Nonparametric inferences for ramp stress tests under random censoring

This paper considers nonparametric estimation of lifetime distribution of a product under ramp stress accelerated life tests in which the stress on an item increases linearly with time and observations are randomly censored. Assuming that a cumulative exposure model holds, the lifetime distribution of an item under ramp stress is derived. Three nonparametric estimators of the lifetime distribution at use condition stress are obtained for the situation where the time transformation function relating stress to lifetime distributions of a test item is a version of the inverse power law. The proposed estimators are robust to underlying lifetime distribution and are computed in closed form. They are compared with maximum likelihood estimator for small samples under exponential lifetime distribution. The method is extended to the case of competing risks.     

Inverse Gaussian process model with frailty term in reliability analysis

Traditional reliability analysis techniques focus on the occurrence of failures over time. Nevertheless, in certain cases where the occurrence of failures is tiny or almost null, the estimation of the quantities that describe the failure process is compromised. In this context, we introduce a reliability model for systems adopting the degradation process using frailty. The evolved degradation model has as experimental data, not the failure, but a quality feature attached to it. Degradation analysis can provide information about the lifetime distribution components without actually observing failures. In this paper, we propose an inverse Gaussian process model with frailty as a possible tool to investigate the effect of unobserved covariates. Moreover, a comparative study with the classical inverse Gaussian process based on simulated data was performed, revealing that the asymptotic properties of the maximum likelihood estimators are compromised when the presence of frailty is ignored. The application was based on two real data sets in the literature, showing that the inverse Gaussian process frailty models are propitious to use; however, gamma and inverse Gaussian distributions for frailty present similar results.     

Minimum Variance Unbiased Estimators for Poisson Probabilities

This paper considers the problem of estimating Poisson probabilities or relative frequencies and some extensions of that problem. It is shown how minimum variance unbiased estimators based on a simple random sample of 72 observations on a Poisson process may be easily developed. Variances of the estimators and estimators for their variances are derived. Comparisons with maximum likelihood estimators and with distribution-free relative frequency estimators are made and illustrated with two examples.     

Inference on Weibull parameters under a balanced two-sample type II progressive censoring scheme

The progressive censoring scheme has received a considerable amount of attention in the last 15 years. During the last few years, joint progressive censoring scheme has gained some popularity. Recently, the authors Mondal and Kundu (“A New Two Sample Type-II Progressive Censoring Scheme,” Communications in Statistics-Theory and Methods) introduced a balanced two-sample type II progressive censoring scheme and provided the exact inference when the two populations are exponentially distributed. In this article, we consider the case when the two populations follow Weibull distributions with the common shape parameter and different scale parameters. We obtain the maximum likelihood estimators of the unknown parameters. It is observed that the maximum likelihood estimators cannot be obtained in explicit forms; hence, we propose approximate maximum likelihood estimators, which can be obtained in explicit forms. We construct the asymptotic and bootstrap confidence intervals of the population parameters. Further, we derive an exact joint confidence region of the unknown parameters. We propose an objective function based on the expected volume of this confidence region, and using that, we obtain the optimum progressive censoring scheme. Extensive simulations have been performed to see the performances of the proposed method, and one real data set has been analyzed for illustrative purposes.     

Two Sample Tests in the Weibull Distribution

A test based on maximum likelihood estimators is given for testing the equality of the shape parameters in two Weibull distributions with the scale parameters unknown. Tests for the equality of the scale parameters are also presented along with a procedure for selecting the Weibull process with the larger mean life.     

Bayes estimation of residual life by fusing multisource information

Residual life estimation is essential for reliability engineering. Traditional methods may experience difficulties in estimating the residual life of products with high reliability, long life, and small sample. The Bayes model provides a feasible solution and can be a useful tool for fusing multisource information. In this study, a Bayes model is proposed to estimate the residual life of products by fusing expert knowledge, degradation data, and lifetime data. The linear Wiener process is used to model degradation data, whereas lifetime data are described via the inverse Gaussian distribution. Therefore, the joint maximum likelihood (ML) function can be obtained by combining lifetime and degradation data. Expert knowledge is used according to the maximum entropy method to determine the prior distributions of parameters, thereby making this work different from existing studies that use non-informative prior. The discussion and analysis of different types of expert knowledge also distinguish our research from others. Expert knowledge can be classified into three categories according to practical engineering. Methods for determining prior distribution by using the aforementioned three types of data are presented. The Markov chain Monte Carlo is applied to obtain samples of the parameters and to estimate the residual life of products due to the complexity of the joint ML function and the posterior distribution of parameters. Finally, a numerical example is presented. The effectiveness and practicability of the proposed method are validated by comparing it with residual life estimation that uses non-informative prior. Then, its accuracy and correctness are proven via simulation experiments.     

Maximum likelihood Estimation of Parameters in the Inverse Gaussian Distribution,With Unknown Origin

Maximum likelihood estimation is applied to the three-parameter Inverse Gaussian distribution, which includes an unknown shifted origin parameter. It is well known that for similar distributions in which the origin is unknown, such as the lognormal, gamma, and Weibull distributions, maximum likelihood estimation can break down. In these latter cases, the likelihood function is unbounded and this leads to inconsistent estimators or estimators not asymptotically normal. It is shown that in the case of the Inverse Gaussian distribution this difticulty does not arise. The likelihood remains bounded and maximum likelihood estimation yields a consistent estimator with the usual asymptotic normality properties. A simple iterative method is suggested for the estimation procedure. Numerical examples are given in which the estimates in the Inverse Gaussian model are compared with those of the lognormal and Weibull distributions.     

Generalized Marshall Olkin Inverse Lindley Distribution with Applications

In this article, a new generalization of the inverse Lindley distribution is introduced based on Marshall-Olkin family of distributions. We call the new distribution, the generalized Marshall-Olkin inverse Lindley distribution which offers more flexibility for modeling lifetime data. The new distribution includes the inverse Lindley and the Marshall-Olkin inverse Lindley as special distributions. Essential properties of the generalized Marshall-Olkin inverse Lindley distribution are discussed and investigated including, quantile function, ordinary moments, incomplete moments, moments of residual and stochastic ordering. Maximum likelihood method of estimation is considered under complete, Type-I censoring and Type-II censoring. Maximum likelihood estimators as well as approximate confidence intervals of the population parameters are discussed. A comprehensive simulation study is done to assess the performance of estimates based on their biases and mean square errors. The notability of the generalized Marshall-Olkin inverse Lindley model is clarified by means of two real data sets. The results showed the fact that the generalized Marshall-Olkin inverse Lindley model can produce better fits than power Lindley, extended Lindley, alpha power transmuted Lindley, alpha power extended exponential and Lindley distributions.     

Estimations of parameters in Pareto reliability model in the presence of masked data

Estimations of parameters included in the individual distributions of the life times of system components in a series system are considered in this paper based on masked system life test data. We consider a series system of two independent components each has a Pareto distributed lifetime. The maximum likelihood and Bayes estimators for the parameters and the values of the reliability of the system's components at a specific time are obtained. Symmetrical triangular prior distributions are assumed for the unknown parameters to be estimated in obtaining the Bayes estimators of these parameters. Large simulation studies are done in order: (i) explain how one can utilize the theoretical results obtained; (ii) compare the maximum likelihood and Bayes estimates obtained of the underlying parameters; and (iii) study the influence of the masking level and the sample size on the accuracy of the estimates obtained.     

Robust inference for one-shot device testing data under exponential lifetime model with multiple stresses

Introduced robust density-based estimators in the context of one-shot devices with exponential lifetimes under a single stress factor. However, it is usual to have several stress factors in industrial experiments involving one-shot devices. In this paper, the weighted minimum density power divergence estimators (WMDPDEs) are developed as a natural extension of the classical maximum likelihood estimators (MLEs) for one-shot device testing data under exponential lifetime model with multiple stresses. Based on these estimators, Wald-type test statistics are also developed. Through a simulation study, it is shown that some WMDPDEs have a better performance than the MLE in relation to robustness. Two examples with multiple stresses show the usefulness of the model and, in particular, of the proposed estimators, both in engineering and medicine.     

Statistical inference on progressive-stress accelerated life testing for the logistic exponential distribution under progressive type-II censoring

The accelerated life testing (ALT) is an efficient approach and has been used in several fields to obtain failure time data of test units in a much shorter time than testing at normal operating conditions. In this article, a progressive-stress ALT under progressive type-II censoring is considered when the lifetime of test units follows logistic exponential distribution. We assume that the scale parameter of the distribution satisfying the inverse power law. First, the maximum likelihood estimates of the model parameters and their approximate confidence intervals are obtained. Next, we obtain Bayes estimators under squared error loss function with the help of Metropolis-Hasting (MH) algorithm. We also derive highest posterior density (HPD) credible intervals of the model parameters. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation. Finally, one data set has been analyzed for illustrative purposes.     

Evaluation of Phase I analysis scenarios on Phase II performance of control charts for autocorrelated observations

Phase I analysis of a control chart implementation comprises parameter estimation, chart design, and outlier filtering, which are performed iteratively until reliable control limits are obtained. These control limits are then used in Phase II for online monitoring and prospective analyses of the process to detect out-of-control states. Although a Phase I study is required only when the true values of the parameters of a process are unknown, this is the case in many practical applications. In the literature, research on the effects of parameter estimation (a component of Phase I analysis) on the control chart performance has gained importance recently. However, these studies consider availability of complete and clean data sets, without outliers and missing observations, for estimation. In this article, we consider AutoRegressive models of order 1 and study the effects of two extreme cases for Phase I analysis; the case where all outliers are filtered from the data set (parameter estimation from incomplete but clean data) and the case where all outliers remain in the data set during estimation. Performance of the maximum likelihood and conditional sum of squares estimators are evaluated and effects on the Phase II use are investigated. Results indicate that the effect of not detecting outliers in Phase I can be severe on the Phase II application of a control chart. A real-world example is provided to illustrate the importance of an appropriate Phase I analysis.     

E-Bayesian estimation of reliability characteristics of two-parameter bathtub-shaped lifetime distribution with application

This article presents the expected Bayesian (E-Bayesian) estimation of the scale parameter, reliability and failure rate functions of two-parameter bathtub-shaped lifetime distribution under type-II censoring data with. Squared error loss function and gamma distribution as a conjugate prior distribution for the unknown parameter are used to obtain the E-Bayesian estimators. Also, three different prior distributions for the hyperparameters for the E-Bayesian estimators are considered. Some properties of the E-Bayesian estimators are studied. Using minimum mean square error criteria, a simulation study is conducted to compare the performance of the E-Bayesian estimators and the corresponding Bayes and maximum likelihood estimators. A real data set is analysed to show the applicability of the different proposed estimators. The numerical results show that the E-Bayesian estimators perform better than the classical and Bayesian estimators.     

A new approach for estimating parameters of two-parameter bathtub-shaped lifetime distribution under modified progressive hybrid censoring

The lifetime distributions with bathtub-shaped hazard rate functions and censoring scheme have been used widely in life testing and reliability engineering. This paper develops a new approach for estimating parameters of an important two-parameter lifetime data analysis model with bathtub-shaped hazard rate function under the assumption that sample is modified progressively hybrid censored. One of the most frequently used methodologies, maximum likelihood (ML) estimation, is used for estimating unknown parameters. The estimates of unknown parameters are proposed using popular Newton–Raphson algorithm because the estimators cannot be obtained in closed forms. It is well known that the convergence of Newton–Raphson algorithm is affected by an initial point. Therefore, a new Newton–Raphson algorithm with an adaptive initial point within the exact joint confidence region has been suggested to compute the ML estimation. Extensive numerical simulations show that the proposed algorithm converges all the times and it is effective. Finally, one real-world data set from engineering is analysed to illustrate the application of the proposed  method.     

GLM profile monitoring using robust estimators

It is well known that performance of control scheme in phase II of statistical process control depends on the estimators utilized in phase I. Sometimes, outliers may be present in the data, which could seriously impact the performance of the estimators. In some practical situations, generalized linear models (GLMs) are used to model a wide class of response variables. This study deals with the robust estimation and monitoring of parameters in GLM profiles in the presence of outliers. In this study, robust estimators are used to estimate the parameters of logistic and Poisson profiles. The results are compared with the maximum likelihood estimators (MLEs). In a numerical example, the profile parameters are estimated by the MLE and robust estimators and the resulting test statistics are monitored by a control scheme. The phase II control charts are determined based on these two types of estimators and compared for different out-of-control conditions. The simulation results confirm that robust estimators in most cases lead to better estimates in comparison with the MLE estimator in terms of average run length criterion.     

Maximum likelihood estimation of change point from stationary to nonstationary in autoregressive models using dynamic linear model

Change point estimation is a useful concept in time series models that could be applied in several fields such as financing, quality control. It helps to decrease costs of decision making and production by monitoring stock market and production lines, respectively. In this paper, the maximum likelihood technique is developed to estimate change point at which the stationary AR(1) model changes to a nonstationary process. Filtering and smoothing of dynamic linear model are used to estimate unknown parameters after change point. We also assume that correlation exists between samples' statistics. Simulation results show the effectiveness of the proposed estimators to estimate the change point of stationary. In addition based on Shewhart control chart, filtering has a better accuracy in comparison to smoothing. A real example is provided to illustrate the application.     

Reliability analysis of Weibull multicomponent system with stress-dependent parameters from accelerated life data

In this paper, reliability estimation of multicomponent system under a multilevel accelerated life testing. When the lifetime of components follows Weibull distribution, the problem of point and interval estimates are discussed from different perspectives. Under a general life-stress assumption that there are multiple nonconstant and stress-dependent scale and shape parameters, the maximum likelihood estimates of unknown parameters along with associated existence and uniqueness are established. Approximate confidence intervals are constructed correspondingly via expected Fisher information matrix. Furthermore, some pivotal quantities are constructed and alternative generalized point and interval estimates are also proposed for comparison. In addition, predictive intervals for the lifetime of the multicomponent system are discussed under classical and generalized pivotal approaches, respectively. The results show that the proposed generalized estimates are superior to the conventional likelihood approach in terms of the accuracy. A real data example is carried out to illustrate the implementations of the proposed methods.     

Life Testing in Variably Scaled Environments

Typically, accelerated life-testing models postulate a specific functional relationship between the stress level at which an experiment is performed and the parameters of the assumed family of lifetime distributions. These models, and the statistical analyses that accompany them, are often criticized on the basis of the dubious validity of the assumed functional relationship and of the uncertainty involved in the extrapolation of experimental results to low stress levels at which little or no data have been obtained. This study focuses on an exponential factorial model for accelerated life tests that postulates that the lifetime distributions of different component types tested under varying environmental conditions are linked via environmental or component-related scale changes. Necessary and sufficient conditions are given for the identifiability of model parameters. For both censored and complete data, the derivation and properties of maximum likelihood estimates of these parameters are discussed in detail. Under the conditions that guarantee identifiability, the existence and the uniqueness of the maximum likelihood estimators are demonstrated, and their computation and large-sample behavior are discussed. In the final section, the model is fitted to published data from an accelerated life-testing experiment.     

Bayesian Analysis in Partially Accelerated Life Tests for Weighted Lomax Distribution

Accelerated life testing has been widely used in product life testing experiments because it can quickly provide information on the lifetime distributions by testing products or materials at higher than basic conditional levels of stress, such as pressure, temperature, vibration, voltage, or load to induce early failures. In this paper, a step stress partially accelerated life test (SS-PALT) is regarded under the progressive type-II censored data with random removals. The removals from the test are considered to have the binomial distribution. The life times of the testing items are assumed to follow length-biased weighted Lomax distribution. The maximum likelihood method is used for estimating the model parameters of length-biased weighted Lomax. The asymptotic confidence interval estimates of the model parameters are evaluated using the Fisher information matrix. The Bayesian estimators cannot be obtained in the explicit form, so the Markov chain Monte Carlo method is employed to address this problem, which ensures both obtaining the Bayesian estimates as well as constructing the credible interval of the involved parameters. The precision of the Bayesian estimates and the maximum likelihood estimates are compared by simulations. In addition, to compare the performance of the considered confidence intervals for different parameter values and sample sizes. The Bootstrap confidence intervals give more accurate results than the approximate confidence intervals since the lengths of the former are less than the lengths of latter, for different sample sizes, observed failures, and censoring schemes, in most cases. Also, the percentile Bootstrap confidence intervals give more accurate results than Bootstrap-t since the lengths of the former are less than the lengths of latter for different sample sizes, observed failures, and censoring schemes, in most cases. Further performance comparison is conducted by the experiments with real data.     

Field data analyses with additional after-warranty failure data

This paper proposes methods of estimating the lifetime distribution for situations where additional field data can be gathered after the warranty expires in a parametric time to failure distribution. For satisfactory inference about parameters involved, it is desirable to incorporate these after-warranty data in the analysis. It is assumed that after-warranty data are reported with probability p₁(<1), while within-warranty data are reported with probability 1. Methods of obtaining maximum likelihood estimators are outlined, their asymptotic properties are studied, and specific formulas for Weibull distribution are obtained. An estimation procedure using the expectation and maximization algorithm is also proposed when the reporting probability is unknown. Simulation studies are performed to investigate the properties of the estimates.