A Bayes Explanation of an Apparent Failure Rate Paradox

For the exponential life distribution model and any prior distribution for the failure rate parameter, the predictive distribution has a decreasing failure rate. A Bayes explanation is given of why this is logically reasonable.     

Estimation of Periodically Changing Failure Rate

This paper considers two life testing procedures (progressively censored samples and Bartholomew's experiment) under the assumption that the life of an item follows the exponential distribution. The failure rates are different under n different conditions of usage of the item at regular intervals of time. The maximum likelihood estimates of the n failure rates have been derived along with their asymptotic variances for both types of data (when failure times are recorded and when only the number of items failing in each interval are recorded). A numerical example illustrates the type of data and relevant calculations for the experiment involving progressively censored samples.     

Local Polynomial Fitting in Failure Rate Estimation

The failure rate function is one of the most important functions in reliability analysis as it addresses concepts such as scheduling, maintenance, improved system design, cost analysis, etc. Kernel-based estimation of the failure rate function imposes minimal assumptions on the data, and thus offers large flexibility. Therefore, it is very helpful in reliability analysis where the shape of the failure function is a primary target. However, standard kernel estimates are biased more than usual in the endpoints. Motivated by the advantages that the local linear fitting method provides for estimation of densities near the boundary, we employ the technique to the failure rate setting. We show that it produces estimators that have the same amount of boundary bias as in areas outside the boundary. The mean square error properties of the method are demonstrated, and a practically useful procedure for automatic smoothing parameter selection is developed, and studied both theoretically, and by simulations. The result is displayed graphically by distributional, and real life data.     

Nonparametric Bayes estimation of a distribution under nominationsampling

Nomination sampling is a sampling process in which every observation is the maximum of a random sample from some distribution. If the data are {(Y_i, K_i), i =1, . . ., n} where K_i is the size of sample i, and Y_i is the maximum of a random sample of size K_i from an unknown Cdf, F; the Bayes estimator of F is derived by discretizing F over a fixed finite partition of the support of F and taking a Dirichlet distribution as the prior for the probabilities of the partitioning intervals. For the flood data of the Nidd River considered by R.A. Boyles and F.J. Samaniego (J. Am. Stat. Assoc., vol.81, p.1039-45, 1986), the plots of the Bayes estimator of F are obtained for several sets of values of the parameters of the Dirichlet distribution     

Nonparametric Estimation of Conditional Distributions

Estimation of conditional distributions is considered. It is assumed that the conditional distribution is either discrete or that it has a density with respect to the Lebesgue measure. Partitioning estimates of the conditional distribution are constructed and results concerning consistency and rate of convergence of the integrated total variation error of the estimates are presented     

一种新的电子元器件突然失效率估算方法

在电子元器件可靠性特性的基础上,引入负荷系数的概念;同时考虑多种负荷的作用,建立了电子元器件突然失效率的估算模型;通过该模型所得的元器件突然失效率数据对电子设备及系统可靠性的计算更具有实际意义。     

Customer-Rush Near Warranty Expiration Limit, and Nonparametric Hazard Rate Estimation From Known Mileage Accumulation Rates

Time or mileage data obtained from warranty claims are generally more accurate for hard failures than for soft failures. For soft failures, automobile users sometimes delay reporting the warranty claim until the warranty coverage is about to expire. This results in an unusually high number of warranty claims near the end of warranty coverage. Because such a phenomenon of customer-rush near the warranty expiration limit occurs due to user behavior rather than due to the vehicle design, it creates a bias in the warranty dataset. Design improvement activities that use field reliability studies based on such data can potentially obtain a distorted picture of the reality, and lead to unwarranted, costly design changes. Research in the area of field reliability studies using warranty data provides several methods for warranty claims resulting from hard failures, and assumes reported time or mileage as actual time or mileage at failure. In this article, the phenomenon of customer-rush near the warranty expiration limit is addressed for arriving at nonparametric hazard rate estimates. The proposed methodology involves situations where estimates of mileage accumulation rates in the vehicle population are available. The claims influenced by soft failures are treated as left-censored, and are identified using information in technician comments about the repair carried out plus, if required, a more involved engineering analysis of field returned parts. Maximum likelihood estimates for the hazard function and their confidence limits are then obtained using Turnbull's iterative procedure. An application example illustrates use of the proposed methodology.     

Modified Method-of-Moments in Empirical Bayes Estimation

A unit is put on test for a fixed time and the number of failures is observed. The probability distribution of the number of failures is assumed to be Poisson, and the Poisson failure intensity is assumed to be a stochastic variable with gamma prior distribution. Schafer & Feduccia introduced an empirical procedure for estimating the parameters of the prior based on method of moments. We investigate the s-efficiencies of empirical Bayes estimates of Poisson failure intensity and reliability when the prior is estimated by the Schafer & Feduccia method. Mean square errors (MSEs) are compared for a range of parameters which typifies certain military equipment failure data. The empirical Bayes estimates have high s-efficiencies for sample size more than 40. A modification of the Schafer & Feduccia procedure substantially improves s-efficiencies for small sample sizes.     

深入理解失效率和瞬时失效率

对失效率λ(t)和瞬时失效率h(t)所具有的实际含义和表征内容进行了深入的探讨。同时详解了失效密度函数f(t)在可靠性工程计算中具有将随机失效数转变为连续变化量的作用。对这几个可靠性参量在工程应用和计算时的适用场合作了分析。通过简洁易懂的论述和实际的计算实例,重点说明了瞬时失效率的3个特点,希望在可靠性基础理论的学习中正确地理解失效率和瞬时失效率的意义。     

Bayes检测与估计的教学探索

Bayes检测和Bayes估计是“信号检测与估计”课程中的主要内容.本文给出一种闭环分析Bayes检测和Bayes估计的方法.通过由Bayes检测的离散求和方法,过渡到Bayes估计的连续积分方法;又通过Bayes估计的决策空间的二维离散化,推导出Bayes检测器的形式.该方法特点是建立由此及彼的闭环联系和反馈,通过比较的观点,突出了Bayes检测和Bayes估计的本质和联系,便于学生理解和掌握.     

Empirical Bayes Estimation in the Weibull Distribution

In part I empirical Bayes estimation procedures are introduced and employed to obtain an estimator for the unknown random scale parameter of a two-parameter Weibull distribution with known shape parameter. In part II, procedures are developed for estimating both the random scale and shape parameters. These estimators use a sequence of maximum likelihood estimates from related reliability experiments to form an empirical estimate of the appropriate unknown prior probability density function. Monte Carlo simulation is used to compare the performance of these estimators with the appropriate maximum likelihood estimator. Algorithms are presented for sequentially obtaining the reduced sample sizes required by the estimators while still providing mean squared error accuracy compatible with the use of the maximum likelihood estimators. In some cases whenever the prior pdf is a member of the Pearson family of distributions, as much as a 60% reduction in total test units is obtained. A numerical example is presented to illustrate the procedures.     

变区间分组检验相乘积累进动周期估计

 该文首先分析了弹道中段进动锥体目标RCS序列的统计特性及其周期性特点,指出目标本身RCS起伏产生的虚假周期是影响传统进动周期估计方法性能的主要因素.基于非参数统计理论,本文提出了一种变区间分组检验相乘积累的RCS序列进动周期估计方法,仿真结果表明,该方法不仅能够有效克服虚假周期影响,并能明显改善进动周期估计精度.     

基于非参数化概率密度估计的雷达目标识别

针对雷达目标识别中,参数化方法估计高分辨距离像的概率密度存在的模型失配问题,提出一种非参数化方法基于累计量的随机学习算法,估计距离像的概率密度。该算法运用多层感知器估计训练样本的分布函数,然后求导得到概率密度。该算法不仅能全面、精确地估计概率密度,而且回避了许多其他非参数方法面临的窗宽敏感性问题。基于外场实测数据的实验证明了该文方法的有效性。     

基于非参数密度估计的多通道盲解卷积

非参数密度估计方法被用来直接估计在自然梯度盲解郑积算法中遇到的评价函数(score function)。与用一个非线性函数简单地代替评价函数相比较,这种直接估计评价函数的方法的主要优点是:它可以用来对杂系混合信号,即同时包含超高斯和亚高斯的信号,进行盲解卷积。因为评价函数可以被直接的估计出来,因此,就不需要针对不同的源信号选择不同的非线性函数来代替评价函数。这种方法可以用在更加“盲”的情况。     

Blind Source Separation Based on Nonparametric Density Estimation

A nonparametric density estimation method is used to directly estimate the score functions encountered in relative gradient (or natural gradient) adaptation algorithms in the blind source separation problem. Compared to the method where simple nonlinear functions are used to replace the unknown score functions, the key advantage of the direct estimation of the score functions lies in the fact that it enables the algorithm to separate hybrid mixtures of sources that contain both super-Gaussian and sub-Gaussian signals. The source statistics required for the choices of the nonlinear functions is no longer needed, because the score functions are directly estimated. The algorithm is thus expected to be applicable to more blind cases.     

Empirical Bayes Estimators of Reliability for Lognormal Failure Model

Empirical Bayes (EB) procedures are considered for estimating the reliability R(t;?,?) = gaufc[(ln t -?)/?] for the lognormal failure model. EB estimators are obtained for the 2 cases: i)? is unknown and ? is known, and both ? and ? are unknown. The empirical Cdf of the maximum likelihood estimators of the parameters is used to obtain the EB estimators. ii) A smooth EB estimator of R(t;?,?) is developed when ? is unknown and ? is known. A modification of this estimator is proposed for both ? and ? unknown. In both cases, EB estimators are obtained for complete samples at each testing stage. Monte Carlo simulations are presented to compare the EB estimators and the maximum likelihood (ML) estimators of R(t;?,?). The simulations indicate that the smooth EB estimators have smaller mean squared errors than the other EB estimators or the ML estimators.     

Failure Rate/MTBF

Failure rate is the most commonly used term in reliability and related engineering interests, yet it is still not well understood by perhaps the majority of those using it. Indeed, more than one reliability treatise wrongly defines failure rate as a conditional probability density, i.e., wrong by the accepted criterion of what constitutes any probability density. The fact that these two concepts are by no means the same is clarified and made more precise. Mean time between failures and dimensionality are also briefly considered.     

Bayes Estimation of Reliability for Special k-out-of-m:G Systems

Component lifetimes of a k-out-of-m: G system are s-independent and exponentially distributed. The Bayes-quadratic-loss estimator of reliability for the system is obtained using a conjugate (1-parameter case) or a noninformative (2-parameter case) prior when data are type-II censored. A simple approximation formula for the mean square error of the Bayes estimator is proposed. Based on the mean square-error criterion, the performance of the Bayes estimator under a noninformative prior is compared with that of the maximum likelihood and minimum variance s-unbiased estimators.     

Nonparametric Estimation of the Number of Unique Sequences in Biological Samples

Large-scale determination of uniquely expressed genes (or mRNAs) in specific cells and tissues is a challenging problem in computational and functional genomics. We consider nonparametric approaches for estimating the number of unique, nonredundant sequences in biological samples. By introducing the moments of species' abundance in a population, we analyze the relative abundance of species in the population and present a lower bound estimator and so-called medial estimator for the number of distinct species in the population. The lower bound estimate is applicable to populations with small coefficients of variation (CV). The medial estimator works well for the populations with relatively large CV, especially gene expression data. Simulation analysis shows that the medial estimator performs better than existing methods. Finally, we apply our nonparametric approaches to estimate the number of expressed mRNAs in a normal colon epithelial tissue as well as unique clones in an amplified cDNA sample prepared from the CNS of the sea-slug Aplysia.     

基于Bayes和Fiducial方法的指数分布区间估计

在已知产品失效率不超过某个值的条件下,对定时截尾寿命试验,采用Bayes方法进行可靠性评估;对定数截尾寿命试验,采用参数受限制的Fiducial方法进行可靠性评估。根据这两种方法,分别给出了平均寿命单侧置信下限计算公式,同时讨论了先验分布参数的取值方法。最后,通过算例验证了这两种方法的有效性,算例表明两种方法均能够减少试验时间、提高可靠性评估精度。