Moment estimators for the 3-parameter Weibull distribution

Weibull moments are defined generally and then calculated for the 3-parameter Weibull distribution with non-negative location parameter. Sample estimates for these moments are given and used to estimate the parameters. The results of a simulation investigation of the properties of the parameter estimates are discussed briefly. A simple method of deciding whether the location parameter can be considered zero is described     

恒加试验下LED照明灯的三参数Weibull分布的参数估计

三参数Weibull分布拟合LED照明灯寿命的优势较为明显,但要得到三参数Weibull分布参数较为精确的点估计较为困难。目前常用的参数估计方法有极大似然法、矩估计法、Bayes估计法等,由于其计算的方程复杂,导致软件编程繁琐,不易掌握,而且也不一定能得到参数估计。鉴于此,文章针对恒加试验提出一种简便地求解三参数Weibull分布参数估计的方法,该方法不涉及超越方程的求解问题,软件编程相当简单,且统计思想清晰。通过LED照明灯恒加试验下的几个案例数据说明方法的应用,并与已有的方法做了对比分析。     

Some Graphical Techniques for Estimating Weibull Confidence Intervals

Many methods for estimating the parameter and percentile statistical confidence intervals for the Weibull and Gumbel (extreme value) distributions have been described in the literature. Most of these methods depend on extensive computer programs, require reference to tables which do not cover all sample sizes of interest and/or are not widely available. This paper describes a semi-empirical technique which permits rapid estimation of the 2-sided 90% statistical confidence intervals for the Weibull or Gumbel distribution parameters, as well as for the 1, 5, 10 percentiles. The estimates can be obtained for type II censoring and sample sizes to 25. The statistical confidence intervals calculated using this method are not exact, but are very good approximations and are useful to engineers who do not have ready access to programs or lengthy tables, or who require quick estimates. If more accurate statistical confidence intervals are required, then the more complicated methods described in the references should be used.     

A general linear-regression analysis applied to the 3-parameterWeibull distribution

The conventional techniques of linear regression analysis (linear least squares) applied to the 3-parameter Weibull distribution are extended (not modified), and new techniques are developed for the 3-parameter Weibull distribution. The three pragmatic estimation methods in this paper are simple, accurate, flexible, and powerful in dealing with difficult problems such as estimates of the 3 parameters becoming nonpositive. In addition, the inherent disadvantages of the 3-parameter Weibull distribution are revealed; the advantages of a new 3-parameter Weibull-like distribution over the original Weibull distribution are explored; and the potential of a 4-parameter Weibull-like distribution is briefly mentioned. This paper demonstrates how a general linear regression analysis or linear least-squares breaks away from the classical or modern nonlinear regression analysis or nonlinear least-squares. By adding a parameter to the simplest 2-parameter linear regression model (AB-model), two kinds of ABC models (elementary 3-parameter nonlinear regression models) are found, and then a 4-parameter AABC model is built as an example of multi-parameter nonlinear regression models. Although some other techniques are still necessary, additional applications of the ABC models are strongly implied     

A covariance fitting approach to parametric localization of multiple incoherently distributed sources

In this paper, a new algorithm for parametric localization of multiple incoherently distributed sources is presented. This algorithm is based on an approximation of the array covariance matrix using central and noncentral moments of the source angular power densities. Based on this approximation, a new computationally simple covariance fitting-based technique is proposed to estimate these moments. Then, the source parameters are obtained from the moment estimates. Compared with earlier algorithms, our technique has lower computational cost and obtains the parameter estimates in a closed form. In addition, it can be applied to scenarios with multiple sources that may have different angular power densities, while other known methods are not applicable to such scenarios.     

Maximum likelihood estimation of K distribution parameters for SARdata

The K distribution has proven to be a promising and useful model for backscattering statistics in synthetic aperture radar (SAR) imagery. However, most studies to date have relied on a method of moments technique involving second and fourth moments to estimate the parameters of the K distribution. The variance of these parameter estimates is large in cases where the sample size is small and/or the true distribution of backscattered amplitude is highly non-Rayleigh. The present authors apply a maximum likelihood estimation method directly to the K distribution. They consider the situation for single-look SAR data as well as a simplified model for multilook data. They investigate the accuracy and uncertainties in maximum likelihood parameter estimates as functions of sample size and the parameters themselves. They also compare their results with those from a new method given by Raghavan (1991) and from a nonstandard method of moments technique; maximum likelihood parameter estimates prove to be at least as accurate as those from the other estimators in all cases tested, and are more accurate in most cases. Finally, they compare the simplified multilook model with nominally four-look SAR data acquired by the Jet Propulsion Laboratory AIRSAR over sea ice in the Beaufort Sea during March 1988. They find that the model fits data from both first-year and multiyear ice well and that backscattering statistics from each ice type are moderately non-Rayleigh. They note that the distributions for the data set differ too little between ice types to allow discrimination based on differing distribution parameters     

Robust parameter-estimation using the bootstrap method for the2-parameter Weibull distribution

This paper proposes bootstrap robust estimation methods for the Weibull parameters; it applies bootstrap estimators of order statistics to the parametric estimation procedure. Estimates of the Weibull parameters are equivalent to the estimates using the extreme value distribution. Therefore, the bootstrap estimators of order statistics for the parameters of the extreme value distribution are examined. Accuracy and robustness for outliers are examined by Monte Carlo experiments which indicate adequate efficiency of the proposed reliability estimators for data with some outliers     

Proportional difference estimate method for bimodal and multimodalfailure distributions

Peaks appear after proportional differentiation of cumulative distribution functions of Weibull and lognormal distributions. The characteristic parameters can be extracted from the proportional difference peaks because these peaks are related to the characteristic parameters directly. On this basis, a simple method known as the proportional difference estimate (PDE) method for determining the characteristic parameters of multimodal failure distributions was developed. This method can be applied to microelectronics dielectric and interconnect reliability studies     

Improved percentile estimation for the two-parameter Weibull distribution

This paper presents an improvement of a technique recently published to estimate the parameters of the two-parameter Weibull distribution. A simple percentile method is used to estimate the two parameters. Computer simulation is employed to compare the proposed method with the maximum likelihood estimation and graphical methods results. A set of frequently-used and newer expressions for estimating the cumulative density are examined. Comparisons are made with both complete and censored data. The primary advantage of the method is its computational simplicity. Results indicate that with respect to Mean Square Error and estimation of the characteristic value with complete data, the percentile method cannot outperform the maximum likelihood method, although differences are minor in many instances. However, with censored data, improvements over the maximum likelihood are observed. When the shape parameter is estimated, the percentile method is quite competitive with that of maximum likelihood for both complete and censored data under a variety of conditions.     

基于Fisher分布的极化合成孔径雷达统计建模及其参数估计方法

为更准确地描述高分辨极化合成孔径雷达(Synthetic Aperture Radar, SAR)图像的尖峰和长拖尾等统计分布特性, 提出了基于Fisher分布的极化图像多变量乘积模型, 并研究了其参数估计方法.首先给出了柯西分布相干斑噪声等效纹理分量的概率密度函数及其低阶矩特征;然后利用散射因子服从F分布的等效纹理变量与高斯散斑变量相乘形成的多变量乘积统计模型, 得到了Fisher分布模型的概率密度函数, 并推导了其多视协方差矩阵的概率密度函数和矩阵行列式值的低阶矩特征;最后提出了基于矩阵行列式值的矩估计和基于Mellin变换的对数累积量估计等两种参数估计方法, 并进行了对比, 同时通过仿真数据和实测数据验证了理论模型和新参数估计方法的有效性.这为高分辨极化SAR图像建模、目标检测和识别等领域的理论研究和工程实现提供了新途径.     

Variance of system-reliability estimates with arbitrarily repeatedcomponents

A model is developed to determine the variance of system reliability estimates and to estimate confidence intervals for series-parallel systems with arbitrarily repeated components. For these systems, different copies of the same component-type are used several or many times within the system, but only a single reliability estimate is available for each distinct component-type. The single estimate is used everywhere the component appears in the system design, and component estimation-error is then magnified at the system-level. The "system-reliability estimate" variance and confidence intervals are derived when the number of component failures follow the binomial distribution with an unknown, yet estimable, probability of failure. The "system-reliability estimate" variance and confidence intervals are obtained by expressing system reliability as a linear sum of products of higher order moments for component unreliability. The generating function is used to determine the moments of the component-unreliability estimates. This model is preferable for many system reliability estimation problems because it does not require independent component and subsystem reliability estimates; it is demonstrated with an example     

Mean likelihood frequency estimation

Estimation of signals with nonlinear as well as linear parameters in noise is studied. Maximum likelihood estimation has been shown to perform the best among all the methods. In such problems, joint maximum likelihood estimation of the unknown parameters reduces to a separable optimization problem, where first, the nonlinear parameters are estimated via a grid search, and then, the nonlinear parameter estimates are used to estimate the linear parameters. We show that a grid search can be avoided by using the mean likelihood estimator for estimating the unknown nonlinear parameters and how its performance can be made equivalent to that of the maximum likelihood estimator (MLE). The mean likelihood estimator requires computation of a multidimensional integral. However, using the concepts of importance sampling, we obtain the mean likelihood estimate without using integration. The technique is computationally far less burdensome than the direct maximum likelihood method but performs just as well. Simulation examples for estimating frequencies of multiple sinusoids in noise are given. The general technique can be applied to a large class of nonlinear regression problems     

An alternative degradation reliability modeling approach using maximum likelihood estimation

An alternative degradation reliability modeling approach is presented in this paper. This approach extends the graphical approach used by several authors by considering the natural ordering of performance degradation data using a truncated Weibull distribution. Maximum Likelihood Estimation is used to provide a one-step method to estimate the model's parameters. A closed form expression of the likelihood function is derived for a two-parameter truncated Weibull distribution with time-independent shape parameter. A semi-numerical method is presented for the truncated Weibull distribution with a time-dependent shape parameter. Numerical studies of generated data suggest that the proposed approach provides reasonable estimates even for small sample sizes. The analysis of fatigue data shows that the proposed approach yields a good match of the crack length mean value curve obtained using the path curve approach and better results than those obtained using the graphical approach.     

Inference on Weibull Percentiles and Shape Parameter from Maximum Likelihood Estimates

Four functions of the maximum likelihood estimates of the Weibull shape parameter and any Weibull percentile are found. The sampling distributions are independent of the population parameters and depend only upon sample size and the degree of (Type II) censoring. These distributions, once determined by Monte Carlo methods, permit the testing of the following hypotheses: 1) that the Weibull shape parameter is equal to a specified value; 2) that a Weibull percentile is equal to a specified value; 3) that the shape parameters of two Weibull populations are equal; and 4) that a specified percentile of two Weibull populations are equal given that the shape parameters are. The OC curves of the various tests are shown to be readily computed. A by-product of the determination of the distribution of the four functions are the factors required for median unbiased estimation of 1) the Weibull shape parameter, 2) a Weibull percentile, 3) the ratio of shape parameters of two Weibull distributions, and 4) the ratio of a specified percentile of two Weibull distributions.     

Second-order parameter estimation

This work provides a general framework for the design of second-order blind estimators without adopting any approximation about the observation statistics or the a priori distribution of the parameters. The proposed solution is obtained minimizing the estimator variance subject to some constraints on the estimator bias. The resulting optimal estimator is found to depend on the observation fourth-order moments that can be calculated analytically from the known signal model. Unfortunately, in most cases, the performance of this estimator is severely limited by the residual bias inherent to nonlinear estimation problems. To overcome this limitation, the second-order minimum variance unbiased estimator is deduced from the general solution by assuming accurate prior information on the vector of parameters. This small-error approximation is adopted to design iterative estimators or trackers. It is shown that the associated variance constitutes the lower bound for the variance of any unbiased estimator based on the sample covariance matrix. The paper formulation is then applied to track the angle-of-arrival (AoA) of multiple digitally-modulated sources by means of a uniform linear array. The optimal second-order tracker is compared with the classical maximum likelihood (ML) blind methods that are shown to be quadratic in the observed data as well. Simulations have confirmed that the discrete nature of the transmitted symbols can be exploited to improve considerably the discrimination of near sources in medium-to-high SNR scenarios.     

Robust parameter estimation of intensity distributions for brainmagnetic resonance images

Presents two new methods for robust parameter estimation of mixtures in the context of magnetic resonance (MR) data segmentation. The head is constituted of different types of tissue that can be modeled by a finite mixture of multivariate Gaussian distributions. The authors' goal is to estimate accurately the statistics of desired tissues in presence of other ones of lesser interest. These latter can be considered as outliers and can severly bias the estimates of the former. For this purpose, the authors introduce a first method, which is an extension of the expectation-maximization (EM) algorithm, that estimates parameters of Gaussian mixtures but incorporates an outlier rejection scheme which allows to compute the properties of the desired tissues in presence of atypical data. The second method is based on genetic algorithms and is well suited for estimating the parameters of mixtures of different kind of distributions. The authors use this property by adding a uniform distribution to the Gaussian mixture for modeling the outliers. The proposed genetic algorithm can efficiently estimate the parameters of this extended mixture for various initial settings. Also, by changing the minimization criterion, estimates of the parameters can be obtained by histogram fitting which considerably reduces the computational cost. Experiments on synthetic and real MR data show that accurate estimates of the gray and white matters parameters are computed     

脉冲噪声下基于压缩变换函数的LFM信号参数估计

针对现有线性调频(LFM)信号参数估计方法在脉冲噪声下性能退化甚至完全失效的问题,该文提出一种脉冲噪声下估计LFM信号参数的新方法。该文构造了一种新的压缩变换(CT)函数,分析了该函数在零点附近的近线性,推导了任意随机变量经该函数变换后的2阶矩有界,证明了函数变换前后LFM信号的初始频率和调频斜率信息不变。将经过函数变换后的信号进行分数阶傅里叶变换(FrFT),根据FrFT域中峰值坐标和信号参数的关系,寻找变换域中的峰值点,实现信号参数的估计。仿真实验表明,该方法可有效抑制脉冲噪声且能准确估计出信号的参数信息,实现简单,不需要噪声的先验信息,具有良好的稳健性。     

Multiple Comparison for Weibull Parameters

Two problems are considered: 1) testing the hypothesis that the shape parameters of k 2-parameter Weibull populations are equal, given a sample of n observations censored (Type II) at r failures, from each population; and 2) Under the assumption of equal shape parameters, the problem of testing the equality of the p-th percentiles. Test statistics (for these hypotheses), which are simple functions of the maximum likelihood estimates, follow distributions that depend only upon r,n,k,p and not upon the Weibull parameters. Critical values of the test statistics found by Monte Carlo sampling are given for selected values of r,n,k,p. An expression is found and evaluated numerically for the exact distribution of the ratio of the largest to smallest maximum likelihood estimates of the Weibull shape parameter in k samples of size n, Type II censored at r = 2. The asymptotic behavior of this distribution for large n is also found.     

ML parameter estimation for Markov random fields with applicationsto Bayesian tomography

Markov random fields (MRFs) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (sometimes referred to as hyperparameters) is difficult in practice for two reasons: (i) direct parameter estimation for MRFs is known to be mathematically and numerically challenging; (ii) parameters can not be directly estimated because the true image cross section is unavailable. We propose a computationally efficient scheme to address both these difficulties for a general class of MRF models, and we derive specific methods of parameter estimation for the MRF model known as generalized Gaussian MRF (GGMRF). We derive methods of direct estimation of scale and shape parameters for a general continuously valued MRF. For the GGMRF case, we show that the ML estimate of the scale parameter, sigma, has a simple closed-form solution, and we present an efficient scheme for computing the ML estimate of the shape parameter, p, by an off-line numerical computation of the dependence of the partition function on p. We present a fast algorithm for computing ML parameter estimates when the true image is unavailable. To do this, we use the expectation maximization (EM) algorithm. We develop a fast simulation method to replace the E-step, and a method to improve the parameter estimates when the simulations are terminated prior to convergence. Experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets.     

Estimation of Weibull Parameters With Competing-Mode Censoring

Existing results are reviewed for the maximum likelihood (ML) estimation of the parameters of a 2-parameter Weibull life distribution for the case where the data are censored by failures due to an arbitrary number of independent 2-parameter Weibull failure modes. For the case where all distributions have a common but unknown shape parameter the joint ML estimators are derived for i) a general percentile of the j-th distribution, ii) the common shape parameter, and iii) the proportion of failures due to failure mode j. Exact interval estimates of the common shape parameter are constructable in terms of the ML estimates obtained by using i) the data without regard to failure mode, and ii) existing tables of the percentage points of a certain pivotal function. Exact interval estimates for a general percentile of failure-mode-j distribution are calculable when the failure proportion due to failure-mode-j is known; otherwise a joint s-confidence region for the percentile and failure proportion is calculable. It is shown that sudden death endurance test results can be analyzed as a special case of competing-mode censoring. Tabular values for the construction of interval estimates for the 10-th percentile of the failure-mode-j distribution are given for 17 combinations of sample size (from 5 to 30) and number of failures.