Approximations to the Distribution of the Sample Coefficient Alpha Under Nonnormality

Approximate distributions of the sample coefficient alpha under nonnormality as well as normality are derived by using the single- and two-term Edgeworth expansions up to the term of order 1/n. The case of the standardized coefficient alpha including the weights for the components of a test is also considered. From the numerical illustration with simulation using the normal and typical nonnormal distributions with different types/degrees of nonnormality, it is shown that the variances of the sample coefficient alpha under nonnormality can be grossly different from those under normality. The corresponding biases and skewnesses are shown to be negative under various conditions. The method of developing confidence intervals of the population coefficient alpha using the Cornish-Fisher expansion with sample cumulants is presented.

     

删失下的指数分布的贝叶斯估计

本文分别在Ⅱ型删失和随机删失下,表明了共轭先验下的指数分布的刻度参数的贝叶斯估计为具有如下形式的收缩估计(?)_(BE)=a■ bEθ,此处■为依赖样本θ的一个无偏估计且Eθ表示先验分布的期望。当采用平方损失函数时,a b=1;如果用加权平方损失函数,则a b＜1。     

广义立方双色散方程解的渐近性质

本文研究广义立方双色散方程的初边值问题,它包括了几类双色散方程以及Bq,IBq,IMBq方程作为特殊情形,当方程非线性项满足某些条件时,利用等价变换和改进的积分估计法证明了问题整体广义解按时间的指数形式衰减为零.最后,将本文定理结论应用到实际模型中,得到了问题的解具有衰减行为时方程参数应满足的条件.     

多元线性正态SRM参数估计的渐近协方差阵及其估计

针对多元线性正态结构关系模型（SRM）,本文给出了真值向量协方差阵的广义逆构造,并在三种常见误差协方差阵假定下应用标准5法分别给出了模型结构参数约束最小二乘估计的渐近协方差阵及其相合估计。     

Minimum Distance Estimation for the Generalized Pareto Distribution

The generalized Pareto distribution (GPD) is widely used for extreme values over a threshold. Most existing methods for parameter estimation either perform unsatisfactorily when the shape parameter k is larger than 0.5, or they suffer from heavy computation as the sample size increases. In view of the fact that k > 0.5 is occasionally seen in numerous applications, including two illustrative examples used in this study, we remedy the deficiencies of existing methods by proposing two new estimators for the GPD parameters. The new estimators are inspired by the minimum distance estimation and the M-estimation in the linear regression. Through comprehensive simulation, the estimators are shown to perform well for all values of k under small and moderate sample sizes. They are comparable to the existing methods for k < 0.5 while perform much better for k > 0.5.     

Parameter and Quantile Estimation for the Generalized Pareto Distribution

The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.     

Weibull分布兴趣参数的广义置信区间

Weibull分布可以用来描述疲劳失效、真空管失效和轴承失效等,是重要的寿命分布。本文研究了Weibull分布尺度参数、形状参数、分位数及可靠度函数的广义置信区间问题。利用广义枢轴量给出四个兴趣参数的广义置信区间。证明了由广义枢轴量确定的四个兴趣参数的广义置信区间具有频率意义下的实际置信水平。     

Parameter Estimation for a Generalized Gamma Distribution

It is fairly commonplace in reliability analyses to encounter data which is incompatible with the exponential, Weibull, and other familiar probability models. Such data motivates research to enlarge the group of probability distributions which are useful to the reliability analyst.

In this paper, we examine a three-parameter generalization of the gamma distribution and derive parameter estimation techniques for that distribution. Those techniques, in the general case, depend upon method of moments considerations which lead to simultaneous equations for which closed form solutions are not available. Graphic solution is proposed and aids to the computations are provided. Major concepts in the paper are summarized by means of a numerical example.

Details are given for the special case in which only the scale parameter is unknown. Three unbiased estimators for that parameter are derived along with their variance formulas. Minimum variance considerations are discussed by application of the Cramér-Rao Theorem.     

Iteratively Reweighted Partial Least Squares Estimation for Generalized Linear Regression

I extend the concept of partial least squares (PLS) into the framework of generalized linear models. A spectroscopy example in a logistic regression framework illustrates the developments. These models form a sequence of rank 1 approximations useful for predicting the response variable when the explanatory information is severely ill-conditioned. Iteratively reweighted PLS algorithms are presented with various theoretical properties. Connections to principal-component and maximum likelihood estimation are made, as well as suggestions for rules to choose the proper rank of the final model.     

Problems of Nonnormality and Multicollinearity for Forecasting Methods Based on Least Squares

Many popular forecasting and time-series analysis methods assume that the variable to be forecast can be expressed as a linear function of a set of predictors. The predictors may include variables related in either a correlative or causal fashion to the response variable, lagged values of this variable, or known mathematical functions of time. The method of least squares is used almost exclusively to estimate the parameters in these models. This paper discusses two hazards in the indiscriminant use of least squares; nonnormality of the observations on the variable of interest and multicollinearity among the predictors. Robust estimation methods are suggested as alternatives to least squares for nonnormal data, and a robust version of exponential smoothing is developed. A small Monte Carlo study indicates that the robust procedure can be superior to ordinary exponential smoothing in many situations. The sources and effects of multicollinearity are discussed, and several diagnostic statistics are presented. Methods for dealing with multicollinearity are reviewed, including collecting additional data, variable selection, and biased estimation. An example of ridge regression is included.     

Reflection of Magneto-Thermoviscoelastic Waves Under Generalized Thermoviscoelasticity

Considering each of the thermal fields, viscoelastic and electromagnetic fields contribute to the total deformation of a body and interact with each other. Reflection of magneto-thermoelastic waves under generalized thermoviscoelastic theories is employed to study the reflection of plane harmonic waves from a semi-infinite elastic solid in a vacuum. The expressions for the reflection coefficients, which are the ratios of the amplitudes of the reflected waves to the amplitudes of the incident waves, are obtained, and the reflection coefficient ratios variation with the angle of incidence under different conditions are shown graphically when acrylic plastic materials are considered.     

Model Misspecification of Generalized Gamma Distribution for Accelerated Lifetime-Censored Data

Abstract

The performance of reliability inference strongly depends on the modeling of the product’s lifetime distribution. Many products have complex lifetime distributions whose optimal settings are not easily found. Practitioners prefer to use simpler lifetime distribution to facilitate the data modeling process while knowing the true distribution. Therefore, the effects of model mis-specification on the product’s lifetime prediction is an interesting research area. This article presents some results on the behavior of the relative bias (RB) and relative variability (RV) of pth quantile of the accelerated lifetime (ALT) experiment when the generalized Gamma (GG₃) distribution is incorrectly specified as Lognormal or Weibull distribution. Both complete and censored ALT models are analyzed. At first, the analytical expressions for the expected log-likelihood function of the misspecified model with respect to the true model is derived. Consequently, the best parameter for the incorrect model is obtained directly via a numerical optimization to achieve a higher accuracy model than the wrong one for the end-goal task. The results demonstrate that the tail quantiles are significantly overestimated (underestimated) when data are wrongly fitted by Lognormal (Weibull) distribution. Moreover, the variability of the tail quantiles is significantly enlarged when the model is incorrectly specified as Lognormal or Weibull distribution. Precisely, the effect on the tail quantiles is more significant when the sample size and censoring ratio are not large enough. Supplementary materials for this article are available online.     

A Generalized EOQ Model for Items with Weibull Distribution Deterioration

A generalized form of Covert and Philip's EOQ model with Weibull distribution deterioration is developed. Specifically, the 3-parameter Weibull distribution is used to represent the time to deterioration. It is shown that this model can be related to the previously developed simpler models. The solutions to two numerical examples are compared to Covert and Philip's solution.     

删失数据下密度核估计的误差分布的强逼近

在随机右删失数据下构造了概率密度函数的核估计和随机加权估计，得到了核估计的误差分布的正态逼近速度和随机加权逼近速度，改进了孙六权和朱力行(1999)的主要结果。     

T-Stress of a Bi-Material Strip Under Generalized Edge Loads

An expression for the T-stress in a bi-material strip with a semi-finite interfacial crack is derived for general edge loads using a conservation integral method. The expression is explicit except for two non-dimensional constants, which are determined from a finite element analysis and which are tabulated as a function of the Dundurs’ parameters and the thickness ratio of the strip.     

任意随机序列关于广义几何分布的一类强偏差定理

本文引进对数似然比作为任意随机变量序列相对于服从广义几何分布的随机变量序列的偏差的一种度量,给出样本空间的一个子集。在该子集上通过构造非负上鞅和纯分析运算得到了任意随机变量序列关于广义几何分布的一类用不等式表示的强极限定理,即强偏差定理。作为推论得到了已有的随机变量序列关于几何分布的强偏差定理以及服从广义几何分布的任意随机变量序列的一类强大数定律。     

NA样本下固定设计回归估计的渐进正态性

对于固定设计回归模型，本文在NA样本、强平稳及较弱的条件下建立了回归权函数估计的渐近正态性，应用这一结果具体地讨论了Gasser-Muller估计和Priesfley-Chao估计，得到相应的结论。     

Asymptotic Solution of the Problem on the Temperature Field in a Well with Account for the Radial-Velocity Distribution

The problem on the temperature field of a liquid in a well has been asymptotically solved for the case of a constant temperature gradient in it. Expressions for calculating the temperature of a liquid in a well in the zero and first approximations have been obtained. It is shown that the asymptotic solution of the problem considered in the zero approximation makes it possible to obtain an algorithm involving averaging of the temperature of a liquid in a well in the case where the velocity profile of the liquid is constant. The space and time temperature distributions of a liquid in a well have been calculated, and the contribution of various physical processes to them has been analyzed. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 4, pp. 87–96, July–August, 2005.