Expected Test Time Research Articles

Optimal truncated sequential test for exponential distribution

In order to save on test costs, the optimal truncated sequential test (OTST) for the parameter of the exponential distribution is studied. According to Bayes decision theory, dynamic programming methods and procedures are established to solve the OTST. Through comparison with the test plans provided by international standard IEC 61124 and Russian national standard GOST R 27.402, the results show that the OTSTs solved by our new method can control the error probabilities strictly and save more synthetical expected test time (SETT). The OTSTs are also compared with the near-optimal ones solved by the sample space sorting method (SSSM). The results show that the test plans solved by these two methods are almost consistent. However, our new method is much faster in computation, especially when the maximum sample size (MSS) of a truncated sequential test increases. It can save 74.23% of the computational time compared with SSSM when the MSS equals 15. At last, the MSS of OTST is optimized. By optimizing the MSS, the corresponding OTST can save about 5% of the SETT compared to the one with the minimum MSS.

Five-Repetition Sit-to-Stand Test Performance in Healthy Individuals: Reference Values and Predictors From 2 Prospective Cohorts.

ObjectiveThe 5-repetition-sit-to-stand (5R-STS) test is an objective test of functional impairment- commonly used in various diseases, including lumbar degenerative disc diseases. It is used to measure the severity of disease and to monitor recovery. We aimed to evaluate reference values for the test, as well as factors predicting 5R-STS performance in healthy adults. MethodsHealthy adults (> 18 years of age) were recruited, and their 5R-STS time was measured. Their age, sex, weight, height, body mass index (BMI), smoking status, education level, work situation and EuroQOL-5D Healthy & Anxiety category were recorded. Linear regression analysis was employed to identify predictors of 5R-STS performance. ResultsWe included 172 individuals with mean age of 39.4±14.1 years and mean BMI of 24.0 ±4.0 kg/m2. Females constituted 57%. Average 5R-STS time was 6.21 ±1.92 seconds, with an upper limit of normal of 12.39 seconds. In a multivariable model, age (regression coefficient [RC], 0.07; 95% confidence interval [CI], 0.05/0.09; p<0.001), male sex (RC, -0.87; 95% CI, -1.50 to -0.23; p=0.008), BMI (RC, 0.40; 95% CI, 0.10–0.71; p=0.010), height (RC, 0.13; 95% CI, 0.04–0.22; p=0.006), and houseworker status (RC, -1.62; 95% CI, -2.93 to -0.32; p=0.016) were significantly associated with 5R-STS time. Anxiety and depression did not influence performance significantly (RC, 0.82; 95% CI, -0.14 to 1.77; p=0.097). ConclusionThe presented reference values can be applied as normative data for 5R-STS in healthy adults, and are necessary to judge what constitutes abnormal performance. We identified several significant factors associated with 5R-STS performance that may be used to calculate individualized expected test times.

Mixed Weibull distributions for the Bayesian analysis of reliability when failures are progressively censored

The progressive type-II censoring has a wide range of applications in reliability studies and survival analysis. The analysis of the mixture models under progressively type-II censored samples has recently been introduced. The reliability estimation from the mixture model under the optimal progressive censoring schemes is still lacking in the current literature. We have discussed the estimation of parameters and reliability characteristics from the Weibull mixture model when failures are progressively censored. The Weibull mixture is proposed for this case as it is S-shaped, which can appropriately handle the mixture data. We have addressed the problem of determination of optimal censoring schemes and estimation of expected test times using the said mixed model. The optimal censoring schemes were found to be less sensitive hence robust. As the corresponding posterior distributions do not provide explicit solutions, we have considered Lindley’s approximation for the approximate numerical solutions.

Reliability estimation from Topp Leone smixture distribution using optimal progressive censoring schemes: A Bayesian approach

The progressive type II censoring has been frequently used for reliability estimation from the different standard time-to-death models during the last two decades. It has been recently introduced for the estimation of mixture models; however the analysis of the reliability for the mixture models using progressive type II censoring especially under Bayesian framework is still rarely found in the literature. Similarly, the problem of determination of optimum censoring schemes in case of progressively censored samples from mixed life models still needs to be addressed. This paper aims to estimate the parameters and reliability characteristics of the Topp Leone (T-L) mixture model in case of progressive type II censoring. The determination of optimum censoring scheme and expected test time has also been discussed. We also investigated the sensitivity of the optimum censoring scheme and found them natural robust. The likelihood equations and the posterior distributions based on T-L mixture distribution under progressively censored samples do not provide explicit solutions; therefore we have used approximate methods for the numerical solutions.

Shifted Exponential Distribution: Bayesian Estimation, Prediction and Expected Test Time under Progressive Censoring

Abstract In this article, we present Bayesian estimation of a shifted exponential distribution assuming progressive type-II censoring with random removals. Bayes estimators and their respective posterior risks are derived under the squared error loss function and the precautionary loss function. Moreover, the credible intervals and expected experiment time along with the Bayesian predictive intervals for one- and two-sample cases are also discussed. A comparison of the expected experiment time under complete and type-II progressive censoring is also a part of this study. Finally, a real-life example is provided to illustrate the proposed methodology.

Analysis of the progressively Type-II right censored data with dependent random removals

In the progressive Type-II censoring with random removal, choosing a distribution for removing the number of unfailed units which is independent from the lifetime distribution, may be a serious challenge and lead to an unreliable inference. Two approaches based on the time distance between consecutive failure times are proposed to determine the removal numbers. Then, the joint probability mass functions of random removals are derived where the underlying distribution of the lifetime data is assumed to be an exponential. Furthermore, The expected test time is investigated under these two approaches and is subsequently considered as an optimality criterion for identifying progressive Type-II censoring schemes. Some statistical inference results are also obtained by using Monte Carlo simulations. The empirical results indicate that the proposed methods work perfectly well in the sense that, reduce the expected total time on the test. Finally, a numerical example, based on a real data set, is presented and analysed to illustrate the proposed approaches.

Assurance test and its equivalent truncated sequential test

In order to save more test cost, assurance test and its equivalent truncated sequential test are studied. In a commonly used case, the operating characteristic (OC) function and expected test time (ETT) function of an assurance test are derived in a concise way. Equivalent test and relative concepts are defined. The procedures to construct a near equivalent truncated sequential test of an assurance test are established. Computation studies show that the near equivalent truncated sequential tests proposed in this paper keep almost the same OC curves with the assurance tests respectively. However, they can save the ETTs dramatically. In fact, the results show that the near equivalent truncated sequential tests can save around 50% of ETTs than the assurance tests respectively.

Estimation of the Parameters and Expected Test Time of Exponentiated Weibull Lifetimes Under Type II Progressive Censoring Scheme With Random Removals

Based on progressive type-II censored sample with random removals, point and interval estimations for the shape parameters of the exponentiated Weibull distribution are discussed. Computational formula for the expected total test time are derived for different situations of sampling plans. This is useful in planning a life test experiment. The efficiency of the estimators are compared in terms of the root mean square error, the variance and the coverage probability of the corresponding confidence intervals. A simulation study is presented for several values of removal probability and different values of failure percentage. Also, numerical applications are conducted to illustrate and compare the usefulness of the different sampling plans in terms of expected test times for different patterns of failure rates.

A heuristic approach for near optimal truncated sequential test of exponential distribution

Optimal truncated sequential test of the exponential distribution is studied in this paper. Definitions and relative concepts of optimal truncated sequential test are discussed. A heuristic approach, sample space sorting method (SSSM) and the procedures are established to solve a near optimal truncated sequential test. Comparing with the test plans provided by international standard IEC 61124 (2012) and Russian national standard GOST R 27.402 (1995), the results show that the near optimal tests proposed in this paper can keep the error probabilities almost equivalence to the nominal error levels and save more synthetical expected test times and maximum expected test times simultaneously.

Block censoring scheme with two-parameter exponential distribution

ABSTRACTThis paper deals with a new censoring scheme, called ‘Block Censoring’ which reduces considerably the total time on test in the life testing experiments with respect to the common used experimental tests such as rightly censored data. This new scheme is analysed when the lifetimes of products follow the two-parameter exponential distribution. Specially, it is proved that the respective spacings are independently distributed exponential. The problem of estimating parameters is investigated in details.A Monte Carlo simulation is conducted for obtaining the optimal block censoring scheme in the sense of the shortest expected test time. Finally, a real data set on times to breakdown of an insulating fluid between electrodes from Nelson [Applied life data analysis. New York: Wiley; 1982. p.105] is analysed.

Burr-X Model Estimate using Bayesian and non-Bayesian Approaches

The present paper is aimed at developing Bayesian and Maximum Likelihood estimations (ML) of the Burr type-X model of distribution when data are gathered from Type-II cumulative censoring with binomial eliminations. The procedures for getting the (ML) evaluations of the parameters are examined. The Bayes technique to get both point and interval estimators of the parameters are illustrated. The expected termination time for Type-II cumulative censoring with binomial eliminations is analyzed after carrying out the computation. Classical and Bayes procedures are improved in the case of parameter estimation and evaluated the expected test time for Burr-X model under cumulative censoring wit binomial sweep. A simulation study is performed to compare the implementation of the various procedures and for the expected termination time of the test. Finally, illustrative examples are given and the results from emulation studies determining the achievement of the suggested techniques are presented.

Abort-on-Fail Test Scheduling for Modular SOCs without and with Preemption

System-on-chips (SOCs) and 3D stacked ICs are often tested for manufacturing defects in a modular fashion, enabling us to record the module test pass probability. We use this pass probability to exploit the abort-on-fail feature of automatic test equipment (ATE) and hence reduce the expected test time in the context of single-site testing. We present a model for calculation of expected test time, for which the abortable test unit can be a module test, a test pattern or a clock cycle. Given an SOC, with test architecture consisting of module test wrappers and test access mechanisms (TAMs), and given module test pass probabilities, we schedule the tests on each TAM to minimize the expected test time. We describe four scheduling heuristics, one without and three with preemption. Experimental results for the ITC’02 SOC Test Benchmarks show 3.5 and 20 percent reduction of expected test time in SOCs with 0.89 and 0.71 SOC test pass probability respectively, without modification of SOC or ATE. Further experiments show how accurate estimates for the module test pass probability or the distribution of pass probability over test patterns need to be to lead to effective test scheduling.

Inferences using type-II progressively censored data with binomial removals

This paper considers the estimation problem for Burr type-X model, when the lifetimes 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 methods of maximum likelihood as well as the Bayes procedure to derive both point and interval estimates of the parameters are used. The expected test time to complete the censoring test is computed and analyzed for different censoring schemes. The effect of the binomial distribution parameter p on the expected test time under progressive censoring and the relative expected test time over the complete sample are investigated. Monte Carlo simulations are performed to compare and evaluate the performance of different methods. Furthermore, an example with a real data set is presented for illustrative purposes.

Reliability sampling plans for a lognormal distribution under progressive first-failure censoring with cost constraint

In this paper, we establish reliability sampling plans for a two-parameter lognormal distribution when it is known that samples are progressive first-failure censored. An EM algorithm is developed to obtain maximum likelihood estimates of unknown parameters. Reliability sampling plans are obtained using two different criteria, namely minimizing the expected test time and minimizing the determinant of the variance covariance matrix of the maximum likelihood estimates, under a restriction on the total cost associated with the experiment and for given specifications on the operating characteristic curve. Finally, a numerical study is performed and specific comments are given.

Parameter Estimations Based On Kumaraswamy Progressive Type II Censored Data with Random Removals

The estimation of two parameters of the Kumaraswamy distribution is considered under Type II progressive censoring with random removals, where the number of units removed at each failure time has a binomial distribution. The MLE was used to obtain the estimators of the unknown parameters, and the asymptotic variance - covariance matrix was also obtained. The formula to compute the expected test time was derived. A numerical study was carried out for different combinations of model parameters. Different censoring schemes were used for the estimation, and performance of these schemes was compared.

TEST TIME REDUCTION THROUGH OPTIMAL DEGRADATION TESTING

In industry, bogey testing, also known as the zero-failure testing, is often used to demonstrate that a product achieves the required reliability at a high confidence level. This test method is simple to apply; however, it requires excessive test time and/or a large sample size, and thus is usually unaffordable. For some products whose failure is defined as a performance characteristic exceeding a threshold, it is possible to measure the performance characteristic during testing. The measurement data can be employed to predict whether or not a test unit will fail by the end of test. When there are sufficient data to make such a prediction with a high degree of confidence, the test of the unit can be terminated. As a result, the test time is reduced. Yang studies the test time reduction for Weibull and binomial distributions.1 This paper describes the test method for lognormal distribution. In particular, this paper describes the sample size, degradation models, and cost function for the lognormal distribution. Then the paper describes the optimum test plans, which choose the optimal sample size and the expected test time, by minimizing the total test cost and simultaneously satisfying the constraints on the type II error and the available sample size. An example is given to illustrate the test method.

Reliability Demonstration Through Degradation Bogey Testing

Bogey testing, also known as zero-failure testing, is used in industry to demonstrate reliability at a high confidence level. This test method is simple to apply; however, it requires excessive test time and/or a large sample size, and thus is often unaffordable. For some products, a failure is defined in terms of a performance characteristic exceeding a specified threshold. For these products, it is possible to measure the performance characteristic at different times during testing. The measurement data can be employed to predict whether or not a test unit will fail by the end of the test. When there are sufficient data to make such a prediction with a high degree of confidence, the test of the unit can be terminated. As a result, the test time is reduced. This paper develops a method for degradation bogey testing to reduce test time. In particular, the paper describes degradation modeling, and the calculation of the conditional failure probability of a test unit. Then we develop the optimum test plans, which choose the sample size, and the expected test time, by minimizing the total test cost, and simultaneously satisfying the constraints on the type II error and available sample size. Sensitivity analysis shows that the optimum test plans are robust against the preestimates of model parameters. The paper also presents decision rules for terminating the test of a unit. The proposed method is illustrated with an example. The application shows that the method is effective in reducing test time.

On estimation based on progressive first-failure-censored sampling

In this paper, a new life test plan called a progressive first-failure-censoring scheme is introduced. Maximum likelihood estimates, exact and approximate confidence intervals and an exact confidence region for the parameters of the Weibull distribution are discussed for the new censoring scheme. A numerical example is provided to illustrate the proposed censoring scheme. Some simulation results are presented and used to assess the performance of the proposed estimation methods developed here. The expected time required to complete the proposed life test plan is derived. Finally, a numerical study for comparing among different censoring schemes in terms of expected test time is given.

Improvement on “Sequential Testing” in MIL-HDBK-781A and IEC 61124

This paper presents results of an analysis of the sequential test (ST) procedures described in MIL-HDBK-781A, and IEC 61124, intended for checking the mean Time Between Failures (TBF) value under an exponential distribution of the TBF. The methodological basis of the calculations consists in discretization of the ST process through subdivision of the time axis in small segments. By this means, the process is converted into a binomial for which an algorithm, and a fast computer program have been developed; and most important of all, a tool is provided for searching for the optimal truncation. The influence of truncation by time on the Expected Test Time (ETT) characteristics was studied; and an improved truncation method, minimizing this influence, was developed. The distributions of the test times were determined. The type A plan characteristics in IEC 61124:2006 have substantial inconsistencies in the probabilities of types I & II errors (up to a factor of 2), and in the ETT (up to 17%). We checked these results by using the binomial-recursive method, and by simulation. The Type C plans, reproduced from GOST R27.402:2005, are consistent; but there is scope (and need) for substantial improvement of the search algorithm for the optimal parameters.

MLE and the estimated expected test time for the two-parameter Gompertz distribution under progressive censoring with binomial removals

This study considers the estimation of two-parameter Gompertz distribution 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. The maximum likelihood estimates of two parameters and their asymptotic distribution are derived. The expected termination point to complete the censoring test is computed and analyzed for different censoring schemes. For binomial removals, the effect of various p on the expected termination point under progressive censoring and the relative expected termination point over the complete sample (REET1) are investigated in this article. For random removals (PCR), the expected termination point is computed and the relative expected termination point over the complete sample (REET2) are also investigated.