Exact Failure Times Research Articles

Reliability analysis based on doubly-truncated and interval-censored data

Field data provide important information on product reliability. Interval sampling is widely used for collection of field data, which typically report incident cases during a certain time period. Such sampling scheme induces doubly truncated (DT) data if the exact failure time is known. In many situations, the exact failure date is known only to fall within an interval, leading to doubly truncated and interval censored (DTIC) data. This article considers analysis of DTIC data under parametric failure time models. We consider a conditional likelihood approach and propose interval estimation for parameters and the cumulative distribution functions. Simulation studies show that the proposed method performs well for finite sample size.

Robust estimation of dependent competing risk model under interval monitoring and determining optimal inspection intervals

AbstractRecently, a growing interest is evident in modelling dependent competing risks in lifetime prognosis problems. In this work, we propose to model the dependent competing risks by Marshal‐Olkin bivariate exponential distribution. The observable data consists of a number of failures due to different causes across different time intervals. The failure count data is common in instances like one‐shot devices where the state of the subjects is inspected at different inspection times rather than the exact failure times. The point estimation of the lifetime distribution in the presence of competing risk has been studied through a divergence‐based robust estimation method called minimum density power divergence estimation (MDPDE) with and without constraint. The optimal value of the tuning parameter has been obtained. The testing of the hypothesis is performed based on a Wald‐type test statistic. The influence function is derived for the point estimator and the test statistic, reflecting the degree of robustness. Another key contribution of this work is determining the optimal inspection times based on predefined objectives. This article presents the determination of multi‐criteria‐based optimal design. Population‐based heuristic algorithm nondominated sorting‐based multiobjective Genetic algorithm is exploited to solve this optimization problem.

Bayesian filter for failure times identification of moving heat sources in 2D geometry

This study investigates the application of the Bayesian filter method for identifying failure times in a 2D parabolic partial differential equation system. The identification of failure times in thermal systems, which are subject to partial differential equations, presents significant difficulties, especially due to their ill-posed nature, which makes them highly sensitive to measurement errors. A Bayesian inference framework was previously developed in a related study, aiming to solve inverse heat conduction problems by utilising temperature measurements from sensors to estimate failure times or potential restarts of fixed heat sources. This paper focuses on the case of mobile sources, where a set of fixed sensors is considered and the trajectories of the heating sources are known and follow a constant velocity. The main objective is to accurately identify the failing heat sources and determine the exact failure time, as well as the possibility of resuming normal operation. A Monte Carlo simulation is performed to assess the impact of sensor measurements.

Semiparametric probit regression model with misclassified current status data.

Current status data arise when each subject under study is examined only once at an observation time, and one only knows the failure status of the event of interest at the observation time rather than the exact failure time. Moreover, the obtained failure status is frequently subject to misclassification due to imperfect tests, yielding misclassified current status data. This article conducts regression analysis of such data with the semiparametric probit model, which serves as an important alternative to existing semiparametric models and has recently received considerable attention in failure time data analysis. We consider the nonparametric maximum likelihood estimation and develop an expectation-maximization (EM) algorithm by incorporating the generalized pool-adjacent-violators (PAV) algorithm to maximize the intractable likelihood function. The resulting estimators of regression parameters are shown to be consistent, asymptotically normal, and semiparametrically efficient. Furthermore, the numerical results in simulation studies indicate that the proposed method performs satisfactorily in finite samples and outperforms the naive method that ignores misclassification. We then apply the proposed method to a real dataset on chlamydia infection.

Step-stress tests for interval-censored data under gamma lifetime distribution

Inferential methods under extreme form of censoring are of interest in reliability theory because of their applicability to practical engineering problems. interval-censored data naturally appear in many situations wherein the exact failure times cannot be observed, but we can only know if the product has failed before a certain inspection time or not. In addition, some products are highly reliable with large mean lifetimes under normal operating conditions, and so accelerated lifetime tests (ALTs) need to be carried out for inferential purposes. Step-stress ALTs increase such stress factors at certain pre-fixed times, thus inducing early failure. Likelihood-based estimation methods are known to have a very good performance in the absence of contamination in the data, but they perform poorly when few observations do not follow the same distribution as the rest of the observations. In this work, we develop robust estimators and Wald-type test statistics based on density power divergence (DPD) method under step-stress ALT model and gamma lifetime distribution based on interval-censored test data. The performance of the proposed estimators and test statistics are then empirically examined through an extensive simulation study. Finally, the usefulness of the developed inferential methods is illustrated with a real kinetics data.

Attention-based deep survival model for time series data

In the era of internet of things and Industry 4.0, smart products and manufacturing systems emit signals tracking their operating condition in real-time. Survival analysis shows its strength in modeling such signals to determine the condition of in-service equipment and products to yield critical operational decisions, i.e., maintenance and repair. One appealing aspect of survival analysis is the possibility to include subjects in the model which did not have their failure yet or when the exact failure time is unknown.NN-based survival models, i.e., deep survival models, show superior performance in modeling the non-linear relationship between the reliability function and covariates. We propose a novel deep survival model, seq2surv, to incorporate the seq2seq structure and attention mechanism to enhance the ability to analyze a sequence of signals in the survival analysis. Similar to the seq2seq model which shows superior performance in machine translation, we designed the seq2surv model to translate from a sequence of signals to a sequence of survival probabilities and to update the reliability predictions along with real-time monitoring. Our results show that the seq2surv model outperforms existing deep survival approaches in terms of higher prediction accuracy and lower errors in the survival function estimation on both simulated and real-world datasets.

Using the accelerated failure time model to analyze current status data with misclassified covariates

Current status data arise commonly in applications when there is only one feasible observation time to check if the failure time has occurred, but the exact failure time remains unknown. To accommodate the covariate effect on failure time, the accelerated failure time (AFT) model has been widely used to analyze current status data with the distribution of the failure time assumed to be specified or unspecified. In this paper, we consider a logistic regression with a misclassfied covariate from the current status observation scheme. A semiparametric AFT model was built to model current status data to eliminate the bias caused by this misclassification. This model is also robust to the misspecification of the failure time compared to the parametric AFT model, as we assume an unknown distribution of the failure time in the proposed model. Furthermore, incorporating the covariate effect on the failure time increases the flexibility of the model. Finally, we adapt the Expectation-Maximization algorithm for estimation, which guarantees the convergence of the estimate. Both theory and empirical studies show the consistency of the estimator.

Optimal planning of progressively Type-I censored step-stress accelerated life test under interval inspection

For highly reliable products and devices, conducting the life tests at normal usage conditions demands a great deal of time and cost, which can result in missed opportunities to introduce the product to the market in a timely manner and eventually loss of the market share. By subjecting test units at higher stress levels than normal operating conditions, accelerated life tests solve this problem and quickly produce information on the lifetime parameters. The lifetime at the design stress is then estimated through extrapolation using a regression model. Due to technical limitations and/or budgetary constraints, the exact failure times of test units may not be available in practice but only the failure counts are collected at certain time points during the life test. In this work, the design optimization of a general k-level step-stress accelerated life test under progressive Type-I censoring is considered with a uniform step duration Δ when the exact failure times are not observable ( interval inspection). Allowing the intermediate censoring at each stress change time point (viz., ), the optimal Δ is determined under various design criteria including D-optimality, T-optimality, C-optimality, A-optimality, and E-optimality. The existence of these optimal designs is investigated in detail for exponential lifetimes with a single stress variable, and the design efficiency is compared to the situation when the exact failure times are available ( continuous inspection).

Design optimization of a simple step-stress accelerated life test – Contrast between continuous and interval inspections with non-uniform step durations

Thanks to continuously advancing technology and manufacturing processes, the products and devices are becoming highly reliable. However, performing the life tests of these products at normal operating conditions becomes extremely difficult, if not impossible, due to their long lifespans. This can result in missed opportunities to introduce the products to the market in a timely manner and eventually loss of the market share. This problem is solved by accelerated life tests where the test units are subjected to higher stress levels than the normal usage level so that information on the lifetime parameters can be obtained more quickly. The lifetime at the design condition is then estimated through extrapolation using a regression model. Although continuous inspection of the exact failure times is an ideal mode, the exact failure times of test units may not be available in practice due to technical limitations and/or budgetary constraints, but only the failure counts are collected at certain time points during the test (i.e., interval inspection). In this work, the design optimization of a simple step-stress accelerated life test under progressive Type-I censoring is studied with non-uniform step durations for assessing the reliability characteristics of a solar lighting device. Allowing the intermediate censoring to take place at the stress change time point, the nature of the optimal stress duration is demonstrated under various design criteria including D-optimality, T-optimality, C-optimality, A-optimality, and E-optimality. The existence of these optimal designs is investigated in detail for exponential lifetimes with a single stress variable, and the effect of the intermediate censoring proportion on the design efficiency is compared between the continuous and interval inspection modes.

Expected termination times of progressively Type‐I censored step‐stress accelerated life tests under continuous and interval inspections

A step‐stress accelerated life test is a special life test where test units are subjected to higher stress levels than normal operating conditions so that the information on the lifetime parameters of a test unit can be obtained more quickly in a shorter period of time. Also, progressive Type‐I censoring is a generalized form of time censoring where functional test units are withdrawn successively from the life test at some prefixed nonterminal time points. Despite its flexibility and efficient utilization of the available resources, progressively censored sampling has not gained much popularity due to its analytical complexity compared to the conventional censoring schemes. In particular, understanding the mean completion time of a life test is of great practical interest in order to design and manage the life test optimally under frequent budgetary and time constraints. In this work, the expected termination time of a general k‐level step‐stress accelerated life test under progressive Type‐I censoring is derived using a recursive relationship of the stochastic termination time based on the conditional block independence. To be comprehensive, two different modes of failure inspections are considered: continuous inspection, where the exact failure times are observed, and interval inspection, where the exact failure times are not available but only the number of failures that occurred.

Assessing the effect of repair delays on a repairable system

Repair delays are common in systems where failures are not self-revealed or in mission-critical systems where repairs cannot be conducted during a mission. The presence of repair delays may significantly affect the failure behavior of a system, but its impact has been largely overlooked in the literature. In this article, we propose a flexible intensity-based model for repairable systems subject to repair delays. Statistical inference of the model is discussed. In the presence of repair delays, the exact failure times are typically unavailable and the failures are only known to occur at times within intervals with known lower and upper bounds. In this case, the likelihood function is complicated. We introduce several efficient numerical integration methods to evaluate the likelihood function and investigate their performance through comprehensive simulations. Goodness-of-fit tests are used to check the adequacy of the baseline rate of occurrence of failures and the effect of repair delays. The proposed methods are demonstrated using maintenance data of a general infusion pump used in a hospital.

The prediction methodology for tire’s high speed durability regulation test using a finite element method

In the tire industry, indoor accelerated life tests as regulations have been performed to ensure tire durability performances instead of outdoor field test. The finite element method has been widely used to minimize real test time and cost, but prediction method for accelerated life test had hardly been made in the past. This study presents a rational methodology to predict the tire’s failure life at the steel belt edge region due to high speed regulation test. Based on the finite element analysis and fatigue characteristic of rubber material, a method to determine exact failure time is proposed. The steady-state rolling analysis by FEM to get strain energy density range(ΔSED) at the steel belt edge region and fatigue test of rubber compound to obtain ΔSED- number of cycle (Nf) curve were used. The reliability of proposed prediction method was verified by real indoor test.

Comparison among several commonly used sampling methods for a degradation test

AbstractDegradation testing is an effective tool for evaluating the reliability of highly reliable products. There have been many data collection methods proposed in the literature. Some of these assumed that only degradation values are recorded, and some assumed failure times to be available. However, most research has been devoted to proposing parameter estimates or to designing degradation tests for a specific sampling method. The differences between these commonly used methods have rarely been investigated. The lack of comparisons between different sampling methods has made it difficult to select an appropriate means by which to collect data. In addition, it remains unclear whether obtaining extra information (eg, exact failure times) is useful for making statistical inferences. In this paper, we assume that the degradation path of a product follows a Wiener degradation process, and we summarize several data collection methods. Maximum likelihood estimates for parameters and their variance‐covariance matrices are derived for each type of data. Several commonly used optimization criteria for designing a degradation test are used to compare estimation efficiency. Sufficient conditions under which one method could be better than the others are proposed. Upper bounds of estimation efficiency are also investigated. Our results provide useful guidelines by which to choose a sampling method, as well as its design variables, to obtain efficient estimation. A simulated example based on real light‐emitting diodes data is studied to verify our theoretical results under a moderate sample size scenario.

Bayesian estimation of the Cox model under different hazard rate shape assumptions via slice sampling

ABSTRACTIn this paper, we provide a full Bayesian analysis for Cox's proportional hazards model under different hazard rate shape assumptions. To this end, we select the modified Weibull distribution family to model failure rates. A novel Markov chain Monte Carlo method allows one to tackle both exact and right-censored failure time data. Both simulated and real data are used to illustrate the methods.

Goodness-of-fit test for the parametric proportional hazard regression model with interval-censored data

ABSTRACTInterval-censored data are common in medical research. Fully parametric models provide simple and efficient inference for the estimation of survival functions using interval-censored observations. Inference based on a parametric regression model requires the complete specification of the probability density function, and therefore, correctly specifying the model is crucial, while the regression diagnostic is a very important step. However, regression diagnostic methods for use with the interval-censored data have not been completely developed. Here, we developed a model-checking procedure based on the cumulative martingale residuals for the interval-censored observations. We employed the conditional expectation of residuals for the diagnostics, because the data showing the exact failure time cannot be obtained for the interval-censoring analyses, and developed the formal resampling-based supremum-type test and graphical model-checking techniques. A simulation study demonstrated an excellent performance of the proposed method during the detection of a misspecified functional form of covariates in the finite sample. Furthermore, we used this method for the analysis of the medical checkup data obtained in Japan.

Regression analysis of interval censored competing risk data using a pseudo-value approach

Interval censored data often occur in an observational study where the subject is followed periodically. Instead of observing an exact failure time, two inspection times that include it are available. There are several methods to analyze interval censored failure time data (Sun, 2006). However, in the presence of competing risks, few methods have been suggested to estimate covariate effect on interval censored competing risk data. A subdistribution hazard model is a commonly used regression model because it has one-to-one correspondence with a cumulative incidence function. Alternatively, Klein and Andersen (2005) proposed a pseudo-value approach that directly uses the cumulative incidence function. In this paper, we consider an extension of the pseudo-value approach into the interval censored data to estimate regression coefficients. The pseudo-values generated from the estimated cumulative incidence function then become response variables in a generalized estimating equation. Simulation studies show that the suggested method performs well in several situations and an HIV-AIDS cohort study is analyzed as a real data example.

Quantile regression models for current status data

Current status data arise frequently in demography, epidemiology, and econometrics where the exact failure time cannot be determined but is only known to have occurred before or after a known observation time. We propose a quantile regression model to analyze current status data, because it does not require distributional assumptions and the coefficients can be interpreted as direct regression effects on the distribution of failure time in the original time scale. Our model assumes that the conditional quantile of failure time is a linear function of covariates. We assume conditional independence between the failure time and observation time. An M-estimator is developed for parameter estimation which is computed using the concave–convex procedure and its confidence intervals are constructed using a subsampling method. Asymptotic properties for the estimator are derived and proven using modern empirical process theory. The small sample performance of the proposed method is demonstrated via simulation studies. Finally, we apply the proposed method to analyze data from the Mayo Clinic Study of Aging.

Estimation in Maxwell distribution with randomly censored data

In many practical situations, complete data are not available in lifetime studies. Many of the available observations are right censored giving survival information up to a noted time and not the exact failure times. This constitutes randomly censored data. In this paper, we consider Maxwell distribution as a survival time model. The censoring time is also assumed to follow a Maxwell distribution with a different parameter. Maximum likelihood estimators and confidence intervals for the parameters are derived with randomly censored data. Bayes estimators are also developed with inverted gamma priors and generalized entropy loss function. A Monte Carlo simulation study is performed to compare the developed estimation procedures. A real data example is given at the end of the study.

A new approach for Weibull modeling for reliability life data analysis

This paper presents a proposed approach for modeling the life data for system components that have failure modes by different Weibull models. This approach is applied for censored, grouped and ungrouped samples. To support the main idea, numerical applications with exact failure times and censored data are implemented. The parameters are obtained by different computational statistical methods such as graphic method based on Weibull probability plot (WPP), maximum likelihood estimates (MLE), Bayes estimators, non-linear Benard’s median rank regression. This paper also presents a parametric estimation method depends on expectation–maximization (EM) algorithm for estimation the parameters of finite Weibull mixture distributions. GOF is used to determine the best distribution for modeling life data. The performance of the proposed approach to model lifetime data is assessed. It’s an efficient approach for moderate and large samples especially with a heavily censored data and few exact failure times.

Analyzing Highly Censored Reliability Data without Exact Failure Times: An Efficient Tool for Practitioners

ABSTRACT There are many real-life situations where exact failure times are not available or not easily available for reliability analysis. Motivated by this problem, this work is concerned with the analysis of reliability data without exact failure times. The unavailability of exact failure times has compelled us to develop a simplified version of the maximum likelihood (ML) estimation for reliability parameters and measures. The approach is applicable to single-parameter reliability models including the exponential reliability model and the Weibull reliability model with an assumed or known shape parameter value. In the latter case, it takes advantage of the fact that in many practical situations a reasonable estimate of the Weibull shape parameter is attainable by certain means. This is particularly valuable for reliability assessment of highly censored Weibull data with only a few failures, because in such situations it is highly desirable or necessary to exploit prior knowledge of the Weibull shape parameter to compensate for the limited information contained in the data in the hope of making a sound assessment. This is an application in a sense of the ideas and principles of the emerging paradigm of statistical engineering. The methods are developed by practitioners and for practitioners. We show that they offer practitioners a handy and efficient tool for reliability analysis, using hard disk drive product testing as examples.