Lognormal Lifetime Distributions Research Articles

Overdispersion Effects on Reliability Test Planning

The impact of overdispersion on the design of optimal reliability demonstration test plans for beta-binomial models with Weibull, gamma, and lognormal lifetime distributions is analyzed. Assuming limited producer and consumer risks, fixed-duration test plans with minimal sample sizes and optimal-duration test plans with minimum costs based on failure count data are determined by solving the corresponding nonlinear programming problems when the fluctuation of the failure probability is described by a beta distribution. If the test time is fixed, the overdispersion effect on the optimal sample size and decision criterion is important in most situations. The influence is less relevant when the engineer also wants to find the minimum-cost test time. Optimal-duration plans usually outperform fixed-duration schemes in terms of costs and robustness against overdispersion. The use of lot inspection schemes with optimal reliability test times is strongly recommended when the presence of overdispersed failure count data is suspected. Applications of the developed methodology to the manufacturing of microelectronic chips and semiconductor lasers are provided for illustrative and comparative purposes.

Non-destructive one-shot device test under step-stress experiment with lognormal lifetime distribution

Dealing with censored data is an important concern in reliability and survival analysis. Non-destructive one-shot devices are an extreme case of interval censoring, wherein we only know if a device has failed or not before an inspection time. Besides, non-destructive one-shot devices are frequently highly reliable with large lifetimes, and then long experimentation times would be needed for inference under normal operating conditions. Alternatively, accelerated life tests (ALTs) shorten the lifetime of the devices by increasing one or more stress factors causing failure. Then, after suitable inference, results can be extrapolated to normal conditions. In particular, step-stress ALT designs increase the stress level at which devices are tested throughout the experiment at some fixed times. Under the non-destructive one-shot device set-up, the number of failures is recorded at some inspection times, including the times of stress change, then resulting in censored data. Among the most popular lifetime distributions used to analyze survival data, the lognormal distribution has hazard function with an increasing–decreasing behavior, which is encountered often in practice as units usually experience early failure and then stabilize over time in terms of performance. However, the classical maximum likelihood estimator (MLE) of parameters of the lognormal lifetime distribution may get highly influenced by data contamination. In this paper we propose a family of divergence-based robust estimators for non-destructive one-shot device step-stress experiments under the lognormal lifetime distribution. Further, from the robust estimators, a generalization of the popular Wald-type test statistic based on the MLE for testing composite null hypothesis is defined, resulting in a robust divergence-based family of test statistics.

Optimal design of cyclic-stress accelerated life tests for lognormal lifetime distribution

ABSTRACT Accelerated life tests (ALTs) have frequently been used in industry to assess the reliability of products within an affordable amount of time and cost. Most of the previous works on designing ALT plans have dealt with the case of constant-, step- or progressive-stress loading, which may not be adequate for products operating under cyclic-stress loading at the use condition. In this paper, optimal and compromise ALT plans are developed under the assumptions of cyclic-stress loading, lognormal lifetime distribution, log-linear relationship between the scale parameter of the lifetime distribution and stress variable and the cumulative exposure model for the effect of changing stress levels. In particular, the common floor level and the proportion of units assigned to two (for optimal plans) or three (for compromise plans) cyclic-stress conditions are determined such that the asymptotic variance of the maximum likelihood estimator of the q-th quantile of the lifetime distribution at the use condition is minimized. In addition, a sample size determination method is developed, and sensitivity analysis procedures are presented to evaluate the robustness of the proposed ALT plans in the face of uncertainty in the pre-estimation of a model parameter. The whole process is illustrated with an example.

EM‐based likelihood inference for one‐shot device test data under log‐normal lifetimes and the optimal design of a CSALT plan

AbstractOne‐shot devices result in an extreme case of interval censoring, wherein one can only know whether the failure time is either before or after the test time. The study of one‐shot device testing has been developed considerably recently, both in terms of estimation and optimal design under different lifetime distributions. However, one‐shot device testing analysis under lognormal lifetime distribution has not been studied yet. While the hazard function for exponential distribution is always a constant, and that of Weibull and gamma are either increasing or decreasing, the lognormal distribution has increasing ‐ decreasing behavior of hazard which is encountered often in practice as units usually experience early failure and then stabilize over time in terms of performance. In this paper, we develop the EM algorithm for the likelihood estimation based on one‐shot device test data under lognormal distribution and also the design of optimal CSALTs (constant stress accelerated life tests) under this set up with budget constraints. A simulation study is carried out to assess the performance of the methods of inference developed here and some real‐life data are analyzed for illustrative purpose.

Optimal design of cyclic-stress accelerated life tests for lognormal lifetime distribution under type I censoring

In this paper, optimal ALT plans are developed under the assumptions of cyclic-stress loading, Type-I censoring, lognormal lifetime distribution, log-linear relationship between the scale parameter of the lifetime distribution and stress variable, and the cumulative exposure model for the effect of changing stress levels. In particular, the common floor level of, and the proportions of units allocated to two cyclic-stress conditions are determined such that the large-sample variance of the maximum likelihood estimator of the q-th quantile of the lifetime distribution at the use condition is minimized. In addition, a sample size determination method is developed for the practical use.

Bayesian C-optimal life testing plans under progressive type-I interval censoring scheme

• Optimal Bayesian life tests plans are presented under Progressive Type-I Interval Censoring Scheme. • Bayesian C -optimal design criterion is used. • An algorithm is presented for log-location-scale family of distributions. • A comparative study is provided for Weibull and log-normal lifetime distributions. • Sampling variations are visualized through MCMC methods. This work considers optimal planning of progressive type-I interval censoring schemes for log-location-scale family of distributions. Optimum schemes are obtained by using a Bayesian C -optimality design criterion. The C -optimality criterion is formed to attain precision in estimating a particular lifetime quantile . An algorithm is proposed to obtain the optimal censoring schemes. Optimal schemes are obtained under two different scenarios for the Weibull and log-normal models, which are two popular special cases of log-location-scale family of distributions. A sensitivity analysis is conducted to study the effect of various prior inputs on the optimal censoring schemes. Furthermore, a simulation study is undertaken to illustrate the sampling variations resulting from the optimal censoring schemes.

Precision of different fatigue methods for predicting glass-ceramic failure

This study aimed to characterize the fatigue behavior using two fatigue methods, boundary and staircase, and to predict the probability of failure (Pf) of zirconia-reinforced lithium silicate glass-ceramic (ZLS). Bar-shaped specimens of ZLS (18 ×4 ×1.2 mm) were fabricated. Thirty specimens were subjected to a three-point flexural strength test using a universal testing machine with 0.5 mm/min crosshead speed, in 37 °C distilled water. Flexural strength data were analyzed with Weibull statistics. Eighty-six bars were subjected to cyclic fatigue using boundary and staircase methods. Fatigue tests were performed in a pneumatic cycling machine (2 Hz, 37 °C distilled water) for 10 ³ and 104 cycles. Fatigue data were analyzed using an inverse power law relationship and log normal-lifetime distribution. Fracture toughness (KIc) was determined using V-notched specimens (18 ×4 ×3 mm) and the short beam toughness method (n = 7). Vickers hardness (VH) was evaluated (4.9 N, 20 s). Fractographic and EDS analyses were also performed. ZLS showed a characteristic strength of 197 MPa, Weibull modulus of 4, VH of 6.67 GPa and KIc of 1.93 MPa m1/2. After 103 cycles, for both methods, there was a degradation of 78% of the initial strength. There was no significant degradation when the number of cycles increased from 103 to 104. Both methods resulted in similar Pf and precision at 40 MPa (~50% Pf). Yet, staircase shows good accuracy and precision in predicting the stress amplitude for a Pf near 50%; while boundary is also effective for Pf lower than 50%. The fatigue methods evaluated show similar accuracy and precision for predicting the Pf of a glass-ceramic when simulations were made in the range of stress levels and lifetimes used in the fatigue tests.

Bayesian accelerated acceptance sampling plans for a lognormal lifetime distribution under Type-I censoring

Developing an accelerated acceptance sampling plan (AASP) is essential to product reliability demonstration, particularly for a product designed to have high reliability and long lifetime. Because of acceleration, a decision on lot acceptance for the product can be quickly made while significantly reducing the total cost of testing. Traditionally, the parameters of an acceptance probability function used in AASP are assumed to be known in advance. However, it is often difficult, if not impossible, to determine the exact values of these parameters. On the other hand, prior information, including earlier test results and expert opinions, may be available for potential uses in planning such tests. In this paper, Bayesian designs of AASP are considered by taking full advantage of prior information to reduce the uncertainty in these important parameters so that significant testing resources (e.g., sample size and testing time) can be saved. In particular, a product with a lognormal lifetime distribution is studied, and two types of Bayesian AASPs under Type-I censoring are developed based on the product's operating characteristic (OC) curve. The first Bayesian AASP considers the risks of the producer and the consumer simultaneously, while the second one considers only the consumer's risk. The optimal sampling plans aim at minimizing the total costs of testing. The proposed AASPs are illustrated using two numerical examples. Sensitivity analysis on the selection of prior distribution is also performed to validate the robustness of the proposed Bayesian AASPs. In addition, a comparison study shows that the proposed test plans outperform the traditional AASPs with respect to the use of testing resources.

Estimating the parameters of lifetime distributions under progressively Type-II censoring from fuzzy data

Abstract The problem of estimating lifetime distribution parameters under progressively Type-II censoring originated in the context of reliability. But traditionally it is assumed that the available data from this censoring scheme are performed in exact numbers. However, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. Thus, it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers. This paper deals with the estimation of lifetime distribution parameters under progressively Type-II censoring scheme when the lifetime observations are reported by means of fuzzy numbers. A new method is proposed to determine the maximum likelihood estimates of the parameters of interest. The methodology is illustrated with two popular models in lifetime analysis, the Rayleigh and Lognormal lifetime distributions.

Statistical Inference of a Two-Component Series System With Correlated Log-Normal Lifetime Distribution Under Multiple Type-I Censoring

In a series system, the system fails if any of the components fails. When the system functions, there may exist correlation among components because they are connected within the same system. In this paper, we consider the reliability analysis of multiple Type-I censored life tests of series systems composed of two components with bivariate log-normal lifetime distributions. The major interest is the inference on the mean lifetimes, and the reliability functions of the system and its components. Given observations of the minimum lifetime of the components of each failed system, location of the MLEs highly relies on the initial values in executing the computation numerically. Alternatively, we apply the Bayesian approach after a re-parametrization of the parameters of interest. A simulation study is conducted which shows that the Bayesian approach provides considerably accurate inference. The proposed approach is successfully applied to a real data set.

Statistical Inference of a Series System with Correlated Log-normal Lifetime Distributions under Type-I Censoring(Session 2B(IASC-ARS))

Statistical Inference of a Series System with Correlated Log-normal Lifetime Distributions under Type-I Censoring(Session 2B(IASC-ARS))

Statistical power of goodness-of-fit tests based on the empirical distribution function for type-I right-censored data

In this study, the power of common goodness-of-fit (GoF) statistics based on the empirical distribution function (EDF) was simulated for single type-I right-censored data. The statistical power of the Kolmogorov–Smirnov, Cramér–von Mises and Anderson–Darling statistics was investigated by varying the null and the alternative distributions, the sample size, the degree of censoring and the significance level. The exponential, Weibull, log-logistic and log-normal lifetime distributions were considered as they are among the most frequently distributions used when modelling censored data. We conclude by giving some general recommendations for testing the distributional assumption of parametric survival models in homogeneous populations when using EDF-based GoF statistics.

On D-optimal robust designs for lifetime improvement experiments

Locating the optimal operating conditions of the process parameters is critical in a lifetime improvement experiment. For log-normal lifetime distribution with compound error structure (i.e., symmetry, inter-class and intra-class correlation error structures), we have developed methods for construction of D-optimal robust first order designs. It is shown that D-optimal robust first order designs are always robust first order rotatable but the converse is not always true.

The Local Rate and the Progenitor Lifetimes of Short‐Hard Gamma‐Ray Bursts: Synthesis and Predictions for the Laser Interferometer Gravitational‐Wave Observatory

The recent discovery of the first four afterglows of short-hard gamma-ray bursts (SHBs) suggests that they typically result from long-lived progenitor systems. The most popular progenitor model invokes the merger of either double neutron star (DNS) binaries or neutron star-black hole (NS-BH) systems. Such events are strong sources of gravitational waves (GWs) and might be detected by ground-based GW observatories. In this work we combine the census of SHB observations with refined theoretical analysis to perform a critical evaluation of the compact binary model. We then explore the implications for GW detection of these events. Beginning from the measured star formation rate through cosmic time, we consider what intrinsic luminosity and lifetime distributions can reproduce the known SHB redshifts and luminosities as well as the peak flux distribution of the large BATSE SHB sample. We find the following: (1) The typical progenitor lifetime is long. Assuming lognormal lifetime distribution, the typical lifetime is >4 (1) Gyr (2 σ [3 σ] c.l.). If the lifetime distribution is a power law with index η, then η > -0.5 (-1) (2 σ [3 σ] c.l.). This result is difficult to reconcile with the properties of the observed Galactic DNS population, suggesting that if SHBs do result from DNS mergers, then the observed Galactic binaries do not represent the cosmic one. (2) We find that the local rate of SHBs is larger than 10 Gpc-3 yr-1 and may be higher by several orders of magnitude, significantly above previous estimates. (3) Assuming that SHBs do result from compact binaries, our predictions for the LIGO and VIRGO event rates are encouraging: the chance for detection by current facilities is not negligible, while a coincident detection of GW and electromagnetic radiation from an SHB is guaranteed for next-generation observatories.

Electromigration behavior of hot-sputtered Al(Cu) versus chemical vapor deposition W vias

The electromigration behavior of hot-sputtered aluminum-copper (Al-Cu) vias is compared to that of chemical vapor deposition (CVD) tungsten (W) vias. Al(Cu) vias were prepared by one of two processes: (1) the via was filled and then defined by chemical mechanical polish (CMP) and (2) the via and top metal line were deposited in the same step (this is referred to as the “Sprint” process). During electromigration testing, the W via chains exhibited more uniform resistance changes with time than the CMP Al(Cu) via chains. This had a direct impact on the shape parameter of the log-normal lifetime distribution, where σ was higher for the CMP Al(Cu) via chains. A closer examination of the failure distributions revealed the need to describe the CMP Al(Cu) via lifetimes by a three-parameter log-normal distribution as opposed to the conventional two-parameter log-normal distribution. Failure analysis of the CMP Al(Cu) samples indicated electromigration failures downstream from the via in the direction of the electron flow. Samples prepared by the Sprint process showed >5× electromigration lifetime improvement as compared to either the CVD W or CMP Al(Cu) via samples.

Critical time of the lognormal distribution

The critical time is the time point as the failure rate starts to decrease and also as the mean residual lifetime starts to increase. The estimated critical time is useful for determining the duration of a burn-in process. The method for estimating the critical time of the failure rate for lognormal lifetime distribution is discussed. A single time censored data is used as a example for illustration.