An imprecise statistical method for accelerated life testing using the power-Weibull model

Accelerated life testing provides an interesting challenge for quantification of the uncertainties involved, in particular due to the required linking of the units’ failure times, or failure time distributions, at different stress levels. This paper provides an initial exploration of the use of statistical methods based on imprecise probabilities for accelerated life testing. We apply nonparametric predictive inference at the normal stress level, in combination with an estimated parametric power-Weibull model linking observations at different stress levels. To provide robustness with regard to this assumed link between different stress levels, we introduce imprecision by considering an interval around the parameter estimate, leading to observations at stress levels other than the normal level to be transformed to intervals at the normal level. The width of such intervals is increasing with the difference between the stress level at which a unit is tested and the normal level. The resulting inference method is predictive, so it explicitly considers the random failure time of a future unit tested at the normal level. We perform simulation studies to investigate the performance of our imprecise predictive method and to get insight into a suitable amount of imprecision for the linking between levels. We also explain how simulation studies can assist in choosing imprecision in order to provide robustness against specific biases or model misspecifications

An imprecise statistical method for accelerated life testing using the power-Weibull model.

Accelerated life testing provides an interesting challenge for quantification of the uncertainties involved, in particular due to the required linking of the units’ failure times, or failure time distributions, at different stress levels. This paper provides an initial exploration of the use of statistical methods based on imprecise probabilities for accelerated life testing. We apply nonparametric predictive inference at the normal stress level, in combination with an estimated parametric power-Weibull model linking observations at different stress levels. To provide robustness with regard to this assumed link between different stress levels, we introduce imprecision by considering an interval around the parameter estimate, leading to observations at stress levels other than the normal level to be transformed to intervals at the normal level. The width of such intervals is increasing with the difference between the stress level at which a unit is tested and the normal level. The resulting inference method is predictive, so it explicitly considers the random failure time of a future unit tested at the normal level. We perform simulation studies to investigate the performance of our imprecise predictive method and to get insight into a suitable amount of imprecision for the linking between levels. We also explain how simulation studies can assist in choosing imprecision in order to provide robustness against specific biases or model misspecifications

Imprecise Statistical Methods for Accelerated Life Testing

Accelerated Life Testing (ALT) is frequently used to obtain information on the lifespan of devices. Testing items under normal conditions can require a great deal of time and expense. To determine the reliability of devices in a shorter period of time, and with lower costs, ALT can often be used. In ALT, a unit is tested under levels of physical stress (e.g. temperature, voltage, or pressure) greater than the unit will experience under normal operating conditions. Using this method, units tend to fail more quickly, requiring statistical inference about the lifetime of the units under normal conditions via extrapolation based on an ALT model. This thesis presents a novel method for statistical inference based on ALT data. The method quantifies uncertainty using imprecise probabilities, in particular it uses Nonparametric Predictive Inference (NPI) at the normal stress level, combining data from tests at that level with data from higher stress levels which have been transformed to the normal stress level. This has been achieved by assuming an ALT model, with the relation between different stress levels modelled by a simple parametric link function. We derive an interval for the parameter of this link function, based on the application of classical hypothesis tests and the idea that, if data from a higher stress level are transformed to the normal stress level, then these transformed data and the original data from the normal stress level should not be distinguishable. In this thesis we consider two scenarios of the methods. First, we present this approach with the assumption of Weibull failure time distributions at each stress level using the likelihood ratio test to obtain the interval for the parameter of the link function. Secondly, we present this method without an assumed parametric distribution at each stress level, and using a nonparametric hypothesis test to obtain the interval. To illustrate the possible use of our new statistical method for ALT data, we present an application to support decisions on warranties. A warranty is a contractual commitment between consumer and producer, in which the latter provides post-sale services in case of product failure. We will consider pricing basic warranty contracts based on the information from ALT data and the use of our novel imprecise probabilistic statistical method

Maximum likelihood estimation of exponential distribution under type-ii censoring from imprecise data

Statistical analysis of lifetime distributions under Type-II censoring scheme is based on precise lifetime data. However, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. This paper deals with the estimation of exponential mean parameter under Type-II censoring scheme when the lifetime observations are fuzzy and are assumed to be related to underlying crisp realization of a random sample. Maximum likelihood estimate of the unknown parameter is obtained by using EM algorithm. In addition, a new numerical method for parameter estimation is provided. Using the parametric bootstrap method, the construction of confidence intervals for the mean parameter is discussed. Monte Carlo simulations are performed to investigate performance of the different methods. Finally, an illustrative example is also included.Keywords: Type-II censoring, Imprecise lifetime, Exponential distribution, Maximumlikelihood estimation, Bootstrap confidence interva

Trends in the Statistical Assessment of Reliability

Changes in technology have had and will continue to have a strong effect on changes in the area of statistical assessment of reliability data. These changes include higher levels of integration in electronics, improvements in measurement technology and the deployment of sensors and smart chips into more products, dramatically improved computing power and storage technology, and the development of new, powerful statistical methods for graphics, inference, and experimental design and reliability test planning. This paper traces some of the history of the development of statistical methods for reliability assessment and makes some predictions about the future

Inference for exponential parameter under progressive Type-II censoring from imprecise lifetime

Progressively Type-II censored sampling is an important method ofobtaining data in lifetime studies. Statistical analysis oflifetime distributions under this censoring scheme is based onprecise lifetime data. However, in realsituations all observations and measurements of progressive Type-II censoring scheme are not precise numbers but more or less non-precise, also called fuzzy. In this paper, we consider the estimation of exponential meanparameter under progressive Type-II censoring scheme, when thelifetime observations are fuzzy and are assumed to be related tounderlying crisp realization of a random sample. We propose a newmethod to determine the maximum likelihood estimate (MLE) of theunknown mean parameter. In addition, a new numerical method forparameter estimation based on fuzzy data is provided. Using the parametric bootstrapmethod, we then discuss the construction of confidence intervalsfor the mean parameter. Monte Carlo simulations are performed toinvestigate the performance of all the different proposedmethods. Finally, an illustrative example is also included