Underlying Lifetime Distribution Research Articles

A novel lifetime analysis of repairable systems via Daubechies wavelets

AbstractIt is well-known that minimally repaired systems are characterized by non-homogeneous Poisson processes. When the underlying lifetime distribution function or its associated intensity function is completely known, the statistical inference of the cumulative number of failures/repairs is made through some standard techniques. On the other hand, when the parametric form of the lifetime distribution or the intensity function is not known, non-parametric inference techniques such as non-parametric maximum likelihood estimate (NPMLE) and kernel-based estimates are employed. Since these techniques are based on the underlying lifetime data, the resulting estimates of the cumulative number of failures/repairs with respect to the system operation time are discontinuous and troublesome in assessing the reliability measures such as reliability function and mean time to failure, etc. In this paper, we propose a novel lifetime analysis of the repairable systems via Daubechies wavelets. More specifically, we generalize the seminal Daubechies wavelet estimate by Kuhl and Bhairgond (Winter Simul Conf Proc 1:562–571, 2000) by combining the ideas on NPMLE and KMBs. In both simulation experiments and real data analyses, we investigate applicability of our Daubechies wavelet-based approaches and compare them with the conventional non-parametric methods.

Inference for Type-I and Type-II Hybrid Censored Minimal Repair and Record Data

In this paper, hybrid censoring mechanisms are applied to minimal repair and record data. Based on the derivation of the joint distribution of such data under hybrid censoring, likelihood inference is discussed. For illustration, Type-I and Type-II hybrid censoring schemes are considered for exponential distributions. In particular, the exact (conditional) distribution of the maximum likelihood estimator is obtained for an exponential distribution. This result is used to construct exact (conditional) confidence intervals using the method of pivoting the cumulative distribution function. Finally, the results are illustrated using two data sets taken from the literature on minimal repair models. Although the discussion of the results is in terms of minimal repair models, the results can be applied directly to record value data. By utilizing a connection of minimal repair times to occurrence times of non-homogeneous Poisson processes, a nonparametric estimate for the intensity rate of the process and the underlying lifetime distribution under hybrid censoring is also proposed. The paper is supplemented by simulational results.

Is There a Cap on Longevity? A Statistical Review

There is sustained and widespread interest in understanding the limit, if there is any, to the human life span. Apart from its intrinsic and biological interest, changes in survival in old age have implications for the sustainability of social security systems. A central question is whether the endpoint of the underlying lifetime distribution is finite. Recent analyses of data on the oldest human lifetimes have led to competing claims about survival and to some controversy, due in part to incorrect statistical analysis. This article discusses the particularities of such data, outlines correct ways of handling them, and presents suitable models and methods for their analysis. We provide a critical assessment of some earlier work and illustrate the ideas through reanalysis of semisupercentenarian lifetime data. Our analysis suggests that remaining life length after age 109 is exponentially distributed and that any upper limit lies well beyond the highest lifetime yet reliably recorded. Lower limits to 95% confidence intervals for the human life span are about 130 years, and point estimates typically indicate no upper limit at all.

Adapting The Extended Neyman’s Smooth Test to Be Used in Accelerated Failure Time Models

Accelerated life testing (ALT) has gained greater importance because of dealing with high reliability units. As a result, there is a big need to use a goodness of fit (GOF) technique for testing the underlying lifetime distribution. But there is a difficulty due to the existence of several stress levels with different samples of units at each level. Then, the choice of a certain GOF technique is based on its capability to combine the failure times from all stress levels to reach a conclusion about the adequacy of a certain lifetime distribution at each stress level. In this paper, the extended Neyman’s smooth test (ENST) is chosen. It is then modified in order to be used in validating the distributional assumption of accelerated failure time (AFT) model. This modified method is called; the adapted extended Neyman’s smooth test (AENST). It is applied to test for both Weibull and exponential distributions in case of constant stress under complete sampling. To check the performance of the AENST, a comparison is made with the conditional probability integral transformation test (CPITT) via a simulation study. Moreover, a real data set is provided to illustrate the application of the introduced AENST. The results revealed that the AENST is a powerful test comparing with the CPITT. Thus, the AENST is recommended for testing the AFT models.

A 3-Component Mixture of Exponential Distribution Assuming Doubly Censored Data: Properties and Bayesian Estimation

The output of an engineering process is the result of several inputs, which may be homogeneous or heterogeneous and to study them, we need a model which should be flexible enough to summarize efficiently the nature of such processes. As compared to simple models, mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of a process in survival analysis and reliability studies. Moreover, due to time and cost constraints, in the most lifetime testing experiments, censoring is an unavoidable feature. This article focuses on studying a mixture of exponential distributions, and we considered this particular distribution for three reasons. The first reason is its application in reliability modeling of electronic components and the second important reason is its skewed behavior. Similarly, the third and the most important reason is that exponential distribution has the memory-less property. In particular, we deal with the problem of estimating the parameters of a 3-component mixture of exponential distributions using type-II doubly censoring sampling scheme. The elegant closed-form expressions for the Bayes estimators and their posterior risks are derived under squared error loss function, precautionary loss function and DeGroot loss function assuming the noninformative (uniform and Jeffreys’) and the informative priors. A detailed Monte Carlo simulation and real data studies are carried out to investigate the performance (in terms of posterior risks) of the Bayes estimators. From results, it is observed that the Bayes estimates assuming the informative prior perform better than the noninformative priors.

Classification using sequential order statistics

Whereas discrimination methods and their error probabilities were broadly investigated for common data distributions such as the multivariate normal or t-distributions, this paper considers the case when the recorded data are assumed to be observations from sequential order statistics. Random vectors of sequential order statistics describe, e.g., successive failures in a k-out-of-n system or in other coherent and load sharing systems allowing for changes of underlying lifetime distributions caused by component failures. Within this framework, the Bayesian two-class discrimination approach with known prior probabilities and class parameters is considered, and exact and asymptotic formulas for the error probabilities in terms of Erlang and hypoexponential distributions are derived. Since the Bayesian classifier is closely related to Kullback–Leibler’s information distance, this approach is extended by invoking other divergence measures such as Jeffreys and Renyi’s distance. While exact formulas for the misclassification rates of the resulting distance-based classifiers are not available, inequalities among the corresponding error probabilities are derived. The performance of the applied classifiers is illustrated by some simulation results.

Statistical inference based on left truncated and interval censored data from log-location-scale family of distributions

Here, left truncated and interval censored data are analyzed by assuming that the underlying lifetime distribution belongs to log-location-scale family. In particular, lognormal and Weibull models are considered. Steps of stochastic expectation maximization (St-EM) algorithm are developed for the estimation of model parameters. MLEs are also obtained using Newton–Raphson method. Asymptotic confidence intervals for parameters are constructed using missing information principle, and parametric bootstrap approach. Through a simulation study, performances of proposed inferential methods are assessed. St-EM algorithm for point estimation and parametric bootstrap approach for constructing confidence intervals are recommended under this setup. Two datasets are analyzed for illustrative purpose. A prediction problem is also discussed.

Bayesian analysis for step‐stress accelerated life testing under progressive interval censoring

AbstractA step‐stress accelerated life testing model is considered for progressive type‐I censored experiments when the tested items are not monitored continuously but inspected at prespecified time points, producing thus grouped data. The underlying lifetime distributions belong to a general scale family of distributions. The points of stress‐level change are simultaneously inspection points as well while there is the option of assigning additional inspection points in between the stress‐level change points. In a Bayesian framework, the posterior distributions of the parameters of the model are derived for characteristic choices of prior distributions, as conjugate‐like and normal priors; vague or noninformative. The developed approach is illustrated on a simulated example and on a real data set, both known from the literature. The results are compared to previous analyses; frequentist or Bayes.

Hazard Rate Modeling of Step-Stress Experiments

Step-stress models form an essential part of accelerated life testing procedures. Under a step-stress model, the test units are exposed to stress levels that increase at intermediate time points of the experiment. The goal is to develop statistical inference for, e.g., the mean lifetime under each stress level, targeting to the extrapolation under normal operating conditions. This is achieved through an appropriate link function that connects the stress level to the associated mean lifetime. The assumptions made about the time points of stress level change, the termination point of the experiment, the underlying lifetime distributions, the type of censoring (if present), and the way of monitoring lead to alternative models. Step-stress models can be designed for single or multiple samples. We discuss recent developments in designing and analyzing step-stress models based on hazard rates. The inference approach adopted is mainly the maximum likelihood, but Bayesian approaches are briefly discussed.

Bayesian estimation of finite3-component mixture of Burr Type-XII distributions assuming Type-I right censoring scheme

As compared to simple models, the mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process. This study focuses on the problem of estimating the parameters of a newly developed 3-component mixture of Burr Type-XII distributions using Type-I right censored data. Firstly, considering a Bayesian structure, some mathematical properties of a 3-component mixture of Burr Type-XII distributions are discussed. These mathematical properties include Bayes estimators and posterior risks for the unknown component and proportion parameters using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function. Secondly, in case when no or little prior information is available, elicitation of hyperparameters is given. Also, the posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. Moreover, the limiting expressions for the Bayes estimators and posterior risks are derived. In addition, the performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. Finally, simulated datasets are designed for the different comparisons and the model is illustrated using the real data.

Innovative Planning in-Service Inspections of Fatigued Structures Under Parametric Uncertainty of Lifetime Models

The main aim of this paper is to present more accurate stochastic fatigue models for solving the fatigue reliability problems, which are attractively simple and easy to apply in practice for situations where it is difficult to quantify the costs associated with inspections and undetected cracks. From an engineering standpoint the fatigue life of a structure consists of two periods: (i) crack initiation period, which starts with the first load cycle and ends when a technically detectable crack is presented, and (ii) crack propagation period, which starts with a technically detectable crack and ends when the remaining cross section can no longer withstand the loads applied and fails statically. Periodic inspections of fatigued structures, which are common practice in order to maintain their reliability above a desired minimum level, are based on the conditional reliability of the fatigued structure. During the period of crack initiation, when the parameters of the underlying lifetime distributions are not assumed to be known, for effective in-service inspection planning (with decreasing intervals as alternative to constant intervals often used in practice for convenience in operation), the pivotal quantity averaging (PQA) approach is offered. During the period of crack propagation (when the damage tolerance situation is used), the approach, based on an innovative crack growth equation, to in-service inspection planning (with decreasing intervals between sequential inspections) is proposed to construct more accurate reliability-based inspection strategy in this case. To illustrate the suggested approaches, the numerical examples are given.

Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach

Age replacement strategies, where a unit is replaced upon failure or on reaching a predetermined age, whichever occurs first, provide simple and intuitively attractive replacement guidelines for technical units. Within theory of stochastic processes, the optimal preventive replacement age, in the sense of leading to minimal expected costs per unit of time when the strategy is used for a sequence of similar units over a long period of time, is derived by application of the renewal reward theorem. The mathematical solution to the problem of what is the optimal age for replacement is well known for the case when the parameter values of the underlying lifetime distributions are known with certainty. In actual practice, such is simply not the case. When these models are applied to solve real-world problems, the parameters are estimated and then treated as if they were the true values. The risk associated with using estimates rather than the true parameters is called estimation risk and is often ignored. When data are limited and (or) unreliable, estimation risk may be significant, and failure to incorporate it into the model design may lead to serious errors. Its explicit consideration is important since decision rules that are optimal in the absence of uncertainty need not even be approximately optimal in the presence of such uncertainty. In the present paper, for efficient optimization of statistical decisions under parametric uncertainty, the pivotal quantity averaging (PQA) approach is suggested. This approach represents a new simple and computationally attractive statistical technique based on the constructive use of the invariance principle in mathematical statistics. It allows one to carry out the transition from the original problem to the equivalent transformed problem (in terms of pivotal quantities and ancillary factors) via invariant embedding a sample statistic in the original problem. In this case, the statistical optimization of the equivalent transformed problem is carried out via ancillary factors. Unlike the Bayesian approach, the proposed approach is independent of the choice of priors. This approach allows one to eliminate unknown parameters from the problem and to find the better decision rules, which have smaller risk than any of the well-known decision rules. To illustrate the proposed approach, the numerical examples are given.

A 3-component mixture of Rayleigh distributions: properties and estimation in Bayesian framework.

To study lifetimes of certain engineering processes, a lifetime model which can accommodate the nature of such processes is desired. The mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process as compared to simple models. This paper is about studying a 3-component mixture of the Rayleigh distributionsin Bayesian perspective. The censored sampling environment is considered due to its popularity in reliability theory and survival analysis. The expressions for the Bayes estimators and their posterior risks are derived under different scenarios. In case the case that no or little prior information is available, elicitation of hyperparameters is given. To examine, numerically, the performance of the Bayes estimators using non-informative and informative priors under different loss functions, we have simulated their statistical properties for different sample sizes and test termination times. In addition, to highlight the practical significance, an illustrative example based on a real-life engineering data is also given.

Inference in a model of successive failures with shape-adjusted hazard rates

For successive failure times of components in a technical system, a flexible model based on sequential order statistics is proposed. Beyond the common assumption of proportionality, this model allows for structural adjustments of the hazard rates of the underlying lifetime distributions in situations, where failures have an impact on the entire shape of the hazard rate of remaining components. The hazard rates may be chosen, e.g., as strictly ordered bathtub curves. The general structure is analysed, and maximum likelihood estimators are stated for both, unrestricted and order restricted model parameters, as well as for parameters connected by a linear link function. Several properties of the estimators are obtained. Utilizing the maximum likelihood estimators, simultaneous confidence regions based on the Jeffreys–Kullback–Leibler distance and the Hellinger distance are examined.

Inference in step-stress models based on failure rates

In step-stress modeling with the cumulative exposure model it is well known that, for underlying exponential distributions, explicit expressions can be obtained for maximum likelihood estimators of parameters as well as for their conditional density functions or conditional moment generation functions, given their existence. Applying a failure rate based approach instead, similar results can be also obtained for underlying lifetime distributions out of a general scale family of distributions, which allows for a flexible modeling. Exemplarily, respective results are presented for Type-I and Type-II censored experiments.

Renewal and Renewal-Intensity Functions with Minimal Repair

The renewal and renewal-intensity functions with minimal repair are explored for the Normal, Gamma, Uniform, and Weibull underlying lifetime distributions. The Normal, Gamma, and Uniform renewal, and renewal-intensity functions are derived by the convolution method. Unlike these last three failure distributions, the Weibull except at shape β=1 does not have a closed-form function for the n-fold convolution. Since the Weibull is the most important failure distribution in reliability analyses, the approximate renewal and renewal-intensity functions of Weibull were obtained by the time-discretizing method using the Mean-Value Theorem for Integrals. A Matlab program outputs all reliability and renewal measures.

Analysis of Interval-Censored Data with Weibull Lifetime Distribution

In this work the analysis of interval-censored data, with Weibull distribution as the underlying lifetime distribution has been considered. It is assumed that censoring mechanism is independent and non-informative. As expected, the maximum likelihood estimators cannot be obtained in closed form. In our simulation experiments it is observed that the Newton-Raphson method may not converge many times. An expectation maximization algorithm has been suggested to compute the maximum likelihood estimators, and it converges almost all the times. The Bayes estimates of the unknown parameters under gamma priors are considered. If the shape parameter is known, the Bayes estimate of the scale parameter can be obtained in explicit form. When both the parameters are unknown, the Bayes estimators cannot be obtained in explicit form. Lindley’s approximation, importance sampling procedures and Metropolis Hastings algorithm are used to compute the Bayes estimates. Highest posterior density credible intervals of the unknown parameter are obtained using importance sampling technique. Small simulation experiments are conducted to investigate the finite sample performance of the proposed estimators, and the analysis of two data sets; one simulated and one real life, have been provided for illustrative purposes.

Parameter estimation from load-sharing system data using the expectation–maximization algorithm

This article considers a system of multiple components connected in parallel. As components fail one by one, the remaining working components share the total load applied to the system. This is commonly referred to as load sharing in the reliability engineering literature. This article considers the traditional approach to the modeling of a load-sharing system under the assumption of the existence of underlying hypothetical latent random variables. Using the Expectation–Maximization (EM) algorithm, a methodology is proposed to obtain the maximum likelihood estimates in such a model in the case where the underlying lifetime distribution of the components is lognormal or normal. The proposed EM method is also illustrated and substantiated using numerical examples. The estimates obtained using the EM algorithm are compared with those obtained using the Broyden–Fletcher–Goldfarb–Shanno algorithm, which falls under the class of numerical methods known as Newton or quasi-Newton methods. The results show that the estimates obtained using the proposed EM method always converge to a unique global maximizer, whereas the estimates obtained using the Newton-type method are highly sensitive to the choice of starting values and thus often fail to converge.

A covariance-based test for shared frailty in multivariate lifetime data

We decompose the score statistic for testing for shared finite variance frailty in multivariate lifetime data into marginal and covariance-based terms. The null properties of the covariance-based statistic are derived in the context of parametric lifetime models. Its non-null properties are estimated using simulation and compared with those of the score test and two likelihood ratio tests when the underlying lifetime distribution is Weibull. Some examples are used to illustrate the covariance-based test. A case is made for using the covariance-based statistic as a simple diagnostic procedure for shared frailty in a parametric exploratory analysis of multivariate lifetime data and a link to the bivariate Clayton–Oakes copula model is shown.

A two-stage group sampling plan based on truncated life tests for a general distribution

A two-stage group acceptance sampling plan based on a truncated life test is proposed, which can be used regardless of the underlying lifetime distribution when multi-item testers are employed. The decision upon lot acceptance can be made in the first or second stage according to the number of failures from each group. The design parameters of the proposed plan such as number of groups required and the acceptance number for each of two stages are determined independently of an underlying lifetime distribution so as to satisfy the consumer's risk at the specified unreliability. Single-stage group sampling plans are also considered as special cases of the proposed plan and compared with the proposed plan in terms of the average sample number and the operating characteristics. Some important distributions are considered to explain the procedure developed here.