the ‎Reversed Mean Residual Life Order

Mixture of Shanker Distributions: Estimation, Simulation and Application

The Shanker distribution, a one-parameter lifetime distribution with an increasing hazard rate function, is recommended by Shanker for modelling lifespan data. In this study, we examine the theoretical and practical implications of 2-component mixture of Shanker model (2-CMSM). A significant feature of proposed model’s hazard rate function is that it has rising, decreasing, and upside-down bathtub forms. We investigate the statistical characteristics of a mixed model, such as the probability-generating function, the factorial-moment-generating function, cumulants, the characteristic function, the Mills ratio, the mean residual life, and the mean time to failure. There is a graphic representation of density, mean, hazard rate functions, coefficient of variation, skewness, and kurtosis. Our final approach is to estimate the parameters of the mixture model using appropriate approaches such as maximum likelihood, least squares, and weighted least squares. Using a simulation analysis, we examined how the estimates behaved graphically. The simulation results demonstrated that, in the majority of cases, the maximum likelihood estimates have the smallest mean square errors among all other estimates. Finally, we observed that when the sample size rises, the precision measures decrease for all of the estimation techniques, indicating that all of the estimation approaches are consistent. Through two real data analyses, the suggested model’s validity and adaptability are contrasted with those of other models, including the mixture of the exponential distributions and the Lindley distributions .

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 cubic transmuted power-function distribution: Properties, inference, and applications.

A new three-parameter cubic transmuted power distribution is proposed using the cubic rank transformation. The density and hazard functions of the new distribution provide great flexibility. Some mathematical properties of the new model such as quantile function, moments, dispersion index, mean residual life, and order statistics are derived. The model parameters are estimated using five different estimation methods. A comprehensive simulation study is carried out to understand the behavior of derived estimators and choose the best estimation method. The usefulness of the proposed distribution is illustrated using a real dataset. It is concluded that the proposed distribution is better than some well-known existing distributions.

A new failure times model for one and two failure modes system: A Bayesian study with Hamiltonian Monte Carlo simulation

This paper presents an additive Gompertz-Weibull (AGW) distribution, a four-parameter hybrid probability distribution, and its applications in reliability engineering. The failure rate (FR) function of the proposed model demonstrates an increasing trend and a variety of bathtub shapes with or without a low and yet long-stable segment, making it appropriate for modelling a wide variety of real-world problems. Some relationships between the AGW’s FR and its mean residual life functions are examined. For parameter estimation, maximum likelihood and Bayesian inferences are considered. For posterior simulations, we use Hamiltonian Monte Carlo to evaluate the Bayes estimators of the AGW parameters. We evaluate the performance of the proposed AGW model to that of other recent bathtub distributions constructed following the same approach on three failure time datasets. The first two datasets represent device failure times, while the third represents early cable-joint failure times, all with bathtub FR. For comparison, five parametric and nonparametric evaluation criteria and the fitted FR and mean residual life curves were employed. The results indicated that the AGW model would be the best choice for describing failure times, especially when the bathtub-shaped FR of the presented dataset exhibits its three segments.

Boundary-free estimators of the mean residual life function for data on general interval

We propose two new kernel-type estimators of the mean residual life function m X of bounded or half-bounded interval-supported distributions. Though not as severe as the boundary problems in kernel density estimation, eliminating the boundary bias problems that occur in the naive kernel estimator of the mean residual life function is needed. In this article, we utilize the property of bijective transformation. Furthermore, our proposed methods preserve the mean value property, which cannot be done by the naive kernel estimator. Some simulation results showing the estimators’ performances and real data analysis will be presented in the last part of this article.

On some properties of distributions possessing a bathtub-shaped failure rate average

AbstractThe life distribution of a device subject to shocks governed by a homogeneous Poisson process is shown to have a bathtub failure rate average (BFRA) when the probabilities $\bar{P}_k$ of surviving k shocks possess the corresponding discrete property. We prove closure under the formation of weak limits for BFRA distributions and explore related moment convergence issues within the BFRA family. Similar results for increasing and decreasing failure rate average distributions are obtained either independently or as consequences of our results. We also establish some results outlining the positions of various non-monotonic ageing classes such as bathtub failure rate, increasing initially then decreasing mean residual life, new worse then better than used in expectation, and increasing initially then decreasing mean time to failure in the hierarchy. Finally, an open problem is posed and a partial solution provided.

A discrete mixed distribution: Statistical and reliability properties with applications to model COVID-19 data in various countries.

The aim of this paper is to introduce a discrete mixture model from the point of view of reliability and ordered statistics theoretically and practically for modeling extreme and outliers' observations. The base distribution can be expressed as a mixture of gamma and Lindley models. A wide range of the reported model structural properties are investigated. This includes the shape of the probability mass function, hazard rate function, reversed hazard rate function, min-max models, mean residual life, mean past life, moments, order statistics and L-moment statistics. These properties can be formulated as closed forms. It is found that the proposed model can be used effectively to evaluate over- and under-dispersed phenomena. Moreover, it can be applied to analyze asymmetric data under extreme and outliers' notes. To get the competent estimators for modeling observations, the maximum likelihood approach is utilized under conditions of the Newton-Raphson numerical technique. A simulation study is carried out to examine the bias and mean squared error of the estimators. Finally, the flexibility of the discrete mixture model is explained by discussing three COVID-19 data sets.

Regression Models for Lifetime Data: An Overview

Two methods dominate the regression analysis of time-to-event data: the accelerated failure time model and the proportional hazards model. Broadly speaking, these predominate in reliability modelling and biomedical applications, respectively. However, many other methods have been proposed, including proportional odds, proportional mean residual life and several other “proportional” models. This paper presents an overview of the field and the concept behind each of these ideas. Multi-parameter modelling is also discussed, in which (in contrast to, say, the proportional hazards model) more than one parameter of the lifetime distribution may depend on covariates. This includes first hitting time (or threshold) regression based on an underlying latent stochastic process. Many of the methods that have been proposed have seen little or no practical use. Lack of user-friendly software is certainly a factor in this. Diagnostic methods are also lacking for most methods.

Generalized mean residual life models for survival data with missing censoring indicators.

The mean residual life (MRL) function is an important and attractive alternative to the hazard function for characterizing the distribution of a time-to-event variable. In this article, we study the modeling and inference of a family of generalized MRL models for right-censored survival data with censoring indicators missing at random. To estimate the model parameters, augmented inverse probability weighted estimating equation approaches are developed, in which the non-missingness probability and the conditional probability of an uncensored observation are estimated by parametric methods or nonparametric kernel smoothing techniques. Asymptotic properties of the proposed estimators are established and finite sample performance is evaluated by extensive simulation studies. An application to brain cancer data is presented to illustrate the proposed methods.

The Power Zeghdoudi Distribution: Properties, Estimation, and Applications to Real Right-Censored Data

A new two-parameter power Zeghdoudi distribution (PZD) is suggested as a modification of the Zeghdoudi distribution using the power transformation method. As a result, the PZD may have increasing, decreasing, and unimodal probability density function and decreasing mean residual life function. In addition, other properties are presented, such as moments, order statistics, reliability measures, Bonferroni and Lorenz curves, Gini index, stochastic ordering, mean and median deviations, and quantile function. Following this, a section is devoted to the related model parameters which are estimated using the maximum likelihood estimation method, the weighted least squares and least squares methods, the maximum product of spacing method, the Cramer–von Mises method, and the right-tail and left-tail Anderson–Darling methods, and the Nikulin–Rao–Robson test statistic is considered. A simulation study is conducted to assess these methods and to investigate the distribution properties with right-censored data. The applicability of the proposed model is studied based on three real data sets of failure times, bladder cancer patients, and glass fiber data with a comparison with such competitors as the gamma, xgamma, Lomax, Darna, power Darna, power Lindley, and exponentiated power Lindley models. According to several established criteria, the comparative findings are overwhelmingly favorable to the suggested model.

Mean residual life order among largest order statistics arising from resilience-scale models with reduced scale parameters

In this paper, we identify some conditions to compare the largest order statistics from resilience-scale models with reduced scale parameters in the sense of mean residual life order. As an example of the established result, the exponentiated generalized gamma distribution is examined. Also, for the special case of the scale model, power-generalized Weibull and half-normal distributions are investigated.

Maximum Likelihood Estimation in the Inverse Weibull Distribution with Type II Censored Data

We consider maximum likelihood estimation for the parameters and certain functions of the parameters in the Inverse Weibull (IW) distribution based on type II censored data. The functions under consideration are the Mean Residual Life (MRL), which is very important in reliability studies, and Tail Value at Risk (TVaR), which is an important measure of risk in actuarial studies. We investigated the performance of the MLE of the parameters and derived functions under various experimental conditions using simulation techniques. The performance criteria are the bias and the mean squared error of the estimators. Recommendations on the use of the MLE in this model are given. We found that the parameter estimators are almost unbiased, while the MRL and TVaR estimators are asymptotically unbiased. Moreover, the mean squared error of all estimators decreased for larger sample sizes and it increased when the censoring proportion is increased for a fixed sample size. The conclusion is that the maximum likelihood method of estimation works well for the parameters and the derived functions of the parameter like the MRL and TVaR. Two examples on a real data set are presented to illustrate the application of the methods used in this paper. The first one is on survival time of pigs while the other is on fire losses.

A stochastic approach using Garima distribution

In this article we propose a power Garima distribution which includes Garima distribution as particular case. Its statistical properties including behavior of its probability density function for varying values of parameters, moments, hazard rate function and mean residual life function has been discussed. The estimation of the parameters of the distribution has been discussed using maximum likelihood estimation.

Topp–Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data

In this article, a four parameter lifetime model called the Topp–Leone modified Weibull distribution is proposed. The suggested distribution can be considered as an alternative to Kumaraswamy Weibull, generalized modified Weibull, extend odd Weibull Lomax, Weibull-Lomax, Marshall-Olkin alpha power extended Weibull and exponentiated generalized alpha power exponential distributions, etc. The suggested model includes the Topp-Leone Weibull, Topp-Leone Linear failure rate, Topp-Leone exponential and Topp-Leone Rayleigh distributions as a special case. Several characteristics of the new suggested model including quantile function, moments, moment generating function, central moments, mean, variance, coefficient of skewness, coefficient of kurtosis, incomplete moments, the mean residual life and the mean inactive time are derived. The probability density function of the Topp–Leone modified Weibull distribution can be right skewed and uni-modal shaped but, the hazard rate function may be decreasing, increasing, J-shaped, U-shaped and bathtub on its parameters. Three different methods of estimation as; maximum likelihood, maximum product spacing and Bayesian methods are used to estimate the model parameters. For illustrative reasons, applications of the Topp–Leone modified Weibull model to four real data sets related to medical and engineering sciences are provided and contrasted with the fit reached by several other well-known distributions.

Harmonic Mixture Fréchet Distribution: Properties and Applications to Lifetime Data

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.

Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications

In this paper, a new five-parameter model called alpha power exponentiated new Weibull-Pareto distribution is introduced based on a new developing technique. We derived some properties relating to the proposed distribution, including moments, moment generating function, quantile function, mean residual life and mean waiting time, and order statistics of the new model. The model parameters are estimated using the maximum likelihood method. Some simulation studies are performed to investigate the effectiveness of the estimates. Finally, we used three real-life data sets to show the flexibility of the introduced distribution.

Residual lifetime estimation for the mining truck tires

Reliability analysis has a very important role in supporting the operation of a system in harsh working conditions such as mines. Haul trucks are considering as a main asset of an open pit mine. Dump trucks are comprised of different sub-systems and among them, tires have a significant effect on the haulage economy. This study focused on the remaining life estimation of the mining truck tires considering the environmental operating conditions at an open pit iron ore mine in Iran. To achieve this, the proportional hazard model was used to estimate the conditional reliability functions and then the mean residual life of tires at the various initial survival times. The results of this study show that the ambient temperature has a significant effect on the reliability performance. By increasing the initial life of tires from 500 to 1000 h, the mean residual life will be decreased by 21% and 30% in low (below 15°C) and high (above 15°C) ambient temperature levels. These results are helpful to keep the truck tire performance at a desirable level by considering the appropriate preventive maintenance intervals.

Jackknife empirical likelihood ratio test for testing mean residual life and mean past life ordering

We propose nonparametric tests for testing mean residual life and mean past life ordering. Asymptotic properties of the proposed test statistics are studied. We also propose Jackknife Empirical Likelihood (JEL) ratio tests for testing mean residual life and mean past life ordering. Some numerical results are presented to evaluate the performance of the proposed tests by comparing them with few competing tests available in the literature. Finally, we illustrate the proposed test procedures using real data sets.

A note on the limiting behaviour of hazard rate functions of generalized mixtures

The mixtures are popular models in probability and statistics. They represent populations made up with different groups. Their distributions are linear combinations with positive weights of the baseline distribution functions. They can be extended by allowing that some of these weights could be negative. The linear combinations with some negative weights are called “negative mixtures” and they appear in different contexts. For example, the distribution of a coherent system can be represented as a negative mixture of the distributions of series (or parallel) systems made up with their components. The same happen with the finite sums of random variables both when they are independent (convolution) or dependent (C-convolution). In general, it is not easy to determine the shape of the reliability and hazard rate functions of mixtures. The same can be applied to their mean residual life functions. However, some results have been obtained for the limiting behaviour when the time goes on (i.e. t→∞). Here we provide new results in this direction that can be applied both to the classical case of mixtures with positive weights and the other cases that contains some negative weights. Some illustrative examples show that these new results allow us to solve cases that cannot be solved with the preceding findings.