Inference for constant-stress accelerated degradation test based on Gamma process

This paper proposes a Gamma constant-stress accelerated degradation model based on the principle of the degradation mechanism invariance. The maximum likelihood estimators of the parameters of the proposed model are derived. Based on Cornish–Fisher expansion, the approximate confidence interval for the shape parameter of the Gamma degradation process is developed. Since it is difficult to obtain the exact confidence intervals for other model parameters and some quantities such as the mean degradation in unit time, the quantile and the reliability function of the lifetime at the normal stress level, the generalized confidence intervals for these quantities are proposed. The percentiles of the proposed generalized pivotal quantities can be obtained by the simulation. The performances of the proposed confidence intervals are evaluated by the Monte Carlo simulation method. In the simulation study, the proposed confidence intervals are compared with the Wald and the bootstrap-p confidence intervals. The simulation results show that the proposed confidence intervals outperform the Wald and the bootstrap-p confidence intervals in terms of the coverage percentage. Finally, a real example is used to illustrate the proposed procedures.     

基于混合Ⅰ型删失数据的威布尔模型精确推断（英文）

威布尔分布是可靠性和寿命测试试验中常用的模型.本文中,我们考虑了基于混合Ⅰ型删失数据的威布尔模型精确推断.我们得到了威布尔分布未知参数最大似然估计的精确分布以及基于精确分布的置信区间.由于精确分布函数较为复杂,我们也给出了未知参数的另外几种置信区间,比如,基于近似方法的置信区间,Bootstrap置信区间.为了评价本文的方法,我们给出了一些数值模拟的结果.     

Likelihood Inference under Generalized Hybrid Censoring Scheme with Comp eting Risks

Statistical inference is developed for the analysis of generalized type-II hybrid censoring data under exponential competing risks model. In order to solve the problem that approximate methods make unsatisfactory performances in the case of small sample size, we establish the exact conditional distributions of estimators for parameters by conditional moment generating function(CMGF). Furthermore, confidence intervals(CIs) are constructed by exact distributions, approximate distributions as well as bootstrap method respectively, and their performances are evaluated by Monte Carlo simulations. And finally, a real data set is analyzed to illustrate all the methods developed here.     

The exact bootstrap method shown on the example of the mean and variance estimation

The bootstrap method is based on resampling of the original randomsample drawn from a population with an unknown distribution. In the article it was shown that because of the progress in computer technology resampling is actually unnecessary if the sample size is not too large. It is possible to automatically generate all possible resamples and calculate all realizations of the required statistic. The obtained distribution can be used in point or interval estimation of population parameters or in testing hypotheses. We should stress that in the exact bootstrap method the entire space of resamples is used and therefore there is no additional bias which results from resampling. The method was used to estimate mean and variance. The comparison of the obtained distributions with the limit distributions confirmed the accuracy of the exact bootstrap method. In order to compare the exact bootstrap method with the basic method (with random sampling) probability that 1,000 resamples would allow for estimating a parameter with a given accuracy was calculated. There is little chance of obtaining the desired accuracy, which is an argument supporting the use of the exact method. Random sampling may be interpreted as discretization of a continuous variable.     

Order statistics and region-based evolutionary computation

Trust region algorithms are well known in the field of local continuous optimization. They proceed by maintaining a confidence region in which a simple, most often quadratic, model is substituted to the criterion to be minimized. The minimum of the model in the trust region becomes the next starting point of the algorithm and, depending on the amount of progress made during this step, the confidence region is expanded, contracted or kept unchanged. In the field of global optimization, interval programming may be thought as a kind of confidence region approach, with a binary confidence level: the region is guaranteed to contain the optimum or guaranteed to not contain it. A probabilistic version, known as branch and probability bound, is based on an approximate probability that a region of the search space contains the optimum, and has a confidence level in the interval [0,1]. The method introduced in this paper is an application of the trust region approach within the framework of evolutionary algorithms. Regions of the search space are endowed with a prospectiveness criterion obtained from random sampling possibly coupled with a local continuous algorithm. The regions are considered as individuals for an evolutionary algorithm with mutation and crossover operators based on a transformation group. The performance of the algorithm on some standard benchmark functions is presented.     

Exact Stratified Linear Rank Tests for Ordered Categorical and Binary Data

Abstract

We present an efficient algorithm for generating exact permutational distributions for linear rank statistics defined on stratified 2 × c contingency tables. The algorithm can compute exact p values and confidence intervals for a rich class of nonparametric problems. These include exact p values for stratified two-population Wilcoxon, Logrank, and Van der Waerden tests, exact p values for stratified tests of trend across several binomial populations, exact p values for stratified permutation tests with arbitrary scores, and exact confidence intervals for odds ratios embedded in stratified 2 × c tables. The algorithm uses network-based recursions to generate stratum-specific distributions and then combines them into an overall permutation distribution by convolution. Where only the tail area of a permutation distribution is desired, additional efficiency gains are achieved by backward induction and branch-and-bound processing of the network. The algorithm is especially efficient for highly imbalanced categorical data, a situation where the asymptotic theory is unreliable. The backward induction component of the algorithm can also be used to evaluate the conditional maximum likelihood, and its higher order derivatives, for the logistic regression model with grouped data. We illustrate the techniques with an analysis of two data sets: The leukemia data on survivors of the Hiroshima atomic bomb and data from an animal toxicology experiment provided by the U.S. Food and Drug Administration.     

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The log-normal distribution is a common choice for modeling positively skewed data arising from many practical applications.This article introduces a new method of constructing confidence interval for a common mean shared by several log-normal populations through confidence distributions, which combines all information from independent sources. We develop a non-trivial weighting approach by taking account of the sample variances of related quantities to enhance efficiency. Combined confidence distributions are used to construct confidence intervals for the common mean and a simplified version of one existing method is also proposed. We conduct simulation studies to evaluate the performance of the proposed methods in comparison with existing methods. Our simulation results show that the weighting approach yields shorter interval length than the non-weighting approach. The newly proposed confidence intervals perform very well in terms of empirical coverage probability and average interval length. Finally, applications of the proposed methodology is illustrated through three real data examples.     

基于混合Ⅱ型删失数据的Weibull模型精确推断和可接受抽样计划(英文)

本文考虑基于混合Ⅱ型删失数据的Weibull模型精确推断和可接受抽样计划.得到威布尔分布未知参数最大似然估计的精确分布以及基于精确分布的置信区间.由于精确分布函数较为复杂,给出未知参数的另外几种置信区间,基于近似方法的置信区间.为了评价本文的方法,给出一些数值模拟的结果.且讨论了可靠性中的可接受抽样计划问题.利用参数最大似然估计的精确分布,给出一个可接受抽样计划的执行程序和数值模拟结果.     

并联压力—强度模型可靠性研究

In this paper,the reliability of a parallel stress-strength model of exponentiated Pareto distribution is discussed.Different point estimations and interval estimations are proposed.The point estimators obtained are maximum likelihood and Bayesian estimators.The interval estimations obtained are approximate,exact,bootstrap-p and bootstrap-t confidence intervals and Bayesian credible interval.Different methods and the corresponding confidence intervals are demonstrated using some simulation studies.     

Exact confidence coefficients of simultaneous confidence intervals for multinomial proportions

Simultaneous confidence intervals for multinomial proportions are useful in many areas of science. Since 1964, approximate simultaneous 1-α confidence intervals have been proposed for multinomial proportions. Although at each point in the parameter space, these confidence sets have asymptotic 1-α coverage probability, the exact confidence coefficients of these simultaneous confidence intervals for a fixed sample size are unknown before.In this paper, we propose a procedure for calculating exact confidence coefficients for simultaneous confidence intervals of multinomial proportions for any fixed sample size. With this methodology, exact confidence coefficients can be clearly derived, and the point at which the infimum of the coverage probability occurs can be clearly identified.     

Simple Accurate Approximation of Likelihood Profiles

Likelihood profiles for arbitrary functions of the model parameters are useful for constructing likelihood ratio confidence intervals, diagnosing linear approximation intervals, suggesting linearizing transforms, and many other purposes. This article investigates a simple integration method for producing accurate approximations to likelihood profiles that avoid problems associated with producing exact profiles. A basic theorem guarantees that the approximation can achieve any desired precision. In many cases the method requires no more than adding a few lines of code to that required to produce the maximum likelihood estimate of the parameter vector. Standard methods for computing likelihood profiles are based on solving a sequence of constrained maximum likelihood problems. Each problem generates a point on the profile. An earlier article proposed an integration method that generates the entire profile directly by solving a differential equation. That method, however, requires the Hessian of the log-likelihood which can be difficult to produce. The method considered here requires only the gradient. It may be particularly useful in conjunction with the EM algorithm which typically does not produce any type of confidence interval.     

双边定时截尾样本下广义逆指数分布形状参数的估计

先给出了广义逆指数分布在双边定时截尾样本下形状参数的最大似然估计,并不能得到估计的显式表达式,但证明了参数在(0,+∞)上最大似然估计是唯一存在的.其次提出用EM算法求出形状参数的估计且该估计具有良好的收敛性,还给出了形状参数的EM估计的渐近方差和近似置信区间;最后通过数值模拟,对形状参数的最大似然估计和EM估计的效果进行了比较,说明了用EM算法求形状参数的估计是可行的,并且模拟效果相对比较好.     

Limit distributions of multivariate kurtosis and moments under Watson rotational symmetric distributions

The limit distribution of Mardia's measure of sample multivariate kurtosis is derived for a wide class of multivariate distributions which includes both the family of elliptical and Watson rotational symmetric distributions. Explicit expressions are given for the higher-order moments of Watson rotational symmetric distributions. The problem of constructing approximate confidence intervals for the kurtosis parameter is also discussed.     

逆抽样下流行病发病率的逼近与渐近置信区间

流行病研究的重要任务之一就是较为精确地估计出疾病的流行程度.疾病的流行性通常用发病率来表征.由于置信区间估计是一种体现对发病率估计好坏的途径,所以它是估计边限的重要提示物.作者在逆抽样条件下探究了7种流行病发病率的逼近与渐近的置信区间估计.通过蒙特卡罗方法,广泛地比较了这些方法的表现性能.为了方便今后进一步应用此结果,制做了许多相应的表格.这些表格清楚地表明为了构造出具有指定期望值的置信区所需要的最小病例数.模拟的结果表明:就流行病发病率的区间估计的覆盖率与区间大小的稳定性而言,逼近与渐近方法要优越于精确方法.更多的研究表明:鞍点逼近型置信区间就控制覆盖率和平均区间长度而言表现得最好,因此,在实际应用中如果能得到,建议尽量使用它.     

LOgistic响应分布中刻度参数的中、小样本推断

在火炸药产品的敏感性推断中，对响应分布的标准差给出较精确的推断，是基础性工作之一．为此，本文基于Logistic响应分布，在二元响应数据下，应用鞍点近似方法构造了刻度参数的近似置信区间，并进行了模拟研究．最后，本文将该方法应用于QD-8电雷管．模拟结果和实例分析表明，在中、小样本情形，本文方法对刻度参数的推断结果较为精确，显著改进了现行的基于渐近正态性的方法．     

基于增广Lagrange函数的RQP方法

Recursive quadratic programming is a family of techniques developd by Bartholomew-Biggs and other authors for solving nonlinear programming problems.This paperdescribes a new method for constrained optimization which obtains its search di-rections from a quadratic programming subproblem based on the well-known aug-mented Lagrangian function.It avoids the penalty parameter to tend to infinity.We employ the Fletcher‘s exact penalty function as a merit function and the use of an approximate directional derivative of the function that avoids the need toevaluate the second order derivatives of the problem functions.We prove that thealgorithm possesses global and superlinear convergence properties.At the sametime, numerical results are reported.     

广义Lorenz 曲线的非参数统计推断

本文讨论了广义Lorenz 曲线的经验似然统计推断. 在简单随机抽样、分层随机抽样和整群随机抽样下, 本文分别定义了广义Lorenz 坐标的pro le 经验似然比统计量, 得出这些经验似然比的极限分布为带系数的自由度为1 的χ² 分布. 对于整个Lorenz 曲线, 基于经验似然方法类似地得出相应的极限过程. 根据所得的经验似然理论, 本文给出了bootstrap 经验似然置信区间构造方法, 并通过数据模拟, 对新给出的广义Lorenz 坐标的bootstrap 经验似然置信区间与渐近正态置信区间以及bootstrap 置信区间等进行了对比研究. 对整个Lorenz 曲线, 基于经验似然方法对其置信域也进行了模拟研究. 最后我们将所推荐的置信区间应用到实例中.     

基于截尾数据指数Pareto分布应力-强度模型的可靠性

在II型双截尾删失计划下，讨论了当系统被独立的随机施加指数Pareto （EP）压力时的系统可靠性问题.作者给出了系统可靠性参数的不同点估计和区间估计，其中点估计包括一致最小方差无偏估计（UMVUE）和最大似然估计（MLE）；区间估计包括精确置信区间，近似置信区间和bootstrap的区间估计.为了评价不同估计方法效果，作者提供数值模拟结果；最后提供了一个真实数据的分析结果来演示本文提出的方法.     

Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems

We develop a numerical algorithm for solving singularly perturbed one-dimensional parabolic convection-diffusion problems. The method comprises a standard finite difference to discretize in temporal direction and Sinc-Galerkin method in spatial direction. The convergence analysis and stability of proposed method are discussed in details, it is justifying that the approximate solution converges to the exact solution at an exponential rate. we know that the conventional methods for these problems suffer due to decreasing of perturbation parameter, but the Sinc method handel such difficulty as singularity. This scheme applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence.     

混合线性模型中方差比的置信区间

本文利用信仰推断给出了求混合线性模型中方差比的置信区间的方法，结果表明所得区间估计具有不变性。另外，本文给出了两种求区间估计的近似方法。