Meeker introduced the concept of limited failure population (LFP) while studying integrated circuits, where most devices exhibit a near-zero probability of failure throughout their technological life, while a small subset contains latent defects that manifest only after prolonged operation. In this article, we extend the LFP framework to the context of one-shot device testing, wherein devices get destroyed or must be rebuilt after testing. We propose a model that incorporates a relationship between defect probability and a set of covariates and develop an expectation–maximization algorithm for estimating the model parameters. The effectiveness of the proposed method is then assessed through a Monte Carlo simulation study, assuming the lifespan of defective devices follow a log-location-scale distribution. Finally, the developed methods are applied to three real datasets.

In this work, we first introduce a discrete version of generalized Fisher information measure and develop some new results for it. We then propose Jensen-generalized discrete Fisher (Jensen-GDF) information as a generalized measure, based on the convexity property of generalized discrete Fisher information measure. We further introduce generating functions for generalized discrete Fisher information and Jensen-GDF information measures and use them to develop some results. We also propose a new correlation coefficient in terms of the generalized discrete Fisher information and discuss some of its properties. Finally, to demonstrate the usefulness of the Jensen-generalized discrete Fisher information measure and the proposed correlation coefficient, we apply them to two real-world examples in image processing and forms of contaminated data, and present corresponding numerical results. Our findings show that the Jensen-GDF information measure and the correlation coefficient introduced here are effective criteria for quantifying similarity between two images in image processing settings and for analyzing contaminated data.

The gait index (GI), a valuable metric to assess human gait, incorporates clinically relevant parameters such as walking speed, knee angle, stride length, and stance-to-swing phase ratio. This index offers insights into an individual's gait pattern, aiding in the identification of subtle gait abnormalities and enabling continuous monitoring of gait changes over time. Building upon this foundation, the present study investigated the influence of specific gait parameters and demographic factors on the gait index, alongside their interaction effects. Analyzing data from 120 healthy individuals using beta regression models, we uncovered significant predictors and interaction effects shaping the Index. Our comparative assessment between Variable Dispersion Beta Regression (VDBR) and Fixed Dispersion Beta Regression (FDBR) models revealed VDBR's superiority over FPBR in capturing gait data heterogeneity. Our analysis revealed that while aging was correlated with decreased GI, gender and BMI exhibited limited individual impact. However, gait-specific predictors such as knee angle, stride length, walking speed, and stance-to-swing phase ratio significantly contributed to GI variability. Additionally, significant interaction effects were identified between knee angle and height normalized stride length, age and knee angle, and age and walking speed, highlighting the complex interplay between demographic and gait-related factors. These findings underscore the multifaceted nature of gait dynamics and offer valuable insights for clinicians, aiding in precise gait pattern assessment and informing the development of gait-related clinical practice, preventive care strategies, and rehabilitation programs. Overall, our research contributes to enhancing mobility and functionality in individuals with gait degradation by identifying significant predictors and interaction effects.

Accelerated life testing is widely employed due to the high cost involved in testing high-quality products under normal operating conditions. For products exposed to continuously fluctuating stress in the working environment, cyclic stress tests become necessary. The Coffin–Manson model is commonly used when product failure is solely attributed to temperature changes ($\Delta T$). However, this assumption does not always hold in many practical situations. The Norris–Landzberg model, which considers both maximum temperature and cyclic change frequency, offers much flexibility in modeling fatigue life due to cyclic temperature fluctuations. Several studies have been conducted based on the Norris–Landzberg model. However, using the multiple linear regression method without any distributional assumption may fail to provide satisfactory inferential results. This article assumes the log-location-scale family of distributions and then shows that the weighted least-squares method based on order statistics of failure times yields the best linear unbiased estimators (BLUEs) of parameters based on complete as well as Type-II censored data. We then study some properties of these BLUEs using both theory and Monte Carlo simulations. Next, we present an illustrative example involving solder joint data to demonstrate the model and the associate inferential results developed here. Finally, the optimal design procedure is discussed.

One-shot devices, such as automotive airbags, fire extinguishers and ammunitions, pose significant challenges in their reliability analysis due to their inherently unobservable lifespans. Nondestructive one-shot devices, in particular, offer additional information when they have not failed prior to inspection, yielding interval-censored failure time data. This article addresses the limitations of traditional testing designs for such devices by introducing an adaptive proportion of failure approach within the context of cyclic accelerated life tests, a variant of accelerated life tests characterized by continuously varying stress levels in the operating environment. Using the Norris–Landzberg model for thermal cycling-induced stresses, we propose here an iterative regression algorithm for statistical inference under this adaptive design. Our algorithm provides estimators that possess consistency and asymptotic normality, demonstrating robustness against initial value sensitivity, a common issue with traditional numerical methods used for maximum likelihood estimation. A simulation study and an illustrative example are presented to exemplify the merits of the proposed approach.

Outlier detection is a significant challenge in Internet of Things (IoT)-based systems, which encompass a multitude of sensor nodes deployed for diverse applications. Ensuring accurate data transmission from these nodes to base stations is crucial, as outliers (fault/event/intrusion) can adversely impact data processing accuracy and overall Quality of Service. The principal component analysis (PCA) has gained popularity for outlier detection in IoT, with recursive PCA (R-PCA) being a widely used method. In this article, we explore popular PCA-based approaches and present an optimized, real-time, and reproducible enhancement to the existing R-PCA method. Our proposed improvement focuses on a data-driven approximation of the distribution of squared prediction error (SPE) scores, a fundamental component of PCA-based outlier detection. We address theoretical ambiguities in the assumptions underlying SPE scores in the existing R-PCA method. Through simulations, we demonstrate the inaccurate distributional assumption of SPE scores in the specified scheme. Additionally, we introduce a more suitable Satterthwaite-based approximation of the SPE score distribution, supported by quantile-quantile (Q-Q) plots. The effectiveness of the proposed approximation is validated through performance evaluation metrics, demonstrating its superiority over the Gaussian approximation used in R-PCA schemes. Furthermore, we provide an overview of our proposed scheme, which can be implemented in any PCA-based outlier detection system used by IoT practitioners and engineers. Our research contributes to advancing outlier detection methodologies in IoT-based systems, enabling more reliable anomaly detection and improved system performance.

Throughout human history, deadly infectious diseases emerged occasionally. Even with the present-day advanced healthcare systems, the COVID-19 has caused more than six million deaths worldwide (as of 27 July 2022). Currently, researchers are working to develop tools for better and effective management of the pandemic. “Contact tracing” is one such tool to monitor and control the spread of the disease. However, manual contact tracing is labor-intensive and time-consuming. Therefore, manually tracking all potentially infected individuals is a great challenge, especially for an infectious disease like COVID-19. To date, many digital contact tracing applications were developed and used globally to restrain the spread of COVID-19. In this work, we perform a detailed review of the current digital contact tracing technologies. We mention some of their key limitations and propose a fully integrated system for contact tracing of infectious diseases using COVID-19 as a case study. Our system has four main modules—1) case maps; 2) exposure detection; 3) screening; and 4) health indicators that take multiple inputs like users’ self-reported information, measurement of physiological parameters, and information of the confirmed cases from the public health, and keeps a record of contact histories using Bluetooth technology. The system can potentially evaluate the users’ risk of getting infected and generate notifications to alert them about the exposure events, risk of infection, or abnormal health indicators. The system further integrates the Web-based information on confirmed COVID-19 cases and screening tools, which potentially increases the adoption rate of the system.

In this work, we propose cumulative residual Fisher information and relative cumulative residual Fisher information measures and establish some of their properties. We first show that these cumulative Fisher measures can be expressed based on the hazard function. We then define extended versions of cumulative residual entropy and cumulative residual Fisher information measures based on the Jensen inequality. We also discuss connections between these information measures based on a new version of de Bruijn's identity for survival functions.

In this article, we develop robust estimators and tests for one-shot device testing under proportional hazards assumption based on divergence measures. Through a detailed Monte–Carlo simulation study and a numerical example, the developed inferential procedures are shown to be more robust against data contamination than the classical procedures, based on maximum likelihood estimators.

Classical inferential methods for one-shot device testing data from an accelerated life-test are based on maximum likelihood estimators (MLEs) of model parameters. However, the lack of robustness of MLE is well-known. In this article, we develop robust estimators for one-shot device testing by assuming a Weibull distribution as a lifetime model. Wald-type tests based on these estimators are also developed. Their robustness properties are evaluated both theoretically and empirically, through an extensive simulation study. Finally, the methods of inference proposed are applied to three numerical examples. Results obtained from both Monte Carlo simulations and numerical studies show the proposed estimators to be a robust alternative to MLEs.

This paper develops a new family of estimators, the minimum density power divergence estimators (MDPDEs), for the parameters of the one-shot device model as well as a new family of test statistics, Z-type test statistics based on MDPDEs, for testing the corresponding model parameters. The family of MDPDEs contains as a particular case the maximum likelihood estimator (MLE) considered in Balakrishnan and Ling (2012). Through a simulation study, it is shown that some MDPDEs have a better behavior than the MLE in terms of robustness. At the same time, it can be seen that some Z-type tests based on MDPDEs have a better behavior than the classical Z-test statistic in terms of robustness, as well.

In this paper, we develop likelihood inference based on the expectation maximization algorithm for the Box-Cox transformation cure rate model assuming the lifetimes to follow a Weibull distribution. A simulation study is carried out to demonstrate the performance of the proposed estimation method. Through Monte Carlo simulations, we also study the effect of model misspecification on the estimate of cure rate. Finally, we analyze a well-known data on melanoma with the model and the inferential method developed here.

Observations on degradation performance are often used to analyze the underlying degradation process of highly reliable products. From the two-phase degradation path of the bearing performance observations, we observed that there exists an abrupt increase in degradation measurement at a change point. Then, the following degradation process started with the abrupt degradation measurement will degrade in a higher degradation rate. Here, a stochastic process-based degradation model is constructed to interpret the jump at the change point in the degradation process which is governed by the linear Wiener process. Meanwhile, the distribution of the first passage time over a prespecified threshold for the process is discussed. In addition, to get the estimates of the model parameter, the expectation-maximization algorithm is utilized since the change points are unobservable. Furthermore, to demonstrate the model's advantages over estimate, a comparison is made between the proposed and the existing known models from the literature. The results reveal that considering the jump in the degradation process can improve the accuracy of estimations in real applications.

Model mis-specification is of great importance in reliability assessment. Different choices of probability models for fitting data may result in substantially different inferential results on some lifetime characteristics of interest. Gamma and Weibull models have been used extensively for modeling lifetime data. Hence, accelerated life models have been developed recently for one-shot device test data under both these models for making inference on mean lifetime as well as the reliability at use level. However, model mis-specification analyses between these two models have not been studied in this context. Here, we examine the effect of model mis-specification between gamma and Weibull models on the likelihood estimation and the inference on the mean lifetime and the reliability at some mission times based on one-shot device test data. Moreover, a distance-based test statistic and the Akaike information criterion as specification tests are studied for the purpose of model validation. A simulation study is carried out to evaluate the bias and coverage probabilities of confidence intervals under model mis-specification. The obtained results reveal that the effect of model mis-specification is negligible only when the sample size is small and when the accelerated and use levels are close, and that the use of specification test is quite important for an accurate reliability assessment.

This paper considers a competing risk model for a one-shot device testing analysis under an accelerated life test setting. Due to the consideration of competing risks, the joint posterior distribution becomes quite complicated. The Metropolis-Hastings sampling method is used for the estimation of the posterior means of the variables of interest. A simulation study is carried out to assess the Bayesian approach with different priors, and also to compare it with the EM algorithm for maximum likelihood estimation. Finally, an example from a tumorigenicity experiment is presented.

This paper provides an extension of the work of Balakrishnan and Ling by introducing a competing risks model into a one-shot device testing analysis under an ALT setting. An expectation maximization (EM) algorithm is then developed for the estimation of model parameters. An extensive Monte Carlo simulation study is carried out to assess the performance of the proposed method. The performance of the EM algorithm and the Fisher scoring method are also compared. Finally, the proposed EM algorithm is applied to a modified Class-B insulation data for illustrating the results developed here.

Based on the observed lifetimes of two systems with shared components, we construct exact nonparametric confidence intervals for quantiles of component and system lifetimes using the minimum and maximum lifetimes, as well as by using all component lifetimes. The coverage probabilities and expected widths of these confidence intervals are then compared for several signature matrices and sample sizes.

The failure of any component within a series system results in the failure of the system. Incorporating active redundancy into a system generally improves its reliability and availability. This type of system does not belong to the category of coherent systems, and thus those results on coherent systems are not applicable. Hence, it is of great interest to develop statistical methodologies that perform well for inference on reliability of redundancy systems. In a life-test for multicomponent systems, component lifetimes may not be observable, but one may observe a set of components that failed along with the system. The data available in this form are called autopsy data, and, in this case, lifetime information on some of the components is missing. In this paper, we consider an equal load-sharing model and develop an expectation-maximization algorithm for estimating system reliability characteristics based on such an autopsy data. The load-sharing parameter indicates whether incorporating active redundancy improves system reliability or induces rapid failure of the system. The performance of the proposed methodology is then evaluated through Monte Carlo simulations and then illustrated with two numerical examples.

For devices with long lifetimes, accelerated life-tests are commonly used to induce quick failures. A link function relating stress levels and lifetime is then applied to extrapolate the lifetimes of units from accelerated conditions to normal operating conditions. Because data from one-shot devices do not contain any lifetimes, a standard reliability analysis with a parametric distributional assumption on lifetimes may be sensitive to violations of the model assumption. For this reason, we have proposed here a proportional hazards model for analyzing one-shot device testing data collected from constant-stress accelerated life-tests. The maximum likelihood estimates of the parameters of this semi-parametric model are developed. Confidence intervals for the reliability at an inspection time are constructed through asymptotic and transformation approaches. A Monte Carlo simulation study is then carried out to compare these confidence intervals in terms of coverage probabilities, and average widths. The obtained results show that the proposed flexible semi-parametric model provides a good insight into the estimation of reliability under normal (typical) operating conditions. A distance-based test statistic is also proposed for testing the proportional hazards model, and the exact calculation of its p-value is discussed. Finally, the proposed proportional hazards model is illustrated with real data from a toxicological study.

In this paper, a mixture representation is derived for the pooled system lifetimes arising from a life-test on two or more independent samples. The components of each system are assumed to have the same common absolutely continuous distribution, but the system signature may vary between the samples. These mixtures are then used for developing exact nonparametric inference in the form of confidence intervals for quantiles of component or system lifetimes, as well as prediction intervals for future component or system lifetimes. Examples are finally provided to illustrate the developed methods. It is noted that testing with systems rather than components directly can reduce the expected number of failures while maintaining nominal coverage probability.

In this paper, we first present explicit expressions for the maximum likelihood estimates (MLEs) of the location, and scale parameters of the Laplace distribution based on a Type-II right censored sample under different cases. Then, after giving the exact density functions of the MLEs, and the expectations, we derive the exact density of the MLE of the quantile, and utilize it to develop exact confidence intervals for the population quantile. We also briefly discuss the MLEs of reliability and cumulative hazard functions, and how to develop exact confidence intervals for these functions. These results can also be extended to any linear estimators. Finally, we present two examples to illustrate the inferential methods developed here.

This paper examines the problem of the stochastic comparison of series and parallel systems with s-independent heterogeneous generalized exponential components. The results established here are developed in three directions. First, we consider a system with possibly different shape and scale parameters, and obtain some ordering results when its matrix of parameters changes to another matrix, in the certain mathematical sense. Next, by using the concept of vector majorization and related orders, we establish various ordering results for the comparisons of series and parallel systems, when their component's lifetimes have either the same shape parameters with possibly different scale parameters, or the same scale parameters with possibly different shape parameters. Finally, some of the known results on various stochastic orderings between parallel systems in the exponential case are extended to the case when the lifetimes of components follow the generalized exponential distributions. The results of this paper can be used in practical situations to replace components of series and parallel systems by new components, or to find various bounds for the important aging characteristics of these systems.

Failure of a component in a composite dynamic system often induces a higher load on surviving components, and increases the hazard rate. Statistical inferential procedures on composite dynamic systems are developed here based on a Burr type-XII distribution with a power-trend hazard rate function. Point estimates of the Burr type-XII parameters, and interval estimates of the baseline survival function are obtained based on the maximum-likelihood estimates, and the Fisher information matrix. A test procedure is presented for examining the relationship between the hazard rate function and the number of failed components. The performance of the proposed method is then evaluated by means of an extensive Monte Carlo simulation study. An example is finally presented for illustrative purpose.

As most systems these days are highly reliable with long lifetimes, failures of systems become rare; consequently, traditional failure time analysis may not be able to provide a precise assessment of the system reliability. In this regard, a degradation measure, as a percentage of the initial value, is an alternate way of describing the system health. This paper presents accelerated degradation analysis that characterizes the health and quality of systems with monotonic and bounded degradation. The maximum likelihood estimates (MLEs) of the model parameters are derived, based on a gamma process, time-scale transformation, and a power link function for associating the covariates. Then, methods of estimating the reliability, the mean and median lifetime, the conditional reliability, and the remaining useful life of systems under normal use conditions are all described. Moreover, approximate confidence intervals for the parameters of interest are developed based on the observed Fisher information matrix. A model validation metric with exact power is introduced. A Monte Carlo simulation study is carried out for evaluating the performance of the proposed methods. For an illustration of the proposed model, and the methods of inference developed here, a numerical example involving light intensity of light emitting diodes (LED) is analyzed.

In this article, a general goodness of fit test is developed by using a new estimate of Kullback-Leibler (KL) information based on Type-II censored data. The proposed test is consistent, and the test statistic is nonnegative, just like KL information. Then, the test statistic is used to test for exponentiality based on Type-II censored data. Through a simulation study, power values of the proposed test are compared with some prominent existing tests. A real-life data analysis is finally presented for illustrative purpose.

Acceptance testing is widely used to assess whether a product meets the expectations of customers. Yet, traditional acceptance tests based on time-to-failure data will not be practical because today's highly reliable products may take a long time to fail. It may be good in this case to base a test on a suitable quality characteristic (QC) whose degradation over time is related to the reliability of the product. Motivated by resistor data, we first propose a degradation model to describe the degradation paths of the resistors. Next, we present an accelerated-stress acceptance test to reduce the acceptance testing time, and then derive the optimal accelerated-stress acceptance testing time for a product, and the probability of acceptance of the batch. A model incorporating cost is also used to determine the optimal design for an accelerated-stress acceptance experiment, and a motivating example is then presented to illustrate the proposed procedure. Finally, we examine the performance of the estimators, and the effect of misspecification of the parameters on the optimal test plan through a Monte Carlo simulation study, and a detailed sensitivity analysis.

Statistical inference of the component lifetime distribution is developed when Type-II censored system lifetime data are observed with a known system structure. The component lifetime distributions are assumed to be from either the log-location-scale family of distributions or the location-scale family of distributions. Two estimation methods, the maximum likelihood method, and the regression-based method, are proposed for the model parameters, and the corresponding computational formulae are provided. Construction of confidence intervals for the model parameters is also considered. The methodologies are illustrated with two commonly used lifetime distributions: the Weibull, and the lognormal. Monte Carlo simulations are used to study the performances of the point and interval estimation methods proposed here. Finally, some recommendations are made based on the obtained simulation results.

ABSTRACT In futures markets with price limits, trading halts are triggered by limit hits. Limit hits are rarely observed, perhaps because traders avoid bid‐ask quotes that cause them. If this explanation is true, futures prices would cluster in a narrow region close to the limits. We test this empirically for currency futures contracts and find results consistent with the explanation. The tests require calculations of all combinations of a computationally intensive time series, which are extremely time consuming on a sequential machine and hence limit the practicality of the analysis. Consequently, we investigate parallel computing strategies in partitioning the datasets and solving them in parallel on a high‐end Beowulf cluster. We discuss two different partitioning strategies of the given datasets on the cluster and elaborate the results. Copyright © 2012 John Wiley & Sons, Ltd.

We discuss here the design of constant-stress accelerated life-tests for one-shot device testing by assuming a Weibull distribution as a lifetime model. Because there are no explicit expressions for the maximum likelihood estimators of the model parameters and their variances, we adopt the asymptotic approach here to develop an algorithm for the determination of optimal allocation of devices, inspection frequency, and the number of inspections at each stress level, by assuming a Weibull distribution with non-constant scale and shape parameters as the lifetime distribution. The asymptotic variance of the estimate of reliability of the device at a specified mission time is minimized subject to a pre-fixed experimental budget, and a termination time. Examples are provided to illustrate the proposed algorithm for the determination of the best test plan. A sensitivity analysis of the best test plan is also carried out to examine the effect of misspecification of the model parameters.

In reliability analysis, accelerated life-tests are commonly used for inducing more failures, thus obtaining more lifetime information in a relatively short period of time. In this paper, we study binary response data collected from an accelerated life-test arising from one-shot device testing based on a Weibull lifetime distribution with both scale and shape parameters varying over stress factors. Log-linear link functions are used to connect both scale and shape parameters in the Weibull model with the stress factors. Because no failure times of units are observed, we use the EM algorithm for computing the maximum likelihood estimates (MLEs) of the model parameters. Moreover, we develop inferences on the reliability at a specific time, and the mean lifetime at normal operating conditions. This method of estimation is then compared with Fisher scoring and least-squares methods in terms of mean square error as well as tolerance value, computational time, and number of cases of divergence. The asymptotic confidence intervals and parametric bootstrap confidence intervals are also developed for some parameters of interest. A transformation approach is also proposed for constructing confidence intervals. A simulation study is then carried out to demonstrate that the proposed estimators perform very well for data of the considered form. Such accelerated one-shot device testing data can also be found in survival analysis. For an illustration, we consider here an application of the proposed algorithm to mice tumor toxicology data from a study involving the development of tumors with respect to risk factors such as sex, strain of offspring, and dose effects.

The gamma distribution is used as a lifetime distribution widely in reliability analysis. Lifetime data are often left truncated, and right censored. The EM algorithm is developed here for the estimation of the scale and shape parameters of the gamma distribution based on left truncated and right censored data. The Newton-Raphson method is also used for the same purpose, and then these two methods of estimation are compared through an extensive Monte Carlo simulation study. The asymptotic variance-covariance matrix of the MLEs under the EM framework is obtained by using the missing information principle (Louis, 1982). Then, the asymptotic confidence intervals for the parameters are constructed. The confidence intervals based on the EM algorithm and the Newton-Raphson method are then compared empirically in terms of coverage probabilities. Finally, all the methods of inference discussed here are illustrated through a numerical example.

We discuss meta-analysis of multiple$s$-independent Type-II right censored data. In particular, we consider parametric inference using Best Linear Unbiased Estimation, as well as non-parametric inference. We provide pertinent numerical results and two examples to illustrate all the methods of inference developed here.

The concept of the signature of a coherent system is useful to study the stochastic and aging properties of the system. Let X1m, X2:n, ⋯ Xn:ndenote the ordered lifetimes of the components of a coherent system consisting of n i.i.d components. If T denotes the lifetime of the system, then the signature vector of the system is defined to be a probability vectors = (s1, s2, ⋯, sn) such that si= P(T = Xi:n), i = 1, 2, ⋯,n. Here we consider a coherent system with sig- nature of the form s = (s1, s2, ⋯ Si, 0 ... , 0), where sk>; 0, k = 1,2, ⋯, i. Under the condition that the system is working at time t, we propose a time dependent measure to calculate the probability of residual life of live components of the system, i.e., Xk:n, k = i + 1 ⋯, n. Several stochastic and aging properties of the proposed measure are explored.

Left- and right-censored life time data arise naturally in one-shot device testing. An experimenter is often interested in identifying the effects of several stress variables on the lifetime of a device, and furthermore multiple-stress experiments controlling simultaneously several variables, result in reducing the experimental time as well as the cost of the experiment. Here, we present an expectation-maximization (EM) algorithm for developing inference on the reliability at a specific time, as well as the mean lifetime of the device based on one-shot device testing data under the exponential distribution when there are multiple stress factors. We use the log-linear link function for this purpose. Unlike in the typical EM algorithm, it is not necessary to obtain maximum likelihood estimates (MLEs) of the parameters at each step of the iteration. By using the one-step Newton-Raphson method, we observe that the convergence occurs quickly. We also use the jackknife technique to reduce the bias of the estimate obtained from the EM algorithm. In addition, we discuss the construction of confidence intervals for some reliability characteristics by using the asymptotic properties of the MLEs based on the observed Fisher information matrix, as well as by the jackknife technique, the parametric bootstrap methods, and a transformation technique. Finally, we present an example to illustrate all the inferential methods developed here.

We propose a general purpose approximate goodness-of-fit test that covers several families of distributions under progressive Type-II censored data. The test procedure is based on the empirical distribution function (EDF), and generalizes the goodness-of-fit test proposed by Chen and Balakrishnan [11] to progressively Type-II censored data. The new method requires some tables for critical values, which are constructed by Monte Carlo simulation. The power of the proposed tests are then assessed for several alternative distributions, while testing for normal, Gumbel, and log-normal distributions, through Monte Carlo simulations. It is observed that the proposed tests are quite powerful when compared to an existing goodness-of-fit test proposed for progressively Type-II censored data due to Balakrishnan et al. . The proposed goodness-of-fit test is then illustrated with two real data sets.

Degradation models are usually used to provide information about the reliability of highly reliable products that are not likely to fail within a reasonable period of time under traditional life tests, or even accelerated life tests. The gamma process is a natural model for describing degradation paths, which exhibit a monotone increasing pattern, while the commonly used Wiener process is not appropriate in such a case. We discuss the problem of optimal design for degradation tests based on a gamma degradation process with random effects. To conduct a degradation experiment efficiently, several decision variables (such as the sample size, inspection frequency, and measurement numbers) need to be determined carefully. These decision variables affect not only the experimental cost, but also the precision of the estimates of lifetime parameters of interest. Under the constraint that the total experimental cost does not exceed a pre-specified budget, the optimal decision variables are found by minimizing the asymptotic variance of the estimate of the 100p-th percentile of the lifetime distribution of the product. Laser data are used to illustrate the proposed method. Moreover, we assess analytically the effects of model mis-specification that occur when the random effects are not taken into consideration in the gamma degradation model. The numerical results of these effects reveal that the impact of model mis-specification on the accuracy and precision of the prediction of percentiles of the lifetimes of products are somewhat serious for the tail probabilities. A simulation study also shows that the simulated values are quite close to the asymptotic values.

In a multi-sample experiment, we model the parameters of an equal load-sharing system by means of link functions in sequential order statistics models, and then discuss the estimation of these parameters based on a given link function. Different link functions are examined along with the corresponding maximum likelihood estimators, and their properties are studied both analytically and through Monte Carlo simulations.

In this paper, we discuss linear inference for the lifetime distribution of components based on a Type-II censored lifetime data of reliability systems with known signatures. We derive the best linear unbiased estimators (BLUE) for the parameter(s) in general scale and location-scale parameter families. The exact computational formulas of the BLUE and their variances and covariance are provided. Selected tables of the coefficients of BLUE are presented for the exponential and extreme value distributions. Using these, best linear unbiased predictors of future system failure times are discussed. Finally, two examples are provided to illustrate all the methods of inference developed here.

We express the joint entropy of progressively Type-II censored order statistics in terms of an incomplete integral of the hazard function, and use it to develop a simple estimate of the joint entropy of progressively Type-II censored data, considered earlier by Balakrishnan , IEEE Trans. Reliability, vol. 56, pp. 349-356. We then construct a goodness-of-fit test statistic based on the Kullback-Leibler information for Pareto, log-normal, and Weibull distributions by using maximum likelihood estimates and approximate maximum likelihood estimates of the model parameters. Finally, we use Monte Carlo simulations to evaluate the power of the proposed test for several alternatives under different sample sizes and progressive censoring schemes.

Burn-in test is a manufacturing process applied to products to eliminate latent failures or weak components in the factory before the products reach customers. The traditional burn-in test over a short period of time to collect time-to-failure or go/no-go data is rather inefficient. This decision problem can be solved if there exists a suitable quality characteristic (QC) whose degradation over time can be related to the lifetime of the product. Recently, optimal burn-in policies have been discussed in the literature assuming that the underlying degradation path follows a Wiener process. However, the degradation model of many materials (especially in the case of fatigue data) may be more appropriately modeled by a gamma process that exhibits a monotone-increasing pattern. Here, motivated by laser data, we first -propose a mixed gamma process to describe the degradation path of the product. Next, we present a decision rule for classifying a unit as typical or weak. A cost model is used to determine the optimal termination time of a burn-in test, and a motivating example is then presented to illustrate the proposed procedure. Finally, a simulation study is carried out to examine the effect of wrongly treating a mixed gamma process as a mixed Wiener process, and the obtained results reveal that the effect on the probabilities of misclassification is not negligible.

The objective of this paper is to provide a new estimation method for parametric models under progressive Type-I censoring. First, we propose a Kaplan-Meier nonparametric estimator of the reliability function taken at the censoring times. It is based on the observable number of failures, and the number of censored units occurring from the progressive censoring scheme at the censoring times. This estimator is then shown to asymptotically follow a normal distribution. Next, we propose a minimum-distance method to estimate the unknown Euclidean parameter of a given parametric model. This method leads to consistent, asymptotically normal estimators. The maximum likelihood estimation method based on group-censored samples is discussed next, and the efficiencies of these two methods are compared numerically. Then, based on the established results, we derive a method to obtain the optimal Type-I progressive censoring scheme, Finally we illustrate all these results through a Monte Carlo simulation study, and an illustrative example.

Although the shape of the failure rate function plays an important role in repair and replacement strategies, the mean residual life (mrl) function is more relevant as the latter summarizes the entire residual life function, whereas the former considers only the risk of instantaneous failure. In this paper, we give a review of the different partial ordering results related tomrlorder, andproportional mrlmodel, with some characterization results. Some properties of theshifted mrlorder have been developed along with the closure of theshifted mrlorder under mixture-type operation. We show that redundancy at the component level is not superior to that at the system level, even when the lifetimes of the original and the spare components are iid, although it is well known that the result holds for usual stochastic order. Theshifted mrlorder is characterized in terms of the DMRL class. Some other characterizations are also presented here followed by the closure of the order under Poisson shock model.

In the context of multiple step-stress models, which is a special type of accelerated life-testing model, interest lies on the expected lifetimes of the experimental units under different stress levels. Although the expected lifetime is shortened as the stress level increases, this information has not been incorporated so far into the associated inferential procedures. For this reason, we develop here the order restricted maximum likelihood estimation (MLE) for multiple step-stress models with exponentially distributed lifetimes under Type-I, and Type-II censored sampling situations. Moreover, the existence of the unrestricted MLE for a certain stress level is conditional on observing failures at that particular stress level. Under the order restriction, MLE exist even for stress levels without observed failures, provided that these stress levels are internal. We also discuss hypothesis testing problems under order restrictions.

Step-stress accelerated degradation testing (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products (under a typical-use condition) when the available test items are very few. Recently, an optimal SSADT plan was proposed based on the assumption that the underlying degradation path follows a Wiener process. However, the degradation model of many materials (especially in the case of fatigue data) may be more appropriately modeled by a gamma process which exhibits a monotone increasing pattern. Hence, in practice, designing an efficient SSADT plan for a gamma degradation process is of great interest. In this paper, we first introduce the SSADT model when the degradation path follows a gamma process. Next, under the constraint that the total experimental cost does not exceed a pre-specified budget, the optimal settings such as sample size, measurement frequency, and termination time are obtained by minimizing the approximate variance of the estimated MTTF of the lifetime distribution of the product. Finally, an example is presented to illustrate the proposed method.

We investigate the effect of daily price limits, which trigger a trading halt if the limit is hit, in futures markets. Empirically, it has been observed that futures price limits are rarely hit. This could be because traders avoid putting in bid-ask quotes which could trigger a trading halt. If this is true, futures prices would cluster in a narrow region close to the limits without hitting them. We test this for the British pound futures contract for a period in which limits are imposed, by comparing the number of daily observations of futures prices which fall in the narrow region, with that predicted at the 99% confidence level by the distribution of the 'true' daily futures price. Our tests require that we calculate all possible combinations of a long time series, which we find we can do relatively efficiently with a parallel program.

BACKGROUND: Prediction of transmembrane (TM) helices by statistical methods suffers from lack of sufficient training data. Current best methods use hundreds or even thousands of free parameters in their models which are tuned to fit the little data available for training. Further, they are often restricted to the generally accepted topology "cytoplasmic-transmembrane-extracellular" and cannot adapt to membrane proteins that do not conform to this topology. Recent crystal structures of channel proteins have revealed novel architectures showing that the above topology may not be as universal as previously believed. Thus, there is a need for methods that can better predict TM helices even in novel topologies and families. RESULTS: Here, we describe a new method "TMpro" to predict TM helices with high accuracy. To avoid overfitting to existing topologies, we have collapsed cytoplasmic and extracellular labels to a single state, non-TM. TMpro is a binary classifier which predicts TM or non-TM using multiple amino acid properties (charge, polarity, aromaticity, size and electronic properties) as features. The features are extracted from sequence information by applying the framework used for latent semantic analysis of text documents and are input to neural networks that learn the distinction between TM and non-TM segments. The model uses only 25 free parameters. In benchmark analysis TMpro achieves 95% segment F-score corresponding to 50% reduction in error rate compared to the best methods not requiring an evolutionary profile of a protein to be known. Performance is also improved when applied to more recent and larger high resolution datasets PDBTM and MPtopo. TMpro predictions in membrane proteins with unusual or disputed TM structure (K+ channel, aquaporin and HIV envelope glycoprotein) are discussed. CONCLUSION: TMpro uses very few free parameters in modeling TM segments as opposed to the very large number of free parameters used in state-of-the-art membrane prediction methods, yet achieves very high segment accuracies. This is highly advantageous considering that high resolution transmembrane information is available only for very few proteins. The greatest impact of TMpro is therefore expected in the prediction of TM segments in proteins with novel topologies. Further, the paper introduces a novel method of extracting features from protein sequence, namely that of latent semantic analysis model. The success of this approach in the current context suggests that it can find potential applications in other sequence-based analysis problems. AVAILABILITY: http://linzer.blm.cs.cmu.edu/tmpro/ and http://flan.blm.cs.cmu.edu/tmpro/

The simple step-stress model under type-II censoring based on Weibull lifetimes, which provides a more flexible model than the exponential model, is considered in this paper. For this model, the maximum likelihood estimates (MLE) of its parameters, as well as the corresponding observed Fisher information matrix, are derived. The likelihood equations do not lead to closed-form expressions for the MLE, and they need to be solved by using an iterative procedure, such as the Newton-Raphson method. We also present a simplified estimator, which is easier to compute, and hence is suitable to use as an initial estimate in the iterative process for the determination of the MLE. We then evaluate the bias, and mean square error of these estimates; and provide asymptotic, and bootstrap confidence intervals for the parameters of the Weibull simple step-stress model. Finally, the results are illustrated with some examples.

We present a simple method for testing goodness-of-fit based on type-II right censored samples. Applying the property of order statistics due to Malmquist, we can transform any conventional type-II right censored sample of sizerout ofnfrom a uniform distribution to a complete sample of sizerfrom a uniform distribution. This result is used to develop the proposed goodness-of-fit test procedure. The simulation studies reveal that the proposed approach provides as good or better overall power than the method of Michael & Schucany.

We express the joint entropy of progressively censored order statistics in terms of an incomplete integral of the hazard function, and provide a simple estimate of the joint entropy of progressively Type-II censored data. We then construct a goodness-of-fit test statistic based on Kullback-Leibler information with progressively Type-II censored data. Finally, by using Monte Carlo simulations, the power of the test is estimated, and compared against several alternatives under different progressive censoring schemes

The prediction intervals proposed by J. F. Lawless (1971) and G. S. Lingappaiah (1973) for the exponential distribution are both simple to use. In this note, we make a comparison of these two prediction intervals based on the expected width of the prediction interval, as well as by means of the probability of the width of one being smaller than the other. For the computation of the latter, we use an algorithm, which is described briefly in the Appendix. Numerical results of these comparisons are presented for different choices of the parameters involved. Both these comparisons reveal that the prediction interval in is better than that in in that it has smaller expected width, as well as higher probability of having smaller width. Finally, we present an example to illustrate the results discussed in this paper.

There has been extensive research on goodness-of-fit procedures for testing whether or not a sample comes from a specified distribution. These goodness-of-fit tests range from graphical techniques, to tests which exploit characterization results for the specified underlying model. In this article, we propose a goodness-of-fit test for the location-scale family based on progressively Type-II censored data. The test statistic is based on sample spacings, and generalizes a test procedure proposed by Tiku . The null distribution of the test statistic is shown to be approximated closely by a s-normal distribution. However, in certain situations it would be better to use simulated critical values instead of the s-normal approximation. We examine the performance of this test for the s-normal and extreme-value (Gumbel) models against different alternatives through Monte Carlo simulations. We also discuss two methods of power approximation based on s-normality, and compare the results with those obtained by simulation. Results of the simulation study for a wide range of sample sizes, censoring schemes, and different alternatives reveal that the proposed test has good power properties in detecting departures from the s-normal and Gumbel distributions. Finally, we illustrate the method proposed here using real data from a life-testing experiment. It is important to mention here that this test can be extended to multi-sample situations in a manner similar to that of Balakrishnan et al.

We consider in this work a k-step-stress accelerated test with equal duration steps /spl tau/. Censoring is allowed at each change stress point i/spl tau/, i=1,...k. The problem of choosing the optimal /spl tau/ is addressed using variance optimality as well as determinant-optimality criteria. We investigate in detail the case of progressively Type-I right censored data with a single stress variable.

The likelihood equations based on a progressively Type-II censored sample from a Gaussian distribution do not provide explicit solutions in any situation except the complete sample case. This paper examines numerically the bias and mean square error of the MLE, and demonstrates that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic s-normality are unsatisfactory, and particularly so when the effective sample size is small. Therefore, this paper suggests using unconditional simulated percentage points of these pivotal quantities for constructing s-confidence intervals. An approximation of the Gaussian hazard function is used to develop approximate estimators which are explicit and are almost as efficient as the MLE in terms of bias and mean square error; however, the probability coverages of the corresponding pivotal quantities based on asymptotic s-normality are also unsatisfactory. A wide range of sample sizes and progressive censoring schemes are used in this study.

This paper presents reliability sampling plans for the lognormal distribution based on progressively censored samples. In constructing these sampling plans, large-sample approximations to the best linear unbiased estimators of the location and scale parameters are used. For some selected progressive censoring schemes, reliability sampling plans are tabulated for p/sub /spl alpha// and p/sub /spl beta// to match MIL-STD-105. While in general, variable-sampling plans require smaller sample size when compared with attribute-sampling plans, the ordinary complete and right-censored life test experiments are special cases of the progressively censored experiment. Hence, the progressively censored reliability sampling plans in this paper are widely applicable. General application of the procedure is discussed, and two examples are provided.