Development of life-test sampling plans for exponential distributions based on accelerated life testing

This paper presents a procedure for developing life-test sampling plans for exponential distributions based upon accelerated life testing(ALT). Type II censoring is assumed at each overstress level. The derived test statistic is shown to be a quotient of two independent random variables, each of which is a rational power of a Chi-square random variable. The distribution of the test statistic is characterized by the H-function, which can be numerically evaluated to obtain desired sampling plans.     

Statistical analysis of Weibull distributed lifetime data under Type II progressive censoring with binomial removals

This paper considers the analysis of Weibull distributed lifetime data observed under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial distribution. Maximum likelihood estimators of the parameters and their asymptotic variances are derived. The expected time required to complete the life test under this censoring scheme is investigated.     

Reliability sampling plans for the Weibull distribution under Type II progressive censoring with binomial removals

This paper presents reliability sampling plans for the Weibull distribution under Type II progressive censoring with random removals (PCR), where the number of units removed at each failure time follows a binomial distribution. To construct the sampling plans, the sample size n and the acceptance constant k are determined based on asymptotic distribution theory. The resulting sampling plans are tabulated for selected specifications under the proposed censoring scheme. Furthermore, a Monte Carlo simulation is conducted to validate the true probability of acceptance for the designed sampling plans.     

Inference in the Pareto distribution based on progressive Type II censoring with random removals

This study considers the estimation problem for the Pareto distribution based on progressive Type II censoring with random removals. The number of units removed at each failure time has a discrete uniform distribution. We are going to use the maximum likelihood method to obtain the estimator of parameter. The expectation and variance of the maximum likelihood estimator will be derived. The expected time required to complete such an experiment will be computed. Some numerical results of expected test times are carried out for this type of progressive censoring and other sampling schemes.     

Accelerated life test sampling plans for the Weibull distribution under Type I progressive interval censoring with random removals

This paper considers the design of accelerated life test (ALT) sampling plans under Type I progressive interval censoring with random removals. We assume that the lifetime of products follows a Weibull distribution. Two levels of constant stress higher than the use condition are used. The sample size and the acceptability constant that satisfy given levels of producer's risk and consumer's risk are found. In particular, the optimal stress level and the allocation proportion are obtained by minimizing the generalized asymptotic variance of the maximum likelihood estimators of the model parameters. Furthermore, for validation purposes, a Monte Carlo simulation is conducted to assess the true probability of acceptance for the derived sampling plans.     

Parameters estimation for weibull distributed lifetimes under progressive censoring with random removeals

This paper considers the estimation problem when lifetimes are Weibull distributed and are collected under a Type-II progressive censoring with random removals, where the number of units removed at each failure time follows a uniform discrete distribution. The expected time of this censoring plan is discussed and compared numerically to that under a Type II censoring without removal. Maximum likelihood estimator of the parameters and their asymptotic variances are derived.     

The exact hypothesis test for the shape parameter of a new two-parameter distribution with the bathtub shape or increasing failure rate function under progressive censoring with random removals

This study considers the exact hypothesis test for the shape parameter of a new two-parameter distribution with the shape of a bathtub or increasing failure rate function under type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial or a uniform distribution. Several test statistics are proposed and one numerical example is provided to illustrate the proposed hypothesis test for the shape parameter. Finally, a simulation study is performed to compare the power performances of all proposed test statistics. We concluded that the test statistic w ₁ is more attractive than other methods as it has better performance than other test statistics for most cases based on the criteria of maximum power.     

Optimum constant-stress life test plans for Pareto distribution under type-I censoring

The paper considers the case of constant-stress partially accelerated life testing (CSPALT) when two stress levels are involved under type-I censoring. The lifetimes of test items are assumed to follow a two-parameter Pareto lifetime distribution. Maximum-likelihood method is used to estimate the parameters of CSPALT model. Confidence intervals for the model parameters are constructed. Optimum CSPALT plans that determine the best choice of the proportion of test units allocated to each stress are developed. Such optimum test plans minimize the generalized asymptotic variance of the maximum-likelihood estimators of the model parameters. For illustration, Monte Carlo simulation studies are presented.     

Optimal step-stress test under type I progressive group-censoring with random removals

Some traditional life tests result in no or very few failures by the end of test. In such cases, one approach is to do life testing at higher-than-usual stress conditions in order to obtain failures quickly. This paper discusses a k-level step-stress accelerated life test under type I progressive group-censoring with random removals. An exponential failure time distribution with mean life that is a log-linear function of stress and a cumulative exposure model are considered. We derive the maximum likelihood estimators of the model parameters and establish the asymptotic properties of the estimators. We investigate four selection criteria which enable us to obtain the optimum test plans. One is to minimize the asymptotic variance of the maximum likelihood estimator of the logarithm of the mean lifetime at use-condition, and the other three criteria are to maximize the determinant, trace and the smallest eigenvalue of Fisher's information matrix. Some numerical studies are discussed to illustrate the proposed criteria.     

Bayesian inference for the Pareto lifetime model under progressive censoring with binomial removals

This paper considers the estimation and prediction problems when lifetimes are Pareto-distributed and are collected under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a Binomial distribution. The analysis is carried out within the Bayesian context.     

Expected experiment times for the Weibull distribution under progressive censoring with random removals

SUMMARY This paper considers the expected experiment times for Weibull-distributed lifetimes under type II progressive censoring, with the numbers of removals being random. The formula to compute the expected experiment times is given. A detailed numerical study of this expected time is carried out for different combinations of model parameters. Furthermore, the ratio of the expected experiment time under this type of progressive censoring to the expected experiment time under complete sampling is studied.     

Bayesian modelling of health insurance losses

The purpose of this paper is to build a model for aggregate losses which constitutes a crucial step in evaluating premiums for health insurance systems. It aims at obtaining the predictive distribution of the aggregate loss within each age class of insured persons over the time horizon involved in planning employing the Bayesian methodology. The model proposed using the Bayesian approach is a generalization of the collective risk model, a commonly used model for analysing risk of an insurance system. Aggregate loss prediction is based on past information on size of loss, number of losses and size of population at risk. In modelling the frequency and severity of losses, the number of losses is assumed to follow a negative binomial distribution, individual loss sizes are independent and identically distributed exponential random variables, while the number of insured persons in a finite number of possible age groups is assumed to follow the multinomial distribution. Prediction of aggregate losses is based on the Gibbs sampling algorithm which incorporates the missing data approach.     

Optimal design of accelerated life tests under modified stress loading methods

SUMMARY Most of the previous work on optimal design of accelerated life test (ALT) plans has assumed instantaneous changes in stress levels, which may not be possible or desirable in practice, because of the limited capability of test equipment, possible stress shocks or the presence of undesirable failure modes. We consider the case in which stress levels are changed at a finite rate, and develop two types of ALT plan under the assumptions of exponential lifetimes of test units and type I censoring. One type of plan is the modified step-stress ALT plan, and the other type is the modified constant-stress ALT plan. These two plans are compared in terms of the asymptotic variance of the maximum likelihood estimator of the log mean lifetime for the use condition (i.e. avar\[ln (0)]). Computational results indicate that, for both types of plan, avar\[ln (0)] is not sensitive to the stress-increasing rate R, if R is greater than or equal to 10, say, in the standardized scale. This implies that the proposed stress loading method can be used effectively with little loss in statistical efficiency. In terms of avar\[ln (0)], the modified step-stress ALT generally performs better than the modified constant-stress ALT, unless R or the probability of failure until the censoring time under a certain stress-increasing rate is small. We also compare the progressive-stress ALT plan with the above two modified ALT plans in terms of avar\[ln (0)], using the optimal stress-increasing rate R* determined for the progressivestress ALT plan. We find that the proposed ALTs perform better than the progressivestress ALT for the parameter values considered.     

Optimal design for constant-stress accelerated degradation test based on gamma process

This paper addresses the optimal design problems for constant-stress accelerated degradation test (CSADT) based on gamma processes with fixed effect and random effect. For three optimization criteria, we prove that optimal CSADT plans with multiple stress levels degenerate to two-stress-level test plans only using the minimum and maximum stress levels under model assumptions. Under each optimization criterion, the optimal sample size allocation proportions for the minimum and maximum stress levels are determined theoretically. The effect of the stress level on the objective functions is also discussed. A numerical example and a simulation study are provided to illustrate the obtained results.     

Sequential testing with interval censored data ∗

Let X ₁,…,X _n be the lifetimes of n items put on testing at the same time. It is not possible to observe the actual lifetimes. However, it is possible to inspect the items at a finite number of time intervals. At each time of inspection, the number of failures can be recorded. Only these numbers of failures at times of inspections will be available to make decision on the distribution of the lifetimes. Decision can be made at the time of inspection. A “sequential” statistical test is developed to test the mean levels of the lifetimes when the probability distribution is assumed to be exponential. Some numerical results will be presented. The power and the expected time for the decision are compared with those for the idealized situation when each and every actual lifetime is recorded. They are also compared with those for the case when one and only one inspection is allowed to make the decision.     

Modeling systems with two dependent components under bivariate shock models

Series and parallel systems consisting of two dependent components are studied under bivariate shock models. The random variables N₁ and N₂ that represent respectively the number of shocks until failure of component 1 and component 2 are assumed to be dependent and phase-type. The times between successive shocks are assumed to follow a continuous phase-type distribution, and survival functions and mean time to failure values of series and parallel systems are obtained in matrix forms. An upper bound for the joint survival function of the components is also provided under the particular case when the times between shocks follow exponential distribution.     

Bayesian acceptance sampling plans following economic criteria: An application to paper pulp manufacturing

The main purposes of this paper are to derive Bayesian acceptance sampling plans regarding the number of defects per unit of product, and to illustrate how to apply the methodology to the paper pulp industry. The sampling plans are obtained following an economic criterion: minimize the expected total cost of quality. It has been assumed that the number of defects per unit of product follows a Poisson distribution with process average 5 , whose prior information is described either for a gamma or for a non- informative distribution. The expected total cost of quality is composed of three independent components: inspection, acceptance and rejection. Both quadratic and step-loss functions have been used to quantify the cost incurred for the acceptance of a lot containing units with defects. Combining the prior information on 5 with the loss functions, four different sampling plans are obtained. When the quadratic-loss function is used, an analytical relation between the optimum settings of the sample size and the acceptance number is derived. The robustness analysis indicates that the sampling plans obtained are robust with respect to the prior distribution of the process average as well as to the misspecification of its mean and variance.     

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.     

Bayesian inference using product of spacings function for Progressive hybrid Type-I censoring scheme

This article is devoted to the development of product of spacings estimator for a Progressive hybrid Type-I censoring scheme with binomial removals. The experimental units are assumed to follow inverse Lindley distribution. We propose a Bayes estimator of associated scale parameter based on the product of spacings function and simultaneously compare it with that obtained under a usual Bayesian estimation procedure. The estimators are obtained under the squared error loss function along with corresponding HP intervals evaluated by using the Markov chain Monte-Carlo technique. The classical product of spacings estimator has also been derived and compared with the maximum likelihood estimator in addition to 95% average asymptotic confidence intervals. The applicability of the proposed methods is demonstrated by analysing a real data of guinea pigs affected with tuberculosis for the considered censoring scheme.     

Life test sampling plans for Weibull distributed lifetimes under accelerated hybrid censoring

Life test sampling plans (LSPs) for the Weibull distribution are usually developed under the assumptions that the shape parameter is known and the life test is conducted at an accelerated condition for which the acceleration factor (AF) is known. However, the sensitivities of a plan to the assumed shape parameter and AF have been rarely investigated. This paper considers the case where the life test is hybrid censored and develops attributes LSPs under the above assumptions. Then, sensitivity analyses are conducted to assess the effects of the uncertainties in the assumed AF and shape parameter on the actual producer and consumer risks. A method is also developed for constructing LSPs that can accommodate these uncertainties.