Tests for homogeneity of extreme value scale parameters

It is often assumed in situations in which life data from Weibull or extreme-value distributions are involved that data in different samples come from extreme-value distributions with the same scale parameter (equivalently, Weibull distributions with the same shape parameter). This paper proposes a number of tests for homogeneity for extreme-value scale parameters, based on a number of commonly used estimators for these scale parameters. Previous theoretical work and some simulation results provided here indicate that the null distributions of the test statistics proposed are well approximated by the x2 distribution under a wide range of conditions     

Some observations in testing for nonstandard mixtures

We consider optimal testing procedures for specific models of early and instantaneous failures in reliability studies. These models are typically used to accommodate lifetime data that have a higher concentration of failures near time zero. We show that it is possible to derive uniformly most powerful tests, for testing the mixing parameter in the instantaneous failure model, for general lifetime distributions. A novel procedure to test for early failures, which uses an invariance property of the maximum likelihood estimate of the nuisance parameter, is also presented.     

On optimal tests for separate hypotheses and conditional probability integral transformations

Consider the problem of testing the composite null hypothesis that a random sample X₁,…,X_n is from a parent which is a member of a particular continuous parametric family of distributions against an alternative that it is from a separate family of distributions. It is shown here that in many cases a uniformly most powerful similar (UMPS) test exists for this problem, and, moreover, that this test is equivalent to a uniformly most powerful invariant (UMPI) test. It is also seen in the method of proof used that the UMPS test statistic Is a function of the statistics U₁,…,U_n?k obtained by the conditional probability integral transformations (CPIT), and thus that no Information Is lost by these transformations, It is also shown that these optimal tests have power that is a nonotone function of the null hypothesis class of distributions, so that, for example, if one additional parameter for the distribution is assumed known, then the power of the test can not lecrease. It Is shown that the statistics U₁, …, U_n?k are independent of the complete sufficient statistic, and that these statistics have important invariance properties. Two examples at given. The UMPS tests for testing the two-parameter uniform family against the two-parameter exponential family, and for testing one truncation parameter distribution against another one are derived.     

Optimal progressive group-censoring plans for exponential distribution in presence of cost constraint

This article discusses a life test under progressive type-I group-censoring. We use maximum likelihood method to obtain the point and interval estimators of the parameter of lifetime distribution. In order to obtain a precise estimate of mean life, one needs to design an optimal life test. Thus, this article proposes an approach to determine the number of test units, number of inspections, and length of inspection interval of a life test under a pre-determined budget of experiment such that the asymptotic variance of estimator of mean life is minimum. The method will be applied to two numerical examples and the sensitivity analysis will be investigated.     

Optimal progressive group censoring scheme under cost considerations for pareto distribution

In this article, optimal design under the restriction of pre-determined budget of experiment is developed for the Pareto distribution when the life test is progressively group censored. We use the maximum-likelihood method to obtain the point estimator of the Pareto parameter. We propose two approaches to decide the number of test units, the number of inspections, and the length of inspection interval under limited budget such that the asymptotic variance of estimator of Pareto parameter is minimum. A numerical example is given to illustrate the proposed method. Some sensitivity analysis is also studied.     

Fisher information in generalized progressive hybrid-censored data

This article provides a simple expression of the Fisher information matrix about the unknown parameter(s) of the underlying lifetime model under the generalized progressive hybrid censoring scheme. The expressions of the expected number of failures and the expected duration of life test are also derived. Exponential and Weibull lifetime models are considered for numerical illustrations. Finally, Fisher information-based optimal schemes are discussed for the Weibull lifetime model.     

Bayesian prediction of future number of failures based on finite mixture of general class of distributions

This paper is concerned with the Bayesian prediction problem of the number of components which will fail in a future time interval. The failure times are distributed according to a finite mixture of a general class of distributions. Type-I censored sample from this nonhomogeneous population and a general class of prior density functions are used. A one-sample scheme is used to predict the number of failures in a future time interval. An example of a finite mixture of k exponential components is given to illustrate our results.     

Analysis of simple step-stress model in presence of competing risks

ABSTRACT

In this article, we consider a simple step-stress life test in the presence of exponentially distributed competing risks. It is assumed that the stress is changed when a pre-specified number of failures takes place. The data is assumed to be Type-II censored. We obtain the maximum likelihood estimators of the model parameters and the exact conditional distributions of the maximum likelihood estimators. Based on the conditional distribution, approximate confidence intervals (CIs) of unknown parameters have been constructed. Percentile bootstrap CIs of model parameters are also provided. Optimal test plan is addressed. We perform an extensive simulation study to observe the behaviour of the proposed method. The performances are quite satisfactory. Finally we analyse two data sets for illustrative purposes.     

Estimation procedures for Maxwell distribution under type-I progressive hybrid censoring scheme

The Maxwell (or Maxwell–Boltzmann) distribution was invented to solve the problems relating to physics and chemistry. It has also proved its strength of analysing the lifetime data. For this distribution, we consider point and interval estimation procedures in the presence of type-I progressively hybrid censored data. We obtain maximum likelihood estimator of the parameter and provide asymptotic and bootstrap confidence intervals of it. The Bayes estimates and Bayesian credible and highest posterior density intervals are obtained using inverted gamma prior. The expression of the expected number of failures in life testing experiment is also derived. The results are illustrated through the simulation study and analysis of a real data set is presented.     

A two-stage group sampling plan based on truncated life tests for a general distribution

A two-stage group acceptance sampling plan based on a truncated life test is proposed, which can be used regardless of the underlying lifetime distribution when multi-item testers are employed. The decision upon lot acceptance can be made in the first or second stage according to the number of failures from each group. The design parameters of the proposed plan such as number of groups required and the acceptance number for each of two stages are determined independently of an underlying lifetime distribution so as to satisfy the consumer's risk at the specified unreliability. Single-stage group sampling plans are also considered as special cases of the proposed plan and compared with the proposed plan in terms of the average sample number and the operating characteristics. Some important distributions are considered to explain the procedure developed here.     

Statistical Analysis of Exponential Lifetimes Under an Integrated Type-Ii Interval Censoring Scheme

In this paper, we propose a new type of censoring scheme which combines the features of grouped censoring and Type-II censoring. Statistical analysis of exponentially distributed lifetimes observed under this scheme is considered. The MLE of the model parameter and its asymptotic variance are derived. Furthermore, expressions for the expected experiment time and the expected number of failures are given. A numerical study is conducted to study these quantities and the results are compared with those under a Type-II censored plan. This comparison provides useful insight on the choice of these plans in designing a life test.     

Predictive inference for future responses given a doubly censored sample from a two parameter exponential distribution

In this paper, we derive the predictive distributions of one and several future responses including their average, on the basis of a type II doubly censored sample from a two parameter exponential life testing model. We also determine the highest predictive density interval for a future response. A numerical example is provided to illustrate the results.     

Estimation of Weibull distribution parameters for irregular interval group failure data with unknown failure times

SUMMARY This paper presents three methods for estimating Weibull distribution parameters for the case of irregular interval group failure data with unknown failure times. The methods are based on the concepts of the piecewise linear distribution function (PLDF), an average interval failure rate (AIFR) and sequential updating of the distribution function (SUDF), and use an analytical approach similar to that of Ackoff and Sasieni for regular interval group data. Results from a large number of simulated case problems generated with specified values of Weibull distribution parameters have been presented, which clearly indicate that the SUDF method produces near-perfect parameter estimates for all types of failure pattern. The performances of the PLDF and AIFR methods have been evaluated by goodness-of-fit testing and statistical confidence limits on the shape parameter. It has been found that, while the PLDF method produces acceptable parameter estimates, the AIFR method may fail for low and high shape parameter values that represent the cases of random and wear-out types of failure. A real-life application of the proposed methods is also presented, by analyzing failures of hydrogen make-up compressor valves in a petroleum refinery.     

Statistical inference for the availability of repairable system with general repair,reboot delay,and switching railure

We studied the inferences of an availability system with reboot delay and standby switching failures in which the system consisted of two operating units and one warm standby. The system was studied under the assumption that the time-to-failure and the time-to-repair were assumed to follow an exponential and a general distribution. The reboot times are assumed to be exponentially distributed with parameter β. We constructed a consistent and asymptotically normal estimator of availability for such a repairable system. Based on this estimator, interval estimation and testing hypothesis were developed by using logit transformation. To implement the simulation inference for the system availability, we adopted two repair-time distributions—namely, lognormal and Weibull; and three types of Weibull distributions—characterized by their shape parameters—were considered. Finally, appropriate tables and figures of all simulation results have been included.     

A class of test statistics for testing whether new is better than used

A class of test statistics Is proposed for testing that a life distribution Is exponential against that it is new better than used and not exponential, Consistency, unbiasedness and the asymptotic normality for the one class of class of test are proved. The efflication are compeled and compared with some otner statinties. The proposed test is shown to perform well against other tests.     

A new test for the extreme value distribution

In this paper a test statistic which is a modification of the W statistic for testing the goodness of fit for the two paremeter extreme value (smallest element) distribution is proposed. The test statistic Is obtained as the ratio of two linear estimates of the scale parameter. It Is shown that the suggested statistic is computationally simple and has good power properties. Percentage points of the statistic are obtained by performing Monte Carlo experiments. An example is given to illustrate the test procedure.     

The large sample coverage probability of confidence intervals in general regression models after a preliminary hypothesis test

We derive a computationally convenient formula for the large sample coverage probability of a confidence interval for a scalar parameter of interest following a preliminary hypothesis test that a specified vector parameter takes a given value in a general regression model. Previously, this large sample coverage probability could only be estimated by simulation. Our formula only requires the evaluation, by numerical integration, of either a double or a triple integral, irrespective of the dimension of this specified vector parameter. We illustrate the application of this formula to a confidence interval for the odds ratio of myocardial infarction when the exposure is recent oral contraceptive use, following a preliminary test where two specified interactions in a logistic regression model are zero. For this real‐life data, we compare this large sample coverage probability with the actual coverage probability of this confidence interval, obtained by simulation.     

Optimal progressive interval censoring plan under accelerated life test with limited budget

In this paper, we investigate some inference and design problems related to multiple constant-stress accelerated life test with progressive type-I interval censoring. A Weibull lifetime distribution at each stress-level combination is considered. The scale parameter of Weibull distribution is assumed to be a log-linear function of stresses. We obtain the estimates of the unknown parameters through the method of maximum likelihood, and also derive the Fisher's information matrix. The optimal number of test units, number of inspections, and length of the inspection interval are determined under D-optimality, T-optimality, and E-optimality criteria with cost constraint. An algorithm based on nonlinear mixed-integer programming is proposed to the optimal solution. The sensitivity of the optimal solution to changes in the values of the different parameters is studied.     

Gamma-minimax results for the class of unimodal priors

Olman and Shmundak proved 1985 that in estimating a bounded normal mean under squared error loss the Bayes estimator with respect to the uniform distribution on the parameter interval is gamma-minimax when the parameter interval is sufficiently small and the class of priors consists of all symmetric and unimodal distributions. Recently, one of the authors showed that this result remains valid for quite general families of distributions which satisfy some regularity conditions. In the present paper a generalization to the class of unimodal priors with fixed mode is derived. It is proved that the Bayes estimator with respect to a suitable mixture of two uniform distributions is gamma-minimax for sufficiently small parameter intervals. To that end appropriate characterizations of a saddle point in the corresponding statistical games are established. Some results of a numerical study are presented.     

Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.