Discrete time cold standby repairable system: Combinatorial analysis

ABSTRACT

In this article, we obtain exact expression for the distribution of the time to failure of discrete time cold standby repairable system under the classical assumptions that both working time and repair time of components are geometric. Our method is based on alternative representation of lifetime as a waiting time random variable on a binary sequence, and combinatorial arguments. Such an exact expression for the time to failure distribution is new in the literature. Furthermore, we obtain the probability generating function and the first two moments of the lifetime random variable.     

Nonparametric Quantile Estimation with Correlated Failure Time Data

In biomedical studies, correlated failure time data arise often. Although point and confidence interval estimation for quantiles with independent censored failure time data have been extensively studied, estimation for quantiles with correlated failure time data has not been developed. In this article, we propose a nonparametric estimation method for quantiles with correlated failure time data. We derive the asymptotic properties of the quantile estimator and propose confidence interval estimators based on the bootstrap and kernel smoothing methods. Simulation studies are carried out to investigate the finite sample properties of the proposed estimators. Finally, we illustrate the proposed method with a data set from a study of patients with otitis media.     

Dependent masking and system life data analysis: Bayesian inference for two-component systems

Data from field operations of a system is often used to estimate the reliability of components. Under ideal circumstances, this system field data contains the time to failure along with information on the exact component responsible for the system failure. However, in many cases, the exact component causing the failure of the system cannot be identified, and is considered to be masked. Previously developed models for estimation of component reliability from masked system life data have been based upon the assumption that masking occurs independently of the true cause of system failure. In this paper we develop a Bayesian methodology for estimating component reliabilities from masked system life data when the probability of masking is dependent upon the true cause of system failure. The Bayesian approach is illustrated for the case of a two-component system of exponentially distributed components.     

Small sample study on estimators of life distributions and of mean survival times using randomly censored data

Whereas large-sample properties of the estimators of survival distributions using censored data have been studied by many authors, exact results for small samples have been difficult to obtain. In this paper we obtain the exact expression for the ath moment (a > 0) of the Bayes estimator of survival distribution using the censored data under proportional hazard model. Using the exact expression we compute the exact mean, variance and MSE of the Bayes estimator. Also two estimators ofthe mean survival time based on the Kaplan-Meier estimator and the Bayes estimator are compared for small samples under proportional hazards.     

Statistical Analysis of Linear Degradation and Failure Time Data with Multiple Failure Modes

The paper considers linear degradation and failure time models with multiple failure modes. Dependence of traumatic failure intensities on the degradation level are included into the models. Estimators of traumatic event cumulative intensities, and of various reliability characteristics are proposed. Prediction of residual reliability characteristics given a degradation value at a given moment is discussed. Non-parametric, semiparametric and parametric estimation methods are given. Theorems on simultaneous asymptotic distribution of random functions characterising degradation and intensities of traumatic events are proposed. Asymptotic properties of unconditional and residual reliability characteristics estimators are given. Real tire wear and failure time data are analysed.     

Tests of independence for censored bivariate failure time data

Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of chi2-type tests for independence between pairs of failure times after adjusting for covariates. A bivariate accelerated failure time model is proposed for the joint distribution of bivariate failure times while leaving the dependence structures for related failure times completely unspecified. Theoretical properties of the proposed tests are derived and variance estimates of the test statistics are obtained using a resampling technique. Simulation studies show that the proposed tests are appropriate for practical use. Two examples including the study of infection in catheters for patients on dialysis and the diabetic retinopathy study are also given to illustrate the methodology.     

Nonparametric Test for Doubly Interval-Censored Failure Time Data

This paper considers comparison of discrete failure time distributions when the survival time of interest measures elapsed time between two related events and observations on the occurrences of both events could be interval-censored. This kind of data is often referred to as doubly interval-censored failure time data. If the occurrence of the first event defining the survival time can be exactly observed, the data are usually referred to as interval-censored data. For the comparison problem based on interval-censored failure time data, Sun (1996) proposed a nonparametric test procedure. In this paper we generalize the procedure given in Sun (1996) to doubly interval-censored data case and the generalized test is evaluated by simulations.     

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

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

Inference for cumulative incidence on discrete failure times with competing risks

In the competing risks analysis, most inferences have been developed based on continuous failure time data. However, failure times are sometimes observed as being discrete. We propose nonparametric inferences for the cumulative incidence function for pure discrete data with competing risks. When covariate information is available, we propose semiparametric inferences for direct regression modelling of the cumulative incidence function for grouped discrete failure time data with competing risks. Simulation studies show that the procedures perform well. The proposed methods are illustrated with a study of contraceptive use in Indonesia.     

Failure Inference From a Marker Process Based on a Bivariate Wiener Model

Many models have been proposed that relate failure times and stochastic time-varying covariates. In some of these models, failure occurs when a particular observable marker crosses a threshold level. We are interested in the more difficult, and often more realistic, situation where failure is not related deterministically to an observable marker. In this case, joint models for marker evolution and failure tend to lead to complicated calculations for characteristics such as the marginal distribution of failure time or the joint distribution of failure time and marker value at failure. This paper presents a model based on a bivariate Wiener process in which one component represents the marker and the second, which is latent (unobservable), determines the failure time. In particular, failure occurs when the latent component crosses a threshold level. The model yields reasonably simple expressions for the characteristics mentioned above and is easy to fit to commonly occurring data that involve the marker value at the censoring time for surviving cases and the marker value and failure time for failing cases. Parametric and predictive inference are discussed, as well as model checking. An extension of the model permits the construction of a composite marker from several candidate markers that may be available. The methodology is demonstrated by a simulated example and a case application.     

Rank Regression Analysis of Multivariate Failure Time Data Based on Marginal Linear Models

Abstract.  Multivariate failure time data arises when each study subject can potentially ex-perience several types of failures or recurrences of a certain phenomenon, or when failure times are sampled in clusters. We formulate the marginal distributions of such multivariate data with semiparametric accelerated failure time models (i.e. linear regression models for log-transformed failure times with arbitrary error distributions) while leaving the dependence structures for related failure times completely unspecified. We develop rank-based monotone estimating functions for the regression parameters of these marginal models based on right-censored observations. The estimating equations can be easily solved via linear programming. The resultant estimators are consistent and asymptotically normal. The limiting covariance matrices can be readily estimated by a novel resampling approach, which does not involve non-parametric density estimation or evaluation of numerical derivatives. The proposed estimators represent consistent roots to the potentially non-monotone estimating equations based on weighted log-rank statistics. Simulation studies show that the new inference procedures perform well in small samples. Illustrations with real medical data are provided.     

Semiparametric Likelihood Estimation in the Clayton–Oakes Failure Time Model

Multivariate failure time data arise when the sample consists of clusters and each cluster contains several possibly dependent failure times. The Clayton–Oakes model (Clayton, 1978; Oakes, 1982) for multivariate failure times characterizes the intracluster dependence parametrically but allows arbitrary specification of the marginal distributions. In this paper, we discuss estimation in the Clayton–Oakes model when the marginal distributions are modeled to follow the Cox (1972) proportional hazards regression model. Parameter estimation is based on an approximate generalized maximum likelihood estimator. We illustrate the model's application with example datasets.     

Estimation in Maxwell distribution with randomly censored data

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

A Statistical Model for Shelf Life Estimation Using Sensory Evaluations Scores

Abstract

This article focuses on the problem of estimating the shelf life of food products by modeling the results coming from sensory evaluations. In such studies, trained panelists are asked to judge food attributes by reference to a scale of numbers (scores varying often from 0 to 6). The usual statistical approach for data analysis is to fit a regression line relating the scores and the time of evaluation. The estimate of the shelf life is obtained by solving the regression equation and replacing the score by a cut-off point (which indicates product “failure”) previously chosen by the food company. The procedure used in these sensory evaluations is such that one never knows the exact “time to failure”. Consequently, data arising from these studies are either right or left censored. We propose a model which incorporates these informations and assumes a Weibull for the underlying distribution of the failure time. Simulation studies were implemented. The approach was used in a real data set coming from sensory evaluations of a dehydrated food product.     

Exact inference for competing risks model with generalized type-I hybrid censored exponential data

Recently, exact inference under hybrid censoring scheme has attracted extensive attention in the field of reliability analysis. However, most of the authors neglect the possibility of competing risks model. This paper mainly discusses the exact likelihood inference for the analysis of generalized type-I hybrid censoring data with exponential competing failure model. Based on the maximum likelihood estimates for unknown parameters, we establish the exact conditional distribution of parameters by conditional moment generating function, and then obtain moment properties as well as exact confidence intervals (CIs) for parameters. Furthermore, approximate CIs are constructed by asymptotic distribution and bootstrap method as well. We also compare their performances with exact method through the use of Monte Carlo simulations. And finally, a real data set is analysed to illustrate the validity of all the methods developed here.     

ANALYSIS OF MASKED DATA IN A DEPENDENT COMPETING RISKS MODEL UNDER UNKNOWN ENVIRONMENT

ABSTRACT

System failure data is often analyzed to estimate component reliabilities. Due to cost and time constraints, the exact component causing the failure of the system cannot be identified in some cases. This phenomenon is called masking. Further, it is sometimes necessary for us to take account of the influence of the operating environment. Here we consider a series system, operating under unknown environment, of two components whose failure times follow the Marshall-Olkin bivariate exponential distribution. We present a maximum likelihood approach for obtaining estimators from the masked data for this system. From a simulation study, we found that the relative errors of the estimates are almost well behaved even for small or moderate expected number of systems whose cause of failure is identified.     

Empirical likelihood method for the multivariate accelerated failure time models

In applications, multivariate failure time data appears when each study subject may potentially experience several types of failures or recurrences of a certain phenomenon, or failure times may be clustered. Three types of marginal accelerated failure time models dealing with multiple events data, recurrent events data and clustered events data are considered. We propose a unified empirical likelihood inferential procedure for the three types of models based on rank estimation method. The resulting log-empirical likelihood ratios are shown to possess chi-squared limiting distributions. The properties can be applied to do tests and construct confidence regions without the need to solve the rank estimating equations nor to estimate the limiting variance-covariance matrices. The related computation is easy to implement. The proposed method is illustrated by extensive simulation studies and a real example.     

Progressive censoring with fixed censoring times

In this paper, we revisit the progressive Type-I censoring scheme as it has originally been introduced by Cohen [Progressively censored samples in life testing. Technometrics. 1963;5(3):327–339]. In fact, original progressive Type-I censoring proceeds as progressive Type-II censoring but with fixed censoring times instead of failure time based censoring times. Apparently, a time truncation has been added to this censoring scheme by interpreting the final censoring time as a termination time. Therefore, not much work has been done on Cohens's original progressive censoring scheme with fixed censoring times. Thus, we discuss distributional results for this scheme and establish exact distributional results in likelihood inference for exponentially distributed lifetimes. In particular, we obtain the exact distribution of the maximum likelihood estimator (MLE). Further, the stochastic monotonicity of the MLE is verified in order to construct exact confidence intervals for both the scale parameter and the reliability.     

A modified approximate method for analysis of degradation data

Estimation of the lifetime distribution of industrial components and systems yields very important information for manufacturers and consumers. However, obtaining reliability data is time consuming and costly. In this context, degradation tests are a useful alternative approach to lifetime and accelerated life tests in reliability studies. The approximate method is one of the most used techniques for degradation data analysis. It is very simple to understand and easy to implement numerically in any statistical software package. This paper uses time series techniques in order to propose a modified approximate method (MAM). The MAM improves the standard one in two aspects: (1) it uses previous observations in the degradation path as a Markov process for future prediction and (2) it is not necessary to specify a parametric form for the degradation path. Characteristics of interest such as mean or median time to failure and percentiles, among others, are obtained by using the modified method. A simulation study is performed in order to show the improved properties of the modified method over the standard one. Both methods are also used to estimate the failure time distribution of the fatigue-crack-growth data set.     

A Discrete Time Parametric Model for the Analysis of Failure Time Data

This paper introduces a parametric discrete failure time model which allows a variety of smooth hazard function shapes, including shapes which are not readily available with continuous failure time models. The model is easy to fit, and statistical inference is simple. Further, it is readily extended to allow for differences between subjects while retaining the ease of fit and simplicity of statistical inference. The performance of the discrete time analysis is demonstrated by application to several data sets.