A general accelerated life model for step-stress testing

In this paper we propose a general accelerated life model for step-stress testing and present a general likelihood function formulation for step-stress models that use both Weibull and lognormal distributions. The proposed model is also applicable to any life distribution in which the stress level only changes the scale parameter of the distribution, and can also be extended to multiple-stress as well as profiled testing patterns. Algorithms for fitting and testing such models are described and illustrated. The model provides a convenient way to interpret the step-stress accelerated life testing results. The practical use of the proposed statistical inference model is demonstrated by a case study.     

Beliefs about beliefs, a theory for stochastic assessments of subjective probabilities

Summary Parameterized families of subjective probability distributions can be used to great advantage to model beliefs of experts, especially when such models include dependence on concomitant variables. In one such model, probabilities of simple events can be expressed in loglinear form. In another, a generalization of the multivariatet distribution has concomitant variables entering linearly through the location vector. Interactive interview methods for assessing this second model and matrix extensions thereof were given in recent joint work of the author with A.P. Dawid, J.B. Kadane and others. In any such verbal assessment method, elicited quantiles must be fitted by subjective probability models. The fitting requires the use of a further probability model for errors of elicitation. This paper gives new theory relating the form of the distribution of elicited probabilities and elicited quantiles to the form of the subjective probability distribution. The first and second order moment structures are developed to permit generalized least squares fits. Present affiliation: State University of New York, Albany     

Generalized renewal process for repairable systems based on finite Weibull mixture

Repairable systems can be brought to one of possible states following a repair. These states are: ‘as good as new’, ‘as bad as old’ and ‘better than old but worse than new’. The probabilistic models traditionally used to estimate the expected number of failures account for the first two states, but they do not properly apply to the last one, which is more realistic in practice. In this paper, a probabilistic model that is applicable to all of the three after-repair states, called generalized renewal process (GRP), is applied. Simplistically, GRP addresses the repair assumption by introducing the concept of virtual age into the stochastic point processes to enable them to represent the full spectrum of repair assumptions. The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that a finite Weibull mixture, with positive component weights only, can be used as underlying distribution of the time to first failure (TTFF) of the GRP model, on condition that the unknown parameters can be estimated. To support the main idea, three examples are presented. In order to estimate the unknown parameters of the GRP model with m-fold Weibull mixture, the EM algorithm is applied. The GRP model with m mixture components distributions is compared to the standard GRP model based on two-parameter Weibull distribution by calculating the expected number of failures. It can be concluded that the suggested GRP model with Weibull mixture with an arbitrary but finite number of components is suitable for predicting failures based on the past performance of the system.     

Mathematical Theory of Probability and Statistics

Two test statistics are suggested for discriminating between the exponential model and the more general Weibull or gamma models, and these are compared to some previously used test statistics by Monte Carlo methods. The results of estimating reliability under an exponential assumption when the true model is Weibull is also investigated. These results as well as the tests mentioned above indicate that the exponential model is often not adequate when the more general models hold. In contrast to this result it was found that the Weibull model was quite robust relative to the generalized gamma distribution with regard to reliability estimation. Some general pivotal function properties are presented for the maximum likelihood estimator of reliability for the generalized gamma distribution and similar results also hold for the Weibull procedure under a generalized gamma assumption. These results made a Monte Carlo study of this problem feasible. Since the maximum likelihood estimators are apparently ill-behaved for smaller sample sizes and since the Weibull model is robust it appears little is gained by using the generalized gamma distribution for samples of size less than 200 to 400.     

Reliability analysis incorporating exponentiated inverse Weibull distribution and inverse power law

In reliability research, electronic devices are an important part of our lives and modelling their lives is the most difficult and fascinating area. To investigate the failure functioning of electronic equipments, reliability monitoring of systems is widely used. However, it is stated in the literature that one in five electronic system collapses are a consequence of degradation and saving energy and forecasting future losses, it is necessary to summarize the data through certain versatile models of probability . In current article, a model of reliability formed on inverse power law and generalized inverse Weibull model is suggested. This current distribution presents a clearer framework to modelling the efficiency and functionality lifespan of electronic equipments. In this article, an empirical analysis is discussed related to life cycle of a surface-mounted electrolytic capacitor (SMEC). In addition, it has noticed that evaluation of suggested distribution varies from classical model of inverse Weibull and that influences average time to failure (ATTF) of the studied capacitor.     

On the economics of cutting tools

The determination of the optimal cutting conditions for machine tools is a familiar problem in the engineering management literature. Two relatively recent papers have pointed out that the cost parameters included in models used to suggest cutting speeds and tool life have traditionally been mis-specified. In this paper we develop a generalized model of the situation, from which the approaches taken in the two previous papers and an earlier work are shown to emerge as special cases     

An anisotropic damage model for tensile fatigue

Fatigue damage in materials is considered to be the effect of material degradation, and the dispersion in fatigue life is attributed to variability in microstructure. This paper presents a numerical model to simulate fatigue damage evolution using continuum damage mechanics to characterize material degradation. An explicit microstructure topology representation is achieved using Voronoi tessellations. Unlike conventional models which use a scalar approximation for damage, this model treats the damage variable as an anisotropic tensor. The model is used to simulate tensile fatigue failure in thin steel specimen. The fatigue life estimations from the model compares well with published experimental results. The results predict a high variability in fatigue life that is characteristic of metals and alloys, as compared with the existing isotropic damage models available in the literature. The model was also used to study the influence of material inhomogeneity on fatigue life dispersion.     

A Method for Discriminating Between Failure Density Functions Used In Reliability Predictions

Unless sufficient evidence to the contrary exists, the exponential distribution is often assumed as a model for the failure density function in reliability predictions.

The generalized gamma distribution, with known location parameter, is a three parameter distribution which encompasses the exponential, Weibull, gamma and many others. In this paper, (i) maximum likelihood estimation for the three parameters is indicated, (ii) it is noted that these estimators are asymptotically multivariate normally distributed, and (iii) using the distribution of the estimators, probability regions for the estimators of the parameters of the generalized gamma distribution are established for large sample situations.

In situations where the generalized gamma can be assumed as the correct density function, the exponential and the Weibull are special cases. A method is presented using experimental or life data for rejecting (with a known probability of false rejection) the Weibull and (or) the exponential functions when they do not appear to describe the failure density function of a unit.     

Generalized mixtures of Weibull components

Weibull mixtures have been used extensively in reliability and survival analysis, and they have also been generalized by allowing negative mixing weights, which arise naturally under the formation of some structures of reliability systems. These models provide flexible distributions for modeling dependent lifetimes from heterogeneous populations. In this paper, we study conditions on the mixing weights and the parameters of the Weibull components under which the considered generalized mixture is a well-defined distribution. Specially, we characterize the generalized mixture of two Weibull components. In addition, some reliability properties are established for these generalized two-component Weibull mixture models. One real data set is also analyzed for illustrating the usefulness of the studied model.     

Comparative study of quantitative models for hardware,software and human reliability assessment

Various models which may be used for quantitative assessment of hardware, software and human reliability are compared in this paper. Important comparison criteria are the system life cycle phase in which the model is intended to be used, the failure category and reliability means considered in the model, model purpose, and model characteristic such as model construction approach, model output and model input. The main objective is to present limitations in the use of current models for reliability assessment of computer-based safety shutdown systems in the process industry and to provide recommendations on further model development. Main attention is given to presenting the overall concept of various models from a user's point of view rather than technical details of specific models. A new failure classification scheme is proposed which shows how hardware and software failures may be modelled in a common framework.     

Implicit constitutive modelling for viscoplasticity using neural networks

Up to now, a number of models have been proposed and discussed to describe a wide range of inelastic behaviours of materials. The fatal problem of using such models is however the existence of model errors, and the problem remains inevitably as far as a material model is written explicitly. In this paper, the authors define the implicit constitutive model and propose an implicit viscoplastic constitutive model using neural networks. In their modelling, inelastic material behaviours are generalized in a state-space representation and the state-space form is constructed by a neural network using input–output data sets. A technique to extract the input–output data from experimental data is also described. The proposed model was first generated from pseudo-experimental data created by one of the widely used constitutive models and was found to replace the model well. Then, having been tested with the actual experimental data, the proposed model resulted in a negligible amount of model errors indicating its superiority to all the existing explicit models in accuracy. © 1998 John Wiley & Sons, Ltd.     

Discrimination between accelerated life models via Approximate Bayesian Computation

Accelerated life testing (ALT) is widely used in high-reliability product estimation to get relevant information about an item's performance and its failure mechanisms. To analyse the observed ALT data, reliability practitioners need to select a suitable accelerated life model based on the nature of the stress and the physics involved. A statistical model consists of (i) a lifetime distribution that represents the scatter in product life and (ii) a relationship between life and stress. In practice, several accelerated life models could be used for the same failure mode and the choice of the best model is far from trivial. For this reason, an efficient selection procedure to discriminate between a set of competing accelerated life models is of great importance for practitioners. In this paper, accelerated life model selection is approached by using the Approximate Bayesian Computation (ABC) method and a likelihood-based approach for comparison purposes. To demonstrate the efficiency of the ABC method in calibrating and selecting accelerated life model, an extensive Monte Carlo simulation study is carried out using different distances to measure the discrepancy between the empirical and simulated times of failure data. Then, the ABC algorithm is applied to real accelerated fatigue life data in order to select the most likely model among five plausible models. It has been demonstrated that the ABC method outperforms the likelihood-based approach in terms of reliability predictions mainly at lower percentiles particularly useful in reliability engineering and risk assessment applications. Moreover, it has shown that ABC could mitigate the effects of model misspecification through an appropriate choice of the distance function.     

Reliability Data Analysis for Life Test Designed Experiments with Sub‐Sampling

The ability to model lifetime data from life test experiments is of paramount importance to all manufacturers, engineers and consumers. The Weibull distribution is commonly used to model the data from life tests. Standard Weibull analysis assume completely randomized designs. However, not all life test experiments come from completely randomized designs. Experiments involving sub‐sampling require a method for properly modeling the data. We provide a Weibull nonlinear mixed models (NLLMs) methodology for incorporating random effects in the analysis. We apply this methodology to a reliability life test on glass capacitors. We compare the NLLMs methodology to other available methods for incorporating random effects in reliability analysis. A simulation study reveals the method proposed in this paper is robust to both model misspecification and increasing levels of variance on the random effect. Copyright © 2012 John Wiley & Sons, Ltd.     

Fatigue damage assessment for a spectral model of non-Gaussian random loads

In this paper, a new model for random loads–the Laplace driven moving average–is presented. The model is second order, non-Gaussian, and strictly stationary. It shares with its Gaussian counterpart the ability to model any spectrum but has additional flexibility to model the skewness and kurtosis of the marginal distribution. Unlike most other non-Gaussian models proposed in the literature, such as the transformed Gaussian or Volterra series models, the new model is no longer derivable from Gaussian processes. In the paper, a summary of the properties of the new model is given and its upcrossing intensities are evaluated. Then it is used to estimate fatigue damage both from simulations and in terms of an upper bound that is of particular use for narrowband spectra.     

The weighted priors approach for combining expert opinions in logistic regression experiments

ABSTRACT

When modeling the reliability of a system or component, it is not uncommon for more than one expert to provide very different prior estimates of the expected reliability as a function of an explanatory variable such as age or temperature. Our goal is to incorporate all information from the experts when choosing a design about which units to test. Bayesian design of experiments has been shown to be very successful for generalized linear models, including logistic regression models. We use this approach to develop methodology for the case where there are several potentially non-overlapping priors under consideration. While multiple priors have been used for analysis in the past, they have never been used in a design context. The Weighted Priors method performs well for a broad range of true underlying model parameter choices and is more robust when compared to other reasonable design choices. We illustrate the method through multiple scenarios and a motivating example. Additional figures for this article are available in the online supplementary information.     

Accelerated Life Testing (ALT) Design Based on Computational Reliability Analysis

Accelerated life testing (ALT) design is usually performed based on assumptions of life distributions, stress–life relationship, and empirical reliability models. Time‐dependent reliability analysis on the other hand seeks to predict product and system life distribution based on physics‐informed simulation models. This paper proposes an ALT design framework that takes advantages of both types of analyses. For a given testing plan, the corresponding life distributions under different stress levels are estimated based on time‐dependent reliability analysis. Because both aleatory and epistemic uncertainty sources are involved in the reliability analysis, ALT data is used in this paper to update the epistemic uncertainty using Bayesian statistics. The variance of reliability estimation at the nominal stress level is then estimated based on the updated time‐dependent reliability analysis model. A design optimization model is formulated to minimize the overall expected testing cost with constraint on confidence of variance of the reliability estimate. Computational effort for solving the optimization model is minimized in three directions: (i) efficient time‐dependent reliability analysis method; (ii) a surrogate model is constructed for time‐dependent reliability under different stress levels; and (iii) the ALT design optimization model is decoupled into a deterministic design optimization model and a probabilistic analysis model. A cantilever beam and a helicopter rotor hub are used to demonstrate the proposed method. The results show the effectiveness of the proposed ALT design optimization model. Copyright © 2015 John Wiley & Sons, Ltd.     

A sequential constant‐stress accelerated life testing scheme and its Bayesian inference

In the analysis of accelerated life testing (ALT) data, some stress‐life model is typically used to relate results obtained at stressed conditions to those at use condition. For example, the Arrhenius model has been widely used for accelerated testing involving high temperature. Motivated by the fact that some prior knowledge of particular model parameters is usually available, this paper proposes a sequential constant‐stress ALT scheme and its Bayesian inference. Under this scheme, test at the highest stress is firstly conducted to quickly generate failures. Then, using the proposed Bayesian inference method, information obtained at the highest stress is used to construct prior distributions for data analysis at lower stress levels. In this paper, two frameworks of the Bayesian inference method are presented, namely, the all‐at‐one prior distribution construction and the full sequential prior distribution construction. Assuming Weibull failure times, we (1) derive the closed‐form expression for estimating the smallest extreme value location parameter at each stress level, (2) compare the performance of the proposed Bayesian inference with that of MLE by simulations, and (3) assess the risk of including empirical engineering knowledge into ALT data analysis under the proposed framework. Step‐by‐step illustrations of both frameworks are presented using a real‐life ALT data set. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal design of partially accelerated life tests for the lognormal distribution under type I censoring

This paper considers optimal designs of partially accelerated life tests in which test items are first run simultaneously at use condition for a specified time, and the surviving items are then run at accelerated condition until a predetermined censoring time. For items having lognormally distributed lives maximum likelihood estimators (MLEs) of the location and scale parameters of the lifetime distribution at use condition, and the acceleration factor which is the ratio of the mean life at use condition to that at accelerated condition are obtained. The change time is determined to minimize either the asymptotic variance of MLE of the acceleration factor or the generalized asymptotic variance of MLEs of the model parameters.     

Parameter estimation for bivariate Weibull distribution using generalized moment method for reliability evaluation

Bivariate Weibull distribution can address the life of a system exhibiting 2‐dimensional characteristics in risk and reliability engineering. The applicability of bivariate Weibull distribution has been hindered by its difficulty with parameter estimation, as the number of parameters in bivariate Weibull distribution is more than those in univariate Weibull distribution. Considering a particular structure of a bivariate Weibull distribution model, this paper proposes a generalized moment method (GMM) for parameter estimation. This GMM method is simple, and it has proved to be efficient. The GMM can guarantee the existence and the uniqueness of the solution. A confidence interval for each estimator is derived from the moments of the bivariate distribution. The paper presents a simulation case and 2 real cases to demonstrate the proposed methods.     

Some recent theory for autoregressive count time series

In this paper an overview is given over recent theoretical developments in autoregressive count time series. The focus is on generalized autoregressive models where the autoregressive structure is incorporated via a link function. Starting from an ordinary autoregressive model the difficulties in extending standard theory of statistical inference to count time series are highlighted. Special attention is given to the issues of ergodicity and asymptotic theory of estimation. Two main approaches are mentioned, a perturbation approach and the use of a weak dependence concept. The main emphasis is on the former. Linear as well as log-linear and nonlinear models are treated. It is argued that the developed theory forms a necessary basis for modelling and application of these count time series. The setting of the paper is one of simple models and conditional distributions of Poisson type. But it is claimed that the framework is general enough to handle many extensions with an accompanying flexibility in applications of these models.