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Pts of the EoL test for Rr=Cr=90%, s=0.1, and prior knowledge of b=3. Calculated for sample quantile estimation (sample q.) and MLE quantile estimation (MLE). Although the sample size is to be an integer, the curves are interpolated to obtain a better understanding of the trajectory.
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Statistical power analyses are used in the design of experiments to determine the required number of specimens, and thus the expenditure, of a test. Commonly, when analyzing and planning life tests of technical products, only the confidence level is taken into account for assessing uncertainty. However, due to the sampling error, the confidence int...
Citations
... Grundler et al. [28], [29] present another method, which treats an RDT as a hypothesis test. They compare the given failure distribution, the alternative hypothesis, with a distribution scaled down by a safety margin, forming the null hypothesis. ...
... In this case, the test power is the probability that the null hypothesis will be rejected. While this approach is generally applicable, it is demonstrated for Weibull distributed failure times in a failure-free RDT [29], [30], or in an unaccelerated failure-based RDT [28], [30]. ...
... Alternatively, a simplified calculation is possible, which assumes normally distributed test results and utilizes synthetic failure times based on the Beta distribution used in median rank regression (MRR). A comparison with the exact test process replication in the application-based approach shows a discrepancy for various parameter combinations, especially for small sample sizes [28]. This probably stems from the distribution assumption, which is not accurate in real life test evaluation with small sample sizes. ...
The planning of reliability demonstration tests is a typical task for reliability engineers. To find the optimum test plan, many different scenarios must be considered and compared. Comparing test plans by their estimation variance has several disadvantages, as it only considers the type I error. To overcome these disadvantages, the probability of successfully demonstrating a lifetime target is used to implement a statistical power analysis for test plans from a practical perspective. Typically, this is done through simulation based on existing knowledge about the relevant failure mechanism. Such simulations are inadequate for real-world applications, mostly as their are computationally demanding. To enable wide applicability of optimal test planning methods based on the probability of test success, an analytic approach is presented in this paper. The simulation is replaced with an approximation of the stochastical distribution of the test results. The analytic method consists of two steps. First, the expected mean and covariance of the maximum likelihood estimates for a given Weibull distribution with an integrated lifetime model are derived by using pivotal quantities. Second, the delta method is applied to describe the lifetime demonstrated with a particular test plan. The method is validated against simulation results, showing an average error of 1.3–1.7 %.
... For example, [14] argues that 14 runs may suffice to demonstrate 90% reliability with 90% confidence, while in [15] the number of simulated experimental runs is similarly small to keep the simulation realistic. Thus, as one would expect, the number of experimental runs required to demonstrate a certain reliability with certain confidence is one of the key questions researchers focus on; see [16] for a systematic discussion of zero-failure test design. ...
We propose using performance metrics derived from zero-failure testing to assess binary classifiers. The principal characteristic of the proposed approach is the asymmetric treatment of the two types of error. In particular, we construct a test set consisting of positive and negative samples, set the operating point of the binary classifier at the lowest value that will result to correct classifications of all positive samples, and use the algorithm's success rate on the negative samples as a performance measure. A property of the proposed approach, setting it apart from other commonly used testing methods, is that it allows the construction of a series of tests of increasing difficulty, corresponding to a nested sequence of positive sample test sets. We illustrate the proposed method on the problem of age estimation for determining whether a subject is above a legal age threshold, a problem that exemplifies the asymmetry of the two types of error. Indeed, misclassifying an under-aged subject is a legal and regulatory issue, while misclassifications of people above the legal age is an efficiency issue primarily concerning the commercial user of the age estimation system.
... Therefore, it is crucial to use a sample that is large enough to accurately discriminate between Type and then to rely on the trend shown by that specific sample. We defined this 'large enough' sample size to be , and determined it to be approximately ∼ 700 per bin by statistical power analysis, which is a widely used statistical tool for sample size determination in meta-analyses (e.g., Borenstein et al. 2009;Grundler et al. 2022). In statistical power analysis, the three parameters to be set are statistical power, significance threshold, and effect size. ...
Supermassive Black Holes (SMBHs) are commonly found at the centers of massive galaxies. Estimating their masses () is crucial for understanding galaxy-SMBH co-evolution. We present WISE2MBH, an efficient algorithm that uses cataloged Wide-field Infrared Survey Explorer (WISE) magnitudes to estimate total stellar mass () and scale this to bulge mass (), and , estimating the morphological type () and bulge fraction (B/T) in the process. WISE2MBH uses scaling relations from the literature or developed in this work, providing a streamlined approach to derive these parameters. It also distinguishes QSOs from galaxies and estimates the galaxy using WISE colors with a relation trained with galaxies from the 2MASS Redshift Survey. WISE2MBH performs well up to thanks to K-corrections in magnitudes and colors. WISE2MBH estimates agree very well with those of a selected sample of local galaxies with measurements or reliable estimates: a Spearman score of 0.8 and a RMSE of 0.63 were obtained. When applied to the ETHER sample at , WISE2MBH provides 1.9 million estimates (78.5\% new) and 100 thousand upper limits. The derived local black hole mass function (BHMF) is in good agreement with existing literature BHMFs. Galaxy demographic projects, including target selection for the Event Horizon Telescope, can benefit from WISE2MBH for up-to-date galaxy parameters and estimates. The WISE2MBH algorithm is publicly available on GitHub.
... On the other hand, an overestimated sample size makes it challenging to prove the significance of the results even though the results may have substantial power. Calculating the sample size, therefore, requires various pieces of information to be considered, including the effective sample size, significance level, power, and variability, which are considered parameters [7][8][9][10][11]. Using parameter estimates in sample size calculation requires introducing uncertainty into the sample size itself. ...
Aiming to serve as a preliminary study for South Korea’s national GHG emission factor development, this study reviewed data treatment and sample size determination approaches to establishing the destruction and removal efficiency (DRE) of the semiconductor and display industry. We used field-measured DRE data to identify the optimal sample size that can secure representativeness by employing the coefficient of variation and stratified sampling. Although outlier removal is often a key process in the development of field-based coefficients, it has been underexplored how different outlier treatment options could be useful when data availability is limited. In our analysis, three possible outlier treatment cases were considered: no treatment (using data with outliers as they are) (Case 1), outlier removal (Case 2), and adjustment of outliers to extreme values (Case 3). The results of the sample size calculation showed that a minimum of 17 and a maximum of 337 data (out of a total of 2968 scrubbers) were required for determining a CF4 gas factor and that a minimum of 3 and a maximum of 45 data (out of a total of 2917 scrubbers) were required for determining a CHF3 gas factor. Our findings suggest that (a) outlier treatment can be useful when the coefficient of variation lacks information from relevant data, and (b) the CV method with outlier adjustment (Case 3) can provide the closest result to the sample size resulting from the stratified sampling method with relevant characteristics considered.
... Since the total failure distribution of a battery is unknown, the sample size cannot be determined using a statistical power analysis, as described e.g., by Grundler et al. [18]. Therefore, the only remaining approach is to specify the sample size via binomial distribution. ...
Modern vehicles have increasing safety requirements and a need for reliable low-voltage power supply in their on-board power supply systems. Understanding the causes and probabilities of failures in a 12 V power supply is crucial. Field analyses of aged and failed 12 V lead batteries can provide valuable insights regarding this topic. In a previous study, non-invasive electrical testing was used to objectively determine the reasons for failure and the lifetime of individual batteries. By identifying all of the potential failure mechanisms, the Latin hypercube sampling method was found to effectively reduce the required sample size. To ensure sufficient confidence in validating diagnostic algorithms and calculating time-dependent failure rates, all identified aging phenomena must be considered. This study presents a probability distribution of the failure mechanisms that occur in the field, as well as provides insights into potential opportunities, but it also challenges diagnostic approaches for current and future vehicles.
... 41 Still, using statistical power analysis is challenging for reliability testing because lifetime data are usually Weibull or log-normally distributed while the power analyses studies exist for normally distributed data. 64 All the examples previously mentioned still do not take full advantage of how DoE methods could contribute to ALT plan optimization. The rest of this section will highlight some more useful applications of DoE to ALT by introducing first, one of DoE statistical methods widely applied, although controversial, in various domains, which are Taguchi methods. ...
Accelerated testing has been commonly used for the assessment of reliability of products or systems. In particular, it has been used to produce models for predicting the reliability of electronic components as a function of design and environmental parameters, or to qualify reliability. Extensive literature exists on different aspects ranging from defining type of stresses and type of censoring data, to optimizing test plans for efficient and relevant testing. On the other hand, design of experiments methodology is commonly used for studying the robustness of systems and for quality applications. This being said, combining both approaches, taking into account the system's physics of failure, is scarcely put into practice in a context of reliability prediction. Yet, this could significantly improve reliability prediction, especially in the case of electronic components which constantly go through technological progress with new parameters or properties to consider. After first presenting existing predictive reliability guides, models and parameters related to accelerated life testing, the purpose of this article is to provide a review of what has been done concerning the combination of such approaches.
... Among the main results obtained from this study, four methods of calculating the Probability of Test Success for various test scenarios stand out: a general method that can deal with all possible scenarios, a calculation method that emulates the actual test process, and two analytical approaches for failure-free and failure-based testing that use the central limit theorem and the asymptotic properties of various statistics, and thus simulate the effort involved in lifetime test planning. In addition, the authors compared the calculation methods, analyzing their main advantages and drawbacks [21]. ...
In engineering projects, reliability is conceived as physical equipment’s ability to function without failure [...]
... Early calculation procedures made use of the law of large numbers and by simulating the tests while incorporating prior knowledge about the failure distribution, the probability could be calculated. In order to establish a broader statistical context, Grundler et al. [4] defined the Probability of Test Success as the statistical power of a reliability demonstration test, since all reliability demonstration tests can be approached as hypothesis tests. By making use of this statistical context, new calculation procedures could be developed [22][23][24][25][26] e. g. using the asymptotic variance of the maximum likelihood estimation in [4]. ...
... In order to establish a broader statistical context, Grundler et al. [4] defined the Probability of Test Success as the statistical power of a reliability demonstration test, since all reliability demonstration tests can be approached as hypothesis tests. By making use of this statistical context, new calculation procedures could be developed [22][23][24][25][26] e. g. using the asymptotic variance of the maximum likelihood estimation in [4]. Although several studies have been conducted in order to enable the application of the Probability of Test Success for systems with multiple failure modes [22,25,26], a proper procedure facilitating a holistic view is still necessary for an efficient planning procedure of reliability demonstration tests. ...
... Although several studies have been conducted in order to enable the application of the Probability of Test Success for systems with multiple failure modes [22,25,26], a proper procedure facilitating a holistic view is still necessary for an efficient planning procedure of reliability demonstration tests. In addition to the studies regarding the consideration of uncertainty [23] as well as the combined approaches for using Bayes' theorem [20,21,24,27] and the concept of the Probability of Test Success [4], the combination of all three aspects in a single holistic procedure has not been tackled yet. Other approaches for reliability demonstration test planning solely consider the statistical error of type I (confidence) in order to derive required sample sizes of EoL tests [30]. ...
Empirical life tests are used for reliability demonstration and determination of the actual reliability of the product. Therefore, engineers are faced with the challenge of selecting the most suitable test strategy out of the possible many and also the optimal parameter setting, e.g. sample size, in order to realize reliability demonstration with limited costs, time and with their available testing resources. It becomes even more challenging due to the stochastic nature of failure times and necessary cost and time being dependent on those. The considerations and guidelines in this paper are intended to simplify this process. Even simple products can fail due to several causes and mechanisms and usually have several components and subsystems. Therefore, this paper provides test planning options for single critical failure mechanisms as well as for systems with multiple failure mechanisms. For this purpose, the Probability of Test Success (Statistical Power of a life test) is used as a central, objective assessment metric. It is capable of indicating the probability of a successful reliability demonstration of a test and thus allows, for example, to answer the question of the required sample size for failure-based tests. The main planning resource is prior knowledge, which is mandatory due to the stochastic lifetime, in order to provide estimates for the Probability of Test Success at all. Therefore, it is also shown how to deal with uncertain prior knowledge and how the underlying information can additionally be used to increase the Probability of Test Success using Bayes’ theorem. The guidelines show how the most efficient test can be identified in the individual case and for individual boundary conditions.
Insufficient supply chain management expertise among law enforcement departments during Black Swan events may affect the operationality, functionality, and effectiveness of their goal to keep the communities safe. Grounded in Black Swan theory, the purpose of this study was to explore strategies for building supply chain resilience and sustainability within law enforcement. Data were collected from 11 command staff personnel from three law enforcement departments in South Carolina through semi-structured interviews, member checking, and assessments of organizational documents related to departmental supply chain policies and procedures. Data were analyzed using Braun and Clarke's six-step thematic analysis framework. Six key themes emerged from the thematic analysis: implementing agile strategies for supply resilience, adjusting policies to ensure timely supplies, utilizing diverse suppliers for adaptable procurement, using diverse funding avenues for agile procurement, enhancing communication for the dynamic supply chain, and strategic engagement and proactive resource planning. The key recommendation for action is for command personnel to establish, implement, and integrate effective agile strategies, processes, and protocols and seek alternative funding sources such as state, local, and federal grants to anticipate and mitigate the adverse effects of Black Swan events on their supply chains. The implications for positive social change include the potential for law enforcement agencies to operate more efficiently, thereby rendering more efficient policing and community safety services even during disrupting events like a pandemic.
Supermassive Black Holes (SMBHs) are commonly found at the centers of massive galaxies. Estimating their masses (MBH) is crucial for understanding galaxy-SMBH co-evolution. We present WISE2MBH, an efficient algorithm that uses cataloged Wide-field Infrared Survey Explorer (WISE) magnitudes to estimate total stellar mass (M*) and scale this to bulge mass (MBulge), and MBH, estimating the morphological type (TType) and bulge fraction (B/T) in the process. WISE2MBH uses scaling relations from the literature or developed in this work, providing a streamlined approach to derive these parameters. It also distinguishes QSOs from galaxies and estimates the galaxy TType using WISE colors with a relation trained with galaxies from the 2MASS Redshift Survey. WISE2MBH performs well up to z ∼ 0.5 thanks to K-corrections in magnitudes and colors. WISE2MBH MBH estimates agree very well with those of a selected sample of local galaxies with MBH measurements or reliable estimates: a Spearman score of ∼0.8 and a RMSE of ∼0.63 were obtained. When applied to the ETHER sample at z ≤ 0.5, WISE2MBH provides ∼1.9 million MBH estimates (78.5 per cent new) and ∼100 thousand upper limits. The derived local black hole mass function (BHMF) is in good agreement with existing literature BHMFs. Galaxy demographic projects, including target selection for the Event Horizon Telescope, can benefit from WISE2MBH for up-to-date galaxy parameters and MBH estimates. The WISE2MBH algorithm is publicly available on GitHub.
