New maximum entropy methods for modeling lifetime distributions

This article introduces two new maximum entropy (ME) methods for modeling the distribution of time to an event. One method is within the classical ME framework and provides characterizations of change point models such as the piecewise exponential distribution. The second method uses the entropy of the equilibrium distribution (ED) for the objective function and provides new characterizations of the exponential, Weibull, Pareto, and uniform distributions. With the same moment constraints, the classical ME and the maximum ED entropy algorithms generate different models for the interarrival time. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 427–434, 2014     

三参数威布尔分布在寿命分析中的参数估计

威布尔分布在可靠性工程中已得到了广泛的应用.在对己给定的寿命试验数据进行可靠性分析与评估中,因简单易解多采用二参数威布尔分布,但参数估计会带来较大误差.对具有以损耗失效为特征的某些机械零部件,采用三参数威布尔分布进行拟合及参数估计,可以得到更高的精度,因而较二参数威布尔分布,更能反映产品可靠性的实际情况.     

Normal and weibull distributions

Numerous applications of the Weibull distribution in diverse fields of human endeavor are well known today. Nevertheless, it is not uncommon to find applications of the normal distribution in such fields of studies as agriculture, biology, chemistry, engineering, physics, sociology and others. At the present time we have at our disposal many more refined statistical techniques for analyzing the normal rather than the Weibull data. Consequently, it is important for applied statisticians to know if some of their data which can be described by the Weibull distribution can also be described by the normal distribution. The present investigation of the author reveals that the normal distri,bution can be considered to be a good approximation to the Weibull distribution as long as its shape parameter is in the open interval (3.25, 3.61). This fact enables them to perform a refined statistical analysis of their data. Conversely, they can now easily compute the desired normal cumulative probabilities from the Weibull distribution function, which would be especially helpful for those standard normal deviates whose cumulative probabilities cannot be read from the available tables of normal cumulative probability. In a similar situation they can also use the Weibull distribution to obtain an approximation to any desired normal deviate for a given normal probability which may be better than those obtained by the linear interpolation method.     

An integral equation for the second moment function of a geometric process and its numerical solution

In this article, an integral equation satisfied by the second moment function M₂(t) of a geometric process is obtained. The numerical method based on the trapezoidal integration rule proposed by Tang and Lam for the geometric function M(t) is adapted to solve this integral equation. To illustrate the numerical method, the first interarrival time is assumed to be one of four common lifetime distributions, namely, exponential, gamma, Weibull, and lognormal. In addition to this method, a power series expansion is derived using the integral equation for the second moment function M₂(t), when the first interarrival time has an exponential distribution.     

Mean hazard rate and its application to the normal approximation of the weibull distribution

It is important for applied statisticians to know if data that can be described by the Weibull distribution can also be described by the normal distribution. Some investigations have been published on this subject. In this article, we review the results in papers related to the above problem. Next, we define the mean hazard rate. We then investigate the properties of the mean hazard rate for the Weibull distribution and the normal distribution, respectively. Finally, we discuss the method of approximating the normal distribution by the Weibull distribution using the mean hazard rate.     

一种新的Weibull分布恒定应力加速寿命试验分析方法

分析了Weibull分布恒定应力加速寿命试验常用的二步分析方法存在的不足 ,建立了一种新的构造数据分析方法。考虑到分布参数的相关性 ,该方法引入了Weibull分布特征寿命参数的二次估计 ,在模型拟合优度上高于原来的二步分析方法 ,使分析精度得到了改善。同时 ,该方法避免了原来二步分析方法的查表过程 ,便于软件实现和工程实际应用。     

Statistical analysis for masked system life data from Marshall‐Olkin Weibull distribution under progressive hybrid censoring

This paper considers the statistical analysis of masked data in a series system, where the components are assumed to have Marshall‐Olkin Weibull distribution. Based on type‐I progressive hybrid censored and masked data, we derive the maximum likelihood estimates, approximate confidence intervals, and bootstrap confidence intervals of unknown parameters. As the maximum likelihood estimate does not exist for small sample size, Gibbs sampling is used to obtain the Bayesian estimates and Monte Carlo method is employed to construct the credible intervals based on Jefferys prior with partial information. Numerical simulations are performed to compare the performances of the proposed methods and one data set is analyzed.     

基于局部双参数估计的SAR图像CFAR目标检测技术

提出了在背景杂波呈现Weibull分布条件下的SAR图像恒虚警目标检测方法.从统计分析观点,推导了基于局部窗口的Weibull分布参数估计和自适应CFAR目标检测阈值的计算公式;对检测后的图像进行形态滤波处理以消除检测中的虚假目标;最后提出了应适当选择恒虚警率.仿真结果表明该方法的有效性.     

Tail hazard rate ordering properties of order statistics and coherent systems

We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k‐out‐of‐n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Heuristic modeling of supply availabilities in large networks

The objective of this article is to describe heuristic solutions to the problem of modeling inventories at each node of a large network in the context of a computer simulation model of that network. The heuristic solutions are compared with the mathematical solution which is too unwieldy for use in a simulation model. The Weibull cumulative distribution is used as an approximation for the heuristic models. We question whether the good performance of the Weibull is coincidence or perhaps mathematically justifiable.     

Test for the extreme value and weibull distributions based on normalized spacings

Discussed in this article are tests for the extreme-value distribution, or, equivalently, for the two-parameter Weibull distribution when parameters are unknown and the sample may be censored. The three tests investigated are based on the median, the mean, and the Anderson-Darling A² statistic calculated from a set zⁱ of values derived from the spacings of the sample. The median and the mean have previously been discussed by Mann, Scheuer, and Fertig [10] and by Tiku and Singh [14]. Asymptotic distributions and points are given for the test statistics, based on recently developed theory, and power studies are conducted to compare them with each other and with two other statistics suitable for the test. Of the normalized spacings tests, A² is recommended overall; the mean also gives good power in many situations, but can be nonconsistent.     

A nonparametric approach to accelerated life testing under multiple stresses

Accelerated life testing (ALT) is concerned with subjecting items to a series of stresses at several levels higher than those experienced under normal conditions so as to obtain the lifetime distribution of items under normal levels. A parametric approach to this problem requires two assumptions. First, the lifetime of an item is assumed to have the same distribution under all stress levels, that is, a change of stress level does not change the shape of the life distribution but changes only its scale. Second, a functional relationship is assumed between the parameters of the life distribution and the accelerating stresses. A nonparametric approach, on the other hand, assumes a functional relationship between the life distribution functions at the accelerated and nonaccelerated stress levels without making any assumptions on the forms of the distribution functions. In this paper, we treat the problem nonparametrically. In particular, we extend the methods of Shaked, Zimmer, and Ball [7] and Strelec and Viertl [8] and develop a nonparametric estimation procedure for a version of the generalized Arrhenius model with two stress variables assuming a linear acceleration function. We obtain consistent estimates as well as confidence intervals of the parameters of the life distribution under normal stress level and compare our nonparametric method with parametric methods assuming exponential, Weibull and lognormal life distributions using both real life and simulated data. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 629–644, 1998     

Mean square invariant forecasters for the weibull distribution

Many techniques of forecasting are based upon extrapolation from time series. While such techniques have useful applications, they entail strong assumptions which are not explicitly enunciated. Furthermore, the time series approach not based on an indigenous forecast principle. The first attack from the present point of view was initiated by S. S. Wilks. Of particular interest over a wide range of operational situations in reliability, for example, is the behavior of the extremes of the Weibull and Gumbel distributions. Here we formulate forecasters for the minima of various forms of these distributions. The forecasters are determined for minimization in mean square of the distance. From n original observations the forecaster provides the minimum of the next m observations when the original distribution is maintained. For each of the forecasters developed, tables of efficiency have been calculated and included in the appendix. An explicit example has been included for one of the forecasters. Its performance has been demonstrated by the use of Monte Carlo technique. The results indicate that the forecaster can be used in practice with satisfactory results.     

Some test functions for the parameters of the weibull distributions

Test functions, based on various types of censored and noncensored data, for testing several hypotheses about the location, the scale, and the shape parameters of the Weibull distributions are proposed. The exact sampling distributions of these test statistics are derived and their properties in special cases are discussed. A numerical example is considered to illustrate the application of the test functions. The results of this paper possess good possibility of wide application in view of the fact that hosts of real data arising from diverse fields of human endeavor are adequately described by the Weibull distribution.     

Recurrence relations between moments of order statistics

A general result for obtaining recurrence relations between single moments of order statistics is obtained and has been used to establish the recurrence relations between moments of some doubly truncated distributions. The examples considered are Weibull, exponential, Pareto, power function, Cauchy, and logistic. Recurrence relations are also obtained for nontruncated gamma and beta distributions.     

Weibull型产品的可靠性验证

讨论了Weibull型产品的可靠性验证问题 ,首先分析了Weibull型产品分布参数的无信息验前分布问题 ,利用验前信息可以得到分布参数的验前概率密度函数 ,进而分析了产品的可靠性验证问题 ,顾及了使用方利益和生产方利益。仿真算例表明 ,使用方利益和生产方利益是相互折衷的。     

按Weibull分布计算滚动轴承的寿命和强度的可靠性

运用weibull分布推导出任意可靠性指标下滚动轴承的寿命和强度计算公式,给出了多列轴承和轴承系统的可靠性计算方法和公式,并按照工程的方法加以简化处理。     

加速寿命试验下复合失效产品的可靠性评估

产品往往受随机失效和耗损失效两种模式的双重影响,通过构建复合寿命分布模型来表征这类产品失效规律。在加速寿命试验下,基于极大似然估计构造了该类产品寿命参数估计的数值算法;为了验证该寿命分布是否真服从假定的复合分布,构建了常应力等效失效数据的柯尔莫哥诺夫分布检验法;最后,以电连接器这一典型产品为例进行了工程仿真试验并进行数据评估,从而验证了评估方法的正确性。     

Optimum simple step-stress accelerated life tests for weibull distribution and type I censoring

This article presents an optimum simple step-stress accelerated life test for the Weibull distribution under Type I censoring. It is assumed that a log-linear relationship exists between the Weibull scale parameter and the (possibly transformed) stress and that a certain cumulative exposure model for the effect of changing stress holds. The optimum plan—low stress and stress change time—is obtained, which minimizes the asymptotic variance of the maximum likelihood estimator of a stated percentile at design stress. For selected values of the design parameters, nomographs useful for finding the optimum plan are given, and the effects of errors in preestimates of the parameters are investigated. As an alternative to the simple step-stress test, a three-level compromise plan is proposed, and its performance is studied and compared with that of the optimum simple step-stress test. © 1993 John Wiley & Sons. Inc.     

An empirical bayes estimator for the scale parameter of the two-parameter weibull distribution

An empirical Bayes estimator is given for the scale parameter in the two-parameter Weibull distribution. The scale parameter is assumed to vary randomly throughout a sequence of experiments according to a common, but unknown, prior distribution. The shape parameter is assumed to be known, however, it may be different in each experiment. The estimator is obtained by means of a continuous approximation to the unknown prior density function. Results from Monte Carlo simulation are reported which show that the estimator has smaller mean-squared errors than the usual maximum-likelihood estimator.