Representing and Approximating Symmetric and Asymmetric Probability Coverage Intervals by Possibility Distributions

At the application level, it is important to be able to define, around the measurement result, an interval that will contain an important part of the distribution of the measured values, that is, a coverage interval. This practice, which is acknowledged by the ISO Guide, is a major shift from the probabilistic representation. It can be viewed as a probability–possibility transformation by viewing possibility distributions as encoding coverage intervals. In this paper, we extend previous works by proposing exact and approximate closed-form possibility expressions of optimum coverage intervals associated with continuous symmetric and asymmetric unimodal probability densities. The approximation is useful in finding bounds on $p$ -values for common statistical tests for key comparison measurements.      

Transformation of Bimodal Probability Distributions Into Possibility Distributions

At the application level, it is important to be able to define the measurement result as an interval that will contain an important part of the distribution of the measured values, that is, a coverage interval. This practice acknowledged by the International Organization for Standardization (ISO) Guide is a major shift from the probabilistic representation. It can be viewed as a probability/possibility transformation by viewing possibility distributions as encoding coverage intervals. In this paper, we extend previous works on unimodal distributions by proposing a possibility representation of bimodal probability distributions. Indeed, U-shaped distributions or Gaussian mixture distribution are not very rare in the context of physical measurements. Some elements to further propagate such bimodal possibility distributions are also exposed. The proposed method is applied to the case of three independent or positively correlated C-grade resistors in series and compared with the Guide to the Expression of Uncertainty in Measurement (GUM) and Monte Carlo methods.      

渤海北部冰厚重现值的极大似然法区间估计

不同重现期的年极值冰厚是有冰海区建筑物设计的关键指标。目前在确定设计冰厚时,往往只给出点估计。而求解设计冰厚的置信区间,在某一置信水平下获取设计值的范围,可以确定设计重现值的不确定性,给海工结构物的设计与建造提供指导。采用极大似然方法区间估计,给出了Gumbel分布、三参数Weibull分布、三参数对数正态分布和P-III型分布置信区间的求解过程。利用渤海北部营口和葫芦岛海区历年总冰厚极大值的实测数据,基于以上4种分布型式,采用极大似然法求得不同重现期下冰厚重现值的置信区间,并对各分布型式进行了优选比较。结果表明,选取P-III型分布求解这两地海冰厚度重现值的置信区间较优。     

切削过程再生颤振的模糊稳定性分析

在机床切削颤振分析中首次考虑了模糊不确定性因素的影响，利用模糊数学分析方法详细论述了切削过程再生颤振的模糊稳定性分析问题，给出了模糊稳定性极限切削宽度集合的可能性分布及其置信区间表达式，最后依据本文的理论分析，经计算机编程绘制了模糊稳定性图     

液体火箭发动机试验流量测量系统可靠性评定

针对液体发动机可靠性评定的研究现状,从性能可靠性和结构可靠性两个维度对流量测量系统的可靠性进行评定。给出了测量系统性能可靠性的定义,将测量数据视为正态分布,通过区间估计的方法计算单侧可靠性置信下限。给出了测量系统结构可靠性的定义。根据流量测量系统串联可靠性框图,对将指数寿命失效型试验信息转换为二项成败型试验信息。采用L-M法(Lindstrom-Maddens),结合查表和线性插值法求解结构可靠性置信下限。该测量系统可靠性评估方法对行业内测量技术人员具有重要的参考意义。     

基于模糊可能性测量理论零件的合格测量

在考虑零件样本不足的情况下,根据可能性理论,将零件尺寸视为模糊变量,给出了表征它的模糊约束和模糊可能性分布函数。基于模糊可能性测量理论,分析了当零件的尺寸公差带为清晰区间（普通集）和模糊允许区间（模糊集）时,测量零件合格与不合格的模糊可能性测量方法。提出了测量零件合格模糊可能性测量、零件可修复不合格模糊可能性测量和和零件不可修复不合格模糊可能性测量的计算公式,并给出了零件合格模糊可能性测量的变化规律。最后进行了实例计算。     

An Exact Confidence Set for a Maximum Point of a Univariate Polynomial Function in a Given Interval

Construction of a confidence set for a maximum point of a function is an important statistical problem which has many applications. In this article, an exact 1 ? α confidence set is provided for a maximum point of a univariate polynomial function in a given interval. It is shown how the construction method can readily be applied to many parametric and semiparametric regression models involving a univariate polynomial function. Examples are given to illustrate this confidence set and to demonstrate that it can be substantially narrower and so better than the only other confidence set available in the statistical literature that guarantees 1 ? α confidence level.     

Sequential Bayesian bit error rate measurement

As bit error rates decrease, the time required to measure a bit error rate (BER) or perform a BER test (i.e., to determine that a particular communications device's BER is less than some acceptable limit) increases dramatically. One cause of long measurement times is the difficulty of deciding a priori how many bits to measure to establish the BER to within a predetermined confidence interval width. This paper explores a new approach to deciding how many bits to measure, namely a sequential Bayesian approach. As measurement proceeds, the posterior distribution of BER is checked to see if the conclusion can be made that the BER rate is known to be within the desired range with high enough probability. Desired properties of the posterior distribution such as the maximum a postiori estimate and confidence limits can be computed quickly using off-the-shelf numerical software. Examples are given of using this method on bit error data measured with an Agilent 81250 parallel BER tester.     

Bootstrap Monte Carlo Simulation of Reliability and Confidence Level with Periodical Maintenance

The bootstrap Monte Carlo simulation (BMCS) is presented as a new method for determining the reliability and confidence level of a general technical system with periodical maintenance. The method can be implemented on the basis of pure samples of failure times or of the mean value distributions in combination with the corresponding sample size of reliability. The BMCS does not require special distribution types and facilitates both the prediction and demonstration of reliability.Firstly, the bootstrap method for determining a confidence interval of a distribution function is investigated in order to verify the use of BMCS as a new method. The confidence level calculated by bootstrapping is therefore to be compared with the confidence level based on the beta-distribution. The accuracy of the BMCS shall subsequently be investigated by observing a component with periodical renewal. The confidence level of the reliability with periodical renewal calculated using the BMCS will be compared with the confidence level determined using the method of moments. Finally, a parameter study is carried out, and a case study as well as the potential of the BMCS shall be presented.     

Uncertainty in optical measurement applications: a case study

The uncertainty related to a measurement is at least as important as the measurement itself. Apart from being able to determine intervals of confidence around the final result within which the true measurement value is expected to lie at a certain level of confidence, the rigorous treatment of uncertainty throughout an algorithm allows increasing its robustness against disturbing influences and judging its applicability to a given task. This paper addresses the propagation of uncertainty within a quality control application using image-based sensors. Simulations and real-world results are provided to show the applicability of the proposed application.     

An application of bootstrap method for analysis of particle size distribution

The Bootstrap method, which is used widely in statistics, is a very powerful method that can be applied to the analysis of particle size distribution. For a number based measurement, this method can estimate the statistic uncertainty or confidence interval for any statistical quantities of interest from the distribution, with very simple protocol and without any parametric assumptions. This paper gives a demonstration to introduce the method to the community of particle size analysis.     

Confidence Intervals from Censored Samples,II

In previous work [1] I have presented an interval estimation procedure applicable to singly censored samples from any univariate population depending only on a scale parameter and location parameter; the procedure applies to several censoring criteria. For any parametric function, the procedure gives an interval estimate based on two confidence statements each at the 1 – 2α level; these estimates were shown to be at least at the (1 – 2α)² level for any size sample and converge, for large samples, to population values. The present work concerns itself specifically with censored samples from N(μ, σ) and studies the possibility of reducing the conservatism of the procedure. It is shown that, asymptotically, the procedure is at least at the 1 – 2α level. For some special parametric functions, theoretical considerations and fairly extensive computations strongly suggest that the procedure is at least at the 1 1 – 2α level even for small samples. It is conjectured that this property carries over to other functions and other distributions involving only a location and a scale parameter.     

Limitation of individual internal exposure by consideration of the confidence interval in routine personal dosimetry at the Chernobyl Sarcophagus

In view of the probabilistic nature and very wide uncertainty of internal exposure assessment, its deterministic ('precise') assessment does not protect against not exceeding established reference levels or even the dose limits for a particular individual. Minimising such potential risks can be achieved by setting up a sufficiently wide confidence interval for an expected dose distribution instead of its average ('best' estimate) value, and by setting the limit at the 99% fractile level. The ratio of the 99% level and the mean ('best' estimate) is referred to as the safety coefficient. It is shown for the typical radiological conditions inside the Chernobyl Sarcophagus that the safety coefficient corresponding to the 99% fractile of the expected internal dose distribution varies within the range from 5 to 10. The maintenance of minimum uncertainty and sufficient sensitivity of the indirect dosimetry method requires measurement of individual daily urinary excretion of 239Pu at a level of at least 4 x 10(-5) Bq. For the purpose of reducing the uncertainty of individual internal dose assessment and making dosimetric methods workable. it is suggested that the results of workplace monitoring are combined with the results of periodic urinary and faecal bioassay measurements.     

Measurement of the detective quantum efficiency (DQE) of digital X-ray detectors according to the novel standard IEC 62220-1

A mobile measurement facility which complies with IEC 62220-1 has been set up to determine the detective quantum efficiency (DQE) of digital X-ray detector systems. Exemplary measurements were performed for two similar CR detector systems, a CsI-based indirect detector and an Se-based direct detector. The standardised radiation quality RQA 5 was applied for measurement and for three of these systems RQA 9 was also applied. A pronounced dependence of DQE on radiation quality was observed for the direct detector, where the DQEs for RQA 5 and RQA 9 differ by a factor of approximately 2. The uncertainty (95% confidence interval) associated with the measured DQE values is within 0.01 and 0.04 depending on, for example, the spatial frequency. Thus, it has been demonstrated that the DQE can be measured accurately and reliably with the accuracy required by the international standard IEC 62220-1. It is now possible to objectively measure and compare DQE values of digital X-ray detector systems.     

Spectral attenuation and aerosol particle size distribution

The problem of the reconstruction of the spectrum of a dispersed system from data on its spectral attenuation is studied. The numerical algorithm for obtaining the particle size distribution by the use of the concept of regularization is thoroughly treated. The applicability of this method to the reconstruction of the particle size distribution of a typical marine aerosol is tested. A method of choosing the regularization parameter of the solution for the inverse problem based on an objective estimate of the validity of the obtained solution is proposed. Results are presented for a set of numerical experiments in which the radius interval for which the distribution function can be obtained with a satisfactory accuracy is estimated. The validity of solutions is estimated depending on the measuring spectral range for the attenuation, the radius interval, and the number and position of points within this interval. The possibility of extending the radius interval for which the distribution function can be obtained by the use of extrapolation of the distribution function tail is discussed.     

基于电压串联加法的1000kV国家工频电压计量标准

国家高电压计量站使用了特殊设计的准确度达0.01级的1000kV串联式标准电压互感器成功地进行了1000kV工频电压加法试验,比值差和相位差测量不确定度均小于等于2×10^-5和0.06′。所得量值与800kV电容式工频电压比例标准装置测量结果比对,比值差的最大偏差为2.7×10^-5,相位差的最大偏差为0.11′,均没有超出双方的90%置信概率区间。     

Exact tolerance interval with specified tail probabilities and a control chart for the sample variance

A β-content tolerance interval (TI) is a statistical interval which contains at least some fraction (proportion) β of the population with a given confidence level. When we are interested in the precision of a quality characteristic, a TI for the sample variance is useful. In this paper, we consider an exact two-sided β-content TI for the sample variance from a normal distribution with a specified ratio of the tail probabilities. The proposed tolerance interval allows the practitioner more control over how the probabilities in the tails are distributed, which may be useful in certain applications. A comparison with an existing two-sided β-content TI shows that the proposed TI is better on the basis of expected coverage and standard deviation of the coverage. In addition, the proposed TI is shown to require fewer subgroups to achieve a specific accuracy level. Moreover, a phase II control chart with guaranteed performance is obtained from the proposed TI. Finally, a real and a simulated data are used for illustration.     

Estimation of the uncertainty in internal dose calculation for two contamination cases

The assessment of internal dose is subject to a large uncertainty due to the limits of measuring technique and to the assumptions made by the expert. Here, we propose an approach to report the confidence interval associated with the evaluated dose. The sources of uncertainties considered so far include the date of intake, the physico-chemical characteristics of the radioactive material, the counting error and the stochastic variability of excretion. Three successive levels of approximation are suggested, depending on the expected dose, for which increasingly realistic parameter values should be sought and applied. Finally, the results of a Monte Carlo dose calculation are presented in the form of a statistical distribution of possible dose values. This approach has been applied to two cases of uranium and caesium exposure.     

Inference on the Gamma Distribution

This study develops inferential procedures for a gamma distribution. Based on the Cornish–Fisher expansion and pivoting the cumulative distribution function, an approximate confidence interval for the gamma shape parameter is derived. The generalized confidence intervals for the rate parameter and other quantities such as mean are explored. The proposed generalized inferential procedures are extended to construct prediction limits for a single future measurement and for at least p of m measurements at each of r locations. The performance of the proposed procedures is evaluated using Monte Carlo simulation. The simulation results show that the proposed procedures are very satisfactory. Finally, three real examples are used to illustrate the proposed procedures. Supplementary materials for this article are available online.     

Uncertainties in doses calculated according to ICRP recommendations after inhalation of 239PUO2 and early chest monitoring

This study estimates uncertainties in Pu biokinetics and effective doses calculated after an acute inhalation exposure to (239)PuO(2) according to ICRP recommendations (default values for aerosols size and PuO(2) dissolution parameters). This was performed using the most recently reported variations in model parameters and simulations after a Monte Carlo approach. Without chest monitoring, uncertainties in thoracic retention and plutonium excretion was 8-10 (95% confidence interval as the ratio between 97.5 and 2.5 percentiles of the lognormal distributions) up to 900 d after exposure. Early chest monitoring reduces significantly the uncertainties in plutonium biokinetics and doses which remain within a 95% confidence interval of 2.3 as compared with 6.6, without monitoring. Analysis of bioassay data previously reported shows that the dose delivered to some individuals can be out of the confidence interval, which was mostly due to an inhibition of the late mechanical clearance of the alveolar interstitium.