A Simple Derivation of Moments of the Exponentiated Weibull Distribution

The Exponentiated Weibull family is an extension of the Weibull family obtained by adding an additional shape parameter. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates which are quite common in reliability and biological studies. As with any other distribution, many of its interesting characteristics and features can be studied through moments. Presently, moments of this distribution are available only under certain restrictions. In this paper, a general derivation of moments without any restriction whatsoever is proposed. A compact expression for moments is presented.     

On the modeling of size distributions when technologies are complex

The study considers a stochastic R&D process where the invented production technologies consist of a large number $n$ of complementary components. The degree of complementarity is captured by the elasticity of substitution of the CES aggregator function. Drawing from the Central Limit Theorem and the Extreme Value Theory we find, under very general assumptions, that the cross-sectional distributions of technological productivity are well-approximated either by the lognormal, Weibull, or a novel “CES/Normal” distribution, depending on the underlying elasticity of substitution between technology components. We find the tail of the “CES/Normal” distribution to be fatter than the Weibull tail but qualitatively thinner than the Pareto (power law) one. We also numerically assess the rate of convergence of the true technological productivity distribution to the theoretical limit with $n$ as fast in the body but slow in the tail.     

Slow convergence of the number of near-maxima for Burr XII distributions

A near-maximum is an observation which falls within a distance a of the maximum observation in an independent and identically distributed sample of size n. Subject to some conditions on the tail thickness of the population distribution, the number K _n (a) of near-maxima is known to converge in probability to one or infinity, or in distribution to a shifted geometric law. In this paper we show that for all Burr XII distributions K _n (a) converges almost surely to unity, but this convergence property may not become clear under certain cases even for very large n. We explore the reason of such slow convergence by studying a distributional continuity between Burr XII and Weibull distributions. We have also given a theoretical explanation of slow convergence of K _n (a) for the Burr XII distributions by showing that the rate of convergence in terms of P{K _n (a) > 1} tending to zero changes very little with the sample size n. Illustrations of the limiting behaviour K _n (a) for the Burr XII and the Weibull distributions are given by simulations and real data. The study also raises an important issue that although the Burr XII provides overall better fit to a given data set than the Weibull distribution, cautions should be taken for the extrapolation of the upper tail behaviour in the case of slow convergence.      

Bayesian life test planning for the Weibull distribution with given shape parameter

This paper describes Bayesian methods for life test planning with Type II censored data from a Weibull distribution, when the Weibull shape parameter is given. We use conjugate prior distributions and criteria based on estimating a quantile of interest of the lifetime distribution. One criterion is based on a precision factor for a credibility interval for a distribution quantile and the other is based on the length of the credibility interval. We provide simple closed form expressions for the relationship between the needed number of failures and the precision criteria. Examples are used to illustrate the results.Received: October 2002 / Revised: March 2004     

Estimating the Weight Undersize Distribution for the Wicksell problem

In Wicksell's corpuscle problem one is interested in estimating the distribution of sphere diameters from the diameters of circle profiles obtained by a random section of the body containing the sphericle particles. The problem is known to be an ill posed inverse problem. Several regularization techniques have been applied to find solutions. We will review some of these in this article. In practical situations one often is more interested in the distribution of weight rather than the distribution of diameters. In estimating the weight undersite distribution similar problems are encountered. We will consider an estimator that is obtained by smoothing the distribution function of the circle diameters locally. It will be shown how the bandwidth must be chosen to obtain consistency and mean square error optirnality. Also asymptotic normality will be shown.     

Investigating design for survival models

The aim of this paper is to derive methodology for designing ‘time to event’ type experiments. In comparison to estimation, design aspects of ‘time to event’ experiments have received relatively little attention. We show that gains in efficiency of estimators of parameters and use of experimental material can be made using optimal design theory. The types of models considered include classical failure data and accelerated testing situations, and frailty models, each involving covariates which influence the outcome. The objective is to construct an optimal design based of the values of the covariates and associated model or indeed a candidate set of models. We consider D-optimality and create compound optimality criteria to derive optimal designs for multi-objective situations which, for example, focus on the number of failures as well as the estimation of parameters. The approach is motivated and demonstrated using common failure/survival models, for example, the Weibull distribution, product assessment and frailty models.     

A new family of bivariate max-infinitely divisible distributions

In this article we discuss the asymptotic behaviour of the componentwise maxima for a specific bivariate triangular array. Its components are given in terms of linear transformations of bivariate generalised symmetrised Dirichlet random vectors introduced in Fang and Fang (Statistical inference in elliptically contoured and related distributions. Allerton Press, New York, 1990). We show that the componentwise maxima of such triangular arrays is attracted by a bivariate max-infinitely divisible distribution function, provided that the associated random radius is in the Weibull max-domain of attraction.     

Efficient parameter estimation for independent and INAR(1) negative binomial samples

We consider moment based estimation methods for estimating parameters of the negative binomial distribution that are almost as efficient as maximum likelihood estimation and far superior to the celebrated zero term method and the standard method of moments estimator. Maximum likelihood estimators are difficult to compute for dependent samples such as samples generated from the negative binomial first-order autoregressive integer-valued processes. The power method of estimation is suggested as an alternative to maximum likelihood estimation for such samples and a comparison is made of the asymptotic normalized variance between the power method, method of moments and zero term method estimators.     

Weibull, RRSB or extreme-value theorists?

The Weibull distribution was discovered by Rosin, Rammler, Sperling and Bennett between 1932 and 1936 in the context of particle measurement. Weibull found the same distribution a little later while investigating the strength of materials. More than 10 years after, in 1951, he finally showed that this distribution has the potential for wide applications in statistics. However, does this justify that only his name is used to denote this important probability distribution? A neutral technical name like “powered exponential distribution” might be more suitable. This paper discusses the papers by Rosin, Rammler, Sperling and Bennett as well as Weibull’s work.     

Robust explicit estimators of Weibull parameters

The Weibull distribution plays a central role in modeling duration data. Its maximum likelihood estimator is very sensitive to outliers. We propose three robust and explicit Weibull parameter estimators: the quantile least squares, the repeated median and the median/Q _n estimator. We derive their breakdown point, influence function, asymptotic variance and study their finite sample properties in a Monte Carlo study. The methods are illustrated on real lifetime data affected by a recording error.     

基于智能调度的散点货源动态集散点设置研究

动态集散点是系统应对客户随机需求,为了降低企业成本以及方便客户及时送件而设置的临时、随机的物流集散点。与城市的客源相比,城市货源的分布更加分散,需求随机性更强,因而一般的固定集散点往往不能满足需求。针对此问题,文章对城市散点货源的动态集散点设置进行研究。依托智能车辆自动化取件为背景,运用层次聚类分析思想,并通过Matlab进行仿真,对集散点的设置进行分析及优化。通过算例分析文章得出合理地设置动态集散点能够有效地提高企业取件效率,且能够满足客户的特殊需求。文章研究可以为物流企业在智能取件模式下的货物集散点设置提供方法参考。     

Efficiency Measurement with the Weibull Stochastic Frontier*

In this paper we consider the Weibull distribution as a model for technical efficiency. The distribution has a shape and scale parameter like the gamma distribution and can be a reasonable competitor in practice. The techniques are illustrated using artificial data as well as a panel of Spanish dairy farms.     

Estimation of the parameters of the Weibull distribution for censored samples

The Weibull distribution is an often used model in survival analysis of technical products. In this paper confidence estimations for the parameters of the Weibull distribution are developed for type I censored samples without and with replacements.     

M5 competition uncertainty: Overdispersion,distributional forecasting,GAMLSS, and beyond

The M5 competition uncertainty track aims for probabilistic forecasting of sales of thousands of Walmart retail goods. We show that the M5 competition data face strong overdispersion and sporadic demand, especially zero demand. We discuss modeling issues concerning adequate probabilistic forecasting of such count data processes. Unfortunately, the majority of popular prediction methods used in the M5 competition (e.g. lightgbm and xgboost GBMs) fail to address the data characteristics, due to the considered objective functions. Distributional forecasting provides a suitable modeling approach to overcome those problems. The GAMLSS framework allows for flexible probabilistic forecasting using low-dimensional distributions. We illustrate how the GAMLSS approach can be applied to M5 competition data by modeling the location and scale parameters of various distributions, e.g. the negative binomial distribution. Finally, we discuss software packages for distributional modeling and their drawbacks, like the R package gamlss with its package extensions, and (deep) distributional forecasting libraries such as TensorFlow Probability.     

逐次定数截尾下Weibull分布参数的逆矩估计

基于逐次定数截尾样本下,讨论了两参数Weibull分布的参数估计,得到了两参数的逆矩估计.并利用模拟方法与极大似然估计作比较,模拟结果表明逆矩估计优于极大似然估计。     

On the two-sided power distribution

This paper concerns a two-parameter two-sided power distribution introduced by van Dorp and Kotz on the interval [0,1]. We introduce a reformulated two-sided power distribution (with the same number of parameters) and provide evidence to prove that it is more flexible than the one suggested by van Dorp and Kotz. We derive various properties of the new distribution as well as provide several hitherto unknown properties of the distribution due to van Dorp and Kotz. We also discuss estimation by the method of moments and the method of maximum likelihood.Received September 2003     

基于整体性的制造企业供应链可靠性分析

制造企业要获得高的市场反应能力和效益,必须要有高的供应链系统可靠性做保障。针对制造企业供应链系统,结合GO法,建立出基于威布尔分布的制造企业供应链系统GO模型。运用定量分析法,研究系统的整体可靠度函数与失效率函数的表达式和图形,通过实例对表达式和图形进一步证明。结果表明:通过将GO法运用于制造企业的供应链系统可以清晰、直观地表达出供应链系统的可靠度及失效率的变化情况,并从表达式和图形中可以追溯出影响制造企业供应链系统整体可靠性的关键因素,为制造业供应链系统制定出相关可靠性管理措施提供科学的依据。     

Bartlett corrections for the weibull distribution

We describe, for the calculation of Bartlett adjustments, a method which may be of use when a transformation to orthogonalized parameters can be found. This method is used to calculate the Bartlett adjustment for the Weibull distribution if two or only one of the parameters are of interest.     

中国钢铁分销业发展研究综述

通过总结国内学者对我国钢铁分销业的发展研究成果,阐述了钢铁分销业的特点与优势、钢铁分销业的研究现状、厂商与分销商的关系、对国外钢铁分销业的研究借鉴以及未来的发展思路,在此基础上对未来研究方向进行了展望。     

用二叉树解决配送线路问题

物流配送是物流中的核心环节,配送成本在整个物流过程费用中占有很大的比重,因此为了减少配送成本有必要对其配送路线进行合理优化。运用二叉树遍历的知识并结合节约算法的思想,将货物需求点作为叶子结点并适当增加一些需求量为零的叶子结点构造一种有特殊意义的二又树,提出了一种运用这种特殊二叉树在满足车辆额定载货量的前提下寻求最优配送路线的方法．并通过实例证明了其正确性。