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Quality and Warranty: Sensitivity of Warranty Cost Models to Distributional Assumptions
Published in Donald B. Owen, Subir Ghosh, William R. Schucany, William B. Smith, Statistics of Quality, 2020
Wallace R. Blischke, Sushmita Das Vij
The shape parameter is β; λ is a scale parameter. Both parameters are positive. The distribution is DFR if β < 1, IFR if β > 1, and reduces to the exponential if β = 1. The mean of the Weibull distribution is μ = λ−1Γ(1 + 1/β); the variance is σ2 = λ−2 [Γ (1 + 2/β) - Γ2(1 + 1/β)], where Γ(·) is the gamma function.
Sampling
Published in A. C. Faul, A Concise Introduction to Machine Learning, 2019
For k = 1, the Weibull distribution becomes the exponential distribution. The Weibull distribution is used in reliability engineering and failure analysis. There, the parameter k is interpreted in the following way. If k > 1, then the failure rate increases with time as parts are more likely to fail as time goes on. If k = 1, the failure rate is constant, the system is stable and there is no aging process. If k < 1, the failure rate decreases with time. Colloquially, this means that if it has not failed by now, it is less likely to fail in the future. For more information on statistical quality control see [31].
Bayesian Inference on General-Order Statistic Models
Published in Mangey Ram, Modeling and Simulation Based Analysis in Reliability Engineering, 2018
Aniket Jain, Biswabrata Pradhan, Debasis Kundu
Weibull model: Although the exponential distribution is used quite extensively as a lifetime distribution, it has a decreasing PDF and a constant hazard function. These are serious limitations for an exponential distribution. The Weibull distribution has two parameters: one shape parameter and one scale parameter. The presence of the shape parameter makes it a very flexible distribution. The Weibull distribution has a decreasing or an unimodal density function. If the shape parameter is less than or equal to one, it has a decreasing PDF. Otherwise, the PDF is an unimodal function. Furthermore, the hazard function also can take various shapes namely increasing, decreasing, or constant. It can be used quite successfully to analyze lifetime data. The Weibull distribution has the following PDF for α> 0 and λ> 0;
Monitoring the alternating renewal processes with Weibull window-censored data
Published in Quality Technology & Quantitative Management, 2023
In reliability applications, the Weibull distribution is a popular distribution for modeling the times of a process or product because of its flexibility in being able to model multiple types of failure mechanisms. It is the preferred distribution for the time effects and product properties, elongation or resistance. The Weibull distribution is also used in biological and medical applications to model the time to occurrence of diseases. Therefore, analysts prefer to model lifetime data by the two-parameter Weibull distribution (Erto et al., 2018; Guo & Wang, 2014; Pascual & Li, 2012; Pascual & Zhang, 2011; Zhang & Chen, 2004). Suppose follows a Weibull distribution, where and are the shape and scale parameter, respectively. The probability density function (pdf) and cumulative distribution function (cdf) of the Weibull distribution for are
A deep learning approach for integrated production planning and predictive maintenance
Published in International Journal of Production Research, 2023
Hassan Dehghan Shoorkand, Mustapha Nourelfath, Adnène Hajji
Since our framework requires also a model-based approach for the forecast stage, we define the best statistical method that fits the run-to-failure data set FD001. We found that a Weibull distribution fits with this data set. Weibull distribution is a two-parameter distribution function represented by a dimensionless shape parameter and scale parameter in units of hours (Chaurasiya, Ahmed, and Warudkar 2018). The Weibull density function, is as follows: While several different approaches can be taken to estimate the parameters and , the maximum likelihood estimator (MLE) is commonly used, because of its numerical stability (Cohen 1965). The calculated shape and scale parameters, by maximising the likelihood function for our data set, are and . The shape parameter of the Weibull distribution represents the failure rate behaviour; when it is greater than , it means the failure rate increases with time, which is known as wear-out life. In mechanical components, this is mainly due to thermal-mechanical fatigue, stress, corrosion, or the degradation process (Abernethy 2004).
Reliability, maintainability, and availability methodology based on statistical approach for cooling systems design modification and maintenance
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2022
Krešimir Osman, Trpimir Alajbeg, Mato Perić
For the calculation and assessment of system reliability, maintainability, and availability by using statistical methods, the most important assumption is its lifetime probability distribution form. Once this is known, it is basic information about the sample for estimating the distribution parameters, and based on that, the reliability, maintainability, and availability parameters can be calculated. In practice, the most used distributions to describe system life are exponential, such as gamma, Weibull, and lognormal ones. Of these distributions, it is important to note that Weibull distribution is most used in industrial practice to access system reliability because the matching distribution curve is improved by the characteristics of very little components or subsystem failure data. In addition, the range failure probability is limited by the value of the shape parameters that further improves the accuracy of the evaluation range. It can be concluded that this method is applicable to the problems described above because it can analyze the product reliability assessment if there are several failure data, including the parameter estimation point and parameter confidence interval estimation. Furthermore, it was shown in engineering practice (Cheng 1999), that most mechanical (e.g., bearings, shafts, hydraulic components – pumps and compressors) as well as electrical and electronic components (e.g., electric motors and generators) are subject to the Weibull distribution and are well represented by it.