对数Weibull分布的数字特征

文章提出了一个新的寿命分布——称为对数Weibull分布.针对该分布研究了它的密度函数、失效率函数的图像特征,同时讨论了对数Weibull分布k阶矩的存在性.     

Marshall–Olkin bivariate Weibull distributions and processes

In this paper we introduce a new probability model known as type 2 Marshall–Olkin bivariate Weibull distribution as an extension of type 1 Marshall–Olkin bivariate Weibull distribution of Marshall–Olkin (J Am Stat Assoc 62:30–44, 1967). Various properties of the new distribution are considered. Bivariate minification processes with the two types of Weibull distributions as marginals are constructed and their properties are considered. It is shown that the processes are strictly stationary. The unknown parameters of the type 1 process are estimated and their properties are discussed. Some numerical results of the estimates are also given.     

Weibull分布有关参数的近似无偏估计

一、引言Weibull分布是最常用的寿命分布之一,其分布函数为:F(t)=1-exp(-(t/η)m),t≥0其中m是形状参数,η是尺度参数,它们均是大于零的未知参数。已有很多文献讨论了Weibull分布参数的估计方法,本文在导出了基于定     

On the Marshall–Olkin extended Weibull distribution

We study some mathematical properties of the Marshall–Olkin extended Weibull distribution introduced by Marshall and Olkin (Biometrika 84:641–652, 1997). We provide explicit expressions for the moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves, reliability and Rényi entropy. We determine the moments of the order statistics. We also discuss the estimation of the model parameters by maximum likelihood and obtain the observed information matrix. We provide an application to real data which illustrates the usefulness of the model.     

The Weibull–Dagum distribution: Properties and applications

ABSTRACT

In this article, we define a new lifetime model called the Weibull–Dagum distribution. The proposed model is based on the Weibull–G class. It can also be defined by a simple transformation of the Weibull random variable. Its density function is very flexible and can be symmetrical, left-skewed, right-skewed, and reversed-J shaped. It has constant, increasing, decreasing, upside-down bathtub, bathtub, and reversed-J shaped hazard rate. Various structural properties are derived including explicit expressions for the quantile function, ordinary and incomplete moments, and probability weighted moments. We also provide explicit expressions for the Rényi and q-entropies. We derive the density function of the order statistics as a mixture of Dagum densities. We use maximum likelihood to estimate the model parameters and illustrate the potentiality of the new model by means of a simulation study and two applications to real data. In fact, the proposed model outperforms the beta-Dagum, McDonald–Dagum, and Dagum models in these applications.     

Marshall–Olkin gamma–Weibull distribution with applications

Abstract

A Marshall–Olkin variant of the Provost type gamma–Weibull probability distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model two real data sets. The new distribution provides a better fit than related distributions as measured by the Anderson–Darling and Cramér–von Mises statistics. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology, engineering, etc.     

Heterogeneous data modeling with two-component Weibull–Poisson distribution

The mixture distribution models are more useful than pure distributions in modeling of heterogeneous data sets. The aim of this paper is to propose mixture of Weibull–Poisson (WP) distributions to model heterogeneous data sets for the first time. So, a powerful alternative mixture distribution is created for modeling of the heterogeneous data sets. In the study, many features of the proposed mixture of WP distributions are examined. Also, the expectation maximization (EM) algorithm is used to determine the maximum-likelihood estimates of the parameters, and the simulation study is conducted for evaluating the performance of the proposed EM scheme. Applications for two real heterogeneous data sets are given to show the flexibility and potentiality of the new mixture distribution.     

Confidence limits for stress–strength reliability involving Weibull models

The problem of interval estimation of the stress–strength reliability involving two independent Weibull distributions is considered. An interval estimation procedure based on the generalized variable (GV) approach is given when the shape parameters are unknown and arbitrary. The coverage probabilities of the GV approach are evaluated by Monte Carlo simulation. Simulation studies show that the proposed generalized variable approach is very satisfactory even for small samples. For the case of equal shape parameter, it is shown that the generalized confidence limits are exact. Some available asymptotic methods for the case of equal shape parameter are described and their coverage probabilities are evaluated using Monte Carlo simulation. Simulation studies indicate that no asymptotic approach based on the likelihood method is satisfactory even for large samples. Applicability of the GV approach for censored samples is also discussed. The results are illustrated using an example.     

Analysis of survival data by a Weibull–Bessel distribution

In survival analysis and reliability studies, problems with random sample size arise quite frequently. More specifically, in cancer studies, the number of clonogens is unknown and the time to relapse of the cancer is defined by the minimum of the incubation times of the various clonogenic cells. In this article, we have proposed a new model where the distribution of the incubation time is taken as Weibull and the distribution of the random sample size as Bessel, giving rise to a Weibull–Bessel distribution. The maximum likelihood estimation of the model parameters is studied and a score test is developed to compare it with its special submodel, namely, exponential–Bessel distribution. To illustrate the model, two real datasets are examined, and it is shown that the proposed model, presented here, fits better than several other existing models in the literature. Extensive simulation studies are also carried out to examine the performance of the estimates.     

基于定时截尾Weibull分布EM算法的寿命检验

在可靠性统计的定时截尾寿命试验中,最后一个失效时间与定时截尾时刻之间的信息常被忽略,文章利用EM算法来处理这一问题,给出无失效数据下EM算法的Weibull分布的分布参数估计值,实际数据计算结果表明,在产品平均寿命的估计中取得了较高的可靠度。     

A new compounding life distribution: the Weibull–Poisson distribution

In this paper, a new compounding distribution, named the Weibull–Poisson distribution is introduced. The shape of failure rate function of the new compounding distribution is flexible, it can be decreasing, increasing, upside-down bathtub-shaped or unimodal. A comprehensive mathematical treatment of the proposed distribution and expressions of its density, cumulative distribution function, survival function, failure rate function, the kth raw moment and quantiles are provided. Maximum likelihood method using EM algorithm is developed for parameter estimation. Asymptotic properties of the maximum likelihood estimates are discussed, and intensive simulation studies are conducted for evaluating the performance of parameter estimation. The use of the proposed distribution is illustrated with examples.     

基于样本分位数的Weibull分布均值和标准差估计

样本均值和标准差是数据分析中常用的指标,但在部分研究中只记录了样本分位数,并未记录均值和标准差。传统的最小二乘法在估计样本均值和标准差时,通常将样本分位数同等对待,但在实际研究中,数据具有峰度和偏度等特征,样本分位数的重要程度不同。因此,文章在估计模型中引入权重,针对可靠性分析中常见的Weibull分布,提出两种方法估计均值和标准差：利用分布函数的加权最小二乘法（FWLS）进行估计;将Weibull分布转换为指数分布,利用数据的加权最小二乘法（WLS）进行估计。通过仿真和实例结果的比较,表明两种方法的估计效果显著。     

逆Weibull分布参数的点估计和联合置信域估计

文章基于完全样本,针对两参数逆Weibull分布参数的点估计和置信区间估计问题,利用二分法导出了参数的最大似然估计,但最大似然估计法不能给出参数的精确置信区间估计,通过构造一类枢轴量得到了形状参数的精确置信区间估计,同时给出了形状参数和尺度参数的联合置信域估计。     

Weibull分布下基于MCMC的贝叶斯恒加试验数据评估

本文基于贝叶斯生存分析理论,在参数的有信息先验假设条件下,通过运用基于Gibbs抽样的马尔可夫链蒙特卡罗(MCMC)方法动态模拟出相关参数后验分布的马尔可夫链,给出恒加试验模型中各参数的贝叶斯估计;利用BUGS软件包对文献[6]中的实例进行建模分析,并将两种假设条件下MCMC具有显著差异的计算结果与传统BLUE结果进行比较,发现BLUE的计算结果近似等于将产品截尾数据当作失效数据时MCMC的处理结果;进而再次揭示出传统BLUE方法的不足,并证明了该模型在可靠性应用中的直观性与有效性。     

Estimation under inverse Weibull distribution based on Balakrishnan’s unified hybrid censored scheme

In this article, point and interval estimations of the parameters α and β of the inverse Weibull distribution (IWD) have been studied based on Balakrishnan’s unified hybrid censoring scheme (UHCS), see Balakrishnan et al. In point estimation, the maximum likelihood (ML) and Bayes (B) methods have been used. The Bayes estimates have been computed based on squared error loss (SEL) function and Linex loss function and using Markov Chain Monte Carlo (MCMC) algorithm. In interval estimation, a (1 ? τ) × 100% approximate, bootstrap-p, credible and highest posterior density (HPD) confidence intervals (CI′s) for the parameters α and β have been introduced. Based on Monte Carlo simulation, Bayes estimates have been compared with their corresponding maximum likelihood estimates by computing the mean squared errors (MSE′s) of all estimators. Finally, point and interval estimations of all parameters have been studied based on a real data set as an illustrative example.     

The negative binomial–beta Weibull regression model to predict the cure of prostate cancer

In this article, for the first time, we propose the negative binomial–beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.     

A new compound class of log-logistic Weibull–Poisson distribution: model,properties and applications

A new class of distributions called the log-logistic Weibull–Poisson distribution is introduced and its properties are explored. This new distribution represents a more flexible model for lifetime data. Some statistical properties of the proposed distribution including the expansion of the density function, quantile function, hazard and reverse hazard functions, moments, conditional moments, moment generating function, skewness and kurtosis are presented. Mean deviations, Bonferroni and Lorenz curves, Rényi entropy and distribution of the order statistics are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study is conducted to examine the bias, mean square error of the maximum likelihood estimators and width of the confidence interval for each parameter and finally applications of the model to real data sets are presented to illustrate the usefulness of the proposed distribution.