Asymptotic properties in partial linear models under dependence

Consider the regression modely _i =ζ _i ^T β+m(t _i )+ε _i fori=1,...,n. Here (ζ _i ^T ,t _i ) ^T ∈ℝ ^p ×[0,1] are design points, β is an unknownp×1 vector of parameters,m is an unknown smooth function from [0,1] to ℝ andε _i are the unobserved errors. We will assume that these errors are not independent. Under suitable assumptions, we obtain expansions for the bias and the variance of a Generalized Least Squares (GLS) type regression estimator, and for an estimator of the nonparametric functionm(·). Furthermore, we prove the asymptotic normality of the first estimator. The obtained results are a generalization of those contained in Speckman (1988), who studied a similar model with i.i.d. error variables. Research supported by the Xunta de Galicia (Spain) and the DGES (Spain) under research projects XUGA 10503A98 and PB98-0182-c02-01, respectively.     

Recursive local polynomial regression under dependence conditions

In the case of the random design nonparametric regression, one recursive local polynomial smoother is considered. Expressions for the bias and the variance matrix of the estimators of the regression function and its derivatives are obtained under dependence conditions (strongly mixing processes). The obtained Mean Squared Error is shown to be larger than those of the analogous nonrecursive regression estimators, although retaining the same convergence rate. The properties of strong consistency with convergence rates are established for the proposed estimators. Finally, in order to analyse the influence of both the sample size and the dependence in the behaviour of the proposed recursive estimator, a simulation study is performed. This work has been partially supported by DGES Grant PB98-0182-C02-01 and by the Xunta de Galicia Grant XUGA10501B97     

Bootstrap tests for nonparametric comparison of regression curves with dependent errors

In this paper, the problem of testing the equality of regression curves with dependent data is studied. Several methods based on nonparametric estimators of the regression function are described. In this setting, the distribution of the test statistic is frequently unknown or difficult to compute, so an approximate test based on the asymptotic distribution of the statistic can be considered. Nevertheless, the asymptotic properties of the methods proposed in this work have been obtained under independence of the observations, and just one of these methods was studied in a context of dependence as reported by Vilar-Fernández and González-Manteiga (Statistics 58(2):81–99, 2003). In addition, the distribution of these test statistics converges to the limit distribution with convergence rates usually rather slow, so that the approximations obtained for reasonable sample sizes are not satisfactory. For these reasons, many authors have suggested the use of bootstrap algorithms as an alternative approach. Our main concern is to compare the behavior of three bootstrap procedures that take into account the dependence assumption of the observations when they are used to approximate the distribution of the test statistics considered. A broad simulation study is carried out to observe the finite sample performance of the analyzed bootstrap tests.      

一元线性回归方法的应用

按常规方法检测大型钢筋水泥预制件内置钢筋的张紧拉力,其数据很粗略,不能完全满足设计和工艺要求。为科学反映真实内在规律性,达到数据准确可靠且操作快捷方便,提出对待一组常规的测量数据应用一元线性回归数据处理方法,通过最小二乘法原理来处理两个变量之间的关系,最后得出数学模型,借助计算器即可圆满解决。该方法在工程领域有着一定的应用前景。     

The Bayes estimator in a misspecified linear regression model

Summary The paper presents an optimal Bayes estimator in the linear regression model under misspecification. This estimator is compared with the ordinary least squares estimator in terms of the matrix mean square error criterion. Some results on admissibility and prediction are also derived.     

Optimal method of linear regression in laser remote sensing

Remote lidar sensing in the photon-counting mode is now the commonly accepted method for studying atmospheric processes in the lower and free atmosphere. However, when processing signals obtained from lidar measurements, investigators necessarily face the problem of achieving accuracy in reconstructing the atmospheric parameters despite the presence of inhomogeneous noise in the measured signals. We propose an optimal method of linear regression (OMLR) of signals. The accuracy of the the method for the reconstructed signal is estimated. An example of application of the OMLR to the reconstruction of the temperature profile from the data obtained with a Raman lidar at the Siberian Lidar Station of the Institute of Atmospheric Optics (Tomsk, Russia) is given. The proposed method is distinguished by simplicity of interpretation of the criteria used, based on careful adherence to statistical principles. This method is shown to be an efficient auxiliary tool for the processing of measured data.     

Continuous dependence theorems in linear elasticity on exterior domains

By using the weight function approach, we prove some continuous dependence theorems for solutions to the equations of linear elasticity in an exterior domain in $\text{R}$³. Among other things, these theorems allow the elasticities to be unbounded as well as the density to be infinitesimal at large spatial distance from the boundary.     

Unbiased Weibull capabilities indices using multiple linear regression

Although the recently proposed Weibull process capability indices (PCIs) actually measure the times that the standard deviation (σ_x) is within the tolerance specifications, because they not accurately estimate neither the log‐mean (μ_x) nor the σ_x values, then the actual PCIs are biased. This actually because μ_x and σ_x are both estimated without considering the effect that the sample size (n) has over their values. Hence, μ_x is subestimated and σ_x is overestimated. As a response to this issue, in this paper, μ_x and σ_x are estimated in function of n. In particular, the PCIs' efficiency is based on the following facts: (1) the derived n value is unique and it completely determines η, (2) the μ_x value completely determines the η value, and (3) the σ_x value completely determines the β value. Thus, now, since μ_x and σ_x are in function of n and they completely determine β and η, then the proposed PCIs are unbiased, and they completely represent the analyzed process also. Finally, a step by step numerical application is given.     

Estimation of Weibull parameters with linear regression method

Abstract

Based on minimum mean square error, a modified probability estimator is proposed by a Monte Carlo simulation for estimating the Weibull parameters with the linear regression method. It is shown that compared with the commonly used estimators, the modified probability estimator gives a more accurate estimation of the Weibull modulus and the same estimation precision of the scale parameter. Furthermore, it is more conservative than the commonly used estimator recommended by previous authors and hence results in a higher safety in reliability predictions.     

基于偏最小二乘回归的军用飞机采购价格预测

考虑到军用飞机采购价格样本数据少、难于预测的特点和偏最小二乘回归方法在处理小样本多元数据方面的优势，提出一种基于偏最小二乘回归的军用飞机价格预测方法。偏最小二乘回归首先提取第一、第二主成分对采购价格样本的特异点进行剔除；然后进行变量投影重要度分析来筛选变量；最后，偏最小二乘回归对筛选的变量进行回归建立军用飞机价格预测模型，并对军用飞机价格进行预测，结果表明，在军用飞机价格预测方面，与未筛选变量的回归模型和逐步多元回归相比，经过变量筛选的偏最小二乘回归模型预测的精度更高，更能体现采购价格与飞机性能参数乏问的关系。     

A fast algorithm for clusterwise linear absolute deviations regression

Summary A fast algorithm and a corresponding FORTRAN implementation of a former algorithm [6] for clusterwise linear L1 regression is given.

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Zusammenfassung Ein schnelles Verfahren und eine zugehörige FORTRAN Implementation eines früheren Algorithmus [6] zur klassenweisen linearen L1 Regression wird angegeben.

     

On estimating Weibull modulus by the linear regression method

Statistical models were developed to estimate the bias of the shape parameter of a 2-parameter Weibull distribution where the shape parameter was estimated using a linear regression. These models were formulated for 27 sample sizes from 5 to 100 and for 35 probability estimators, , by varying “a” and “b”. In each simulation, 20,000 trials were used. From these models, a class of unbiased estimators were developed for each sample size. The standard deviation and coefficient of variation of these estimators were compared to the bias of the estimators. The standard deviation increased while the coefficient of variation decreased with increasing bias of the shape parameter. Also, the Anderson–Darling statistics was used to determine that the normal, log-normal, 3-parameter Weibull, and 3-parameter log-Weibull distributions did not provide good fit to the estimator of the shape parameter.     

Phase II monitoring of multivariate multiple linear regression profiles

In certain cases, the quality of a process or a product can be effectively characterized by two or more multiple linear regression profiles in which response variables are correlated. This structure can be modeled as multivariate multiple linear regression profiles. When linear profiles are monitored separately, then correlation between response variables is ignored and misleading results could be expected. To overcome this problem, the use of methods that consider the multivariate structure between response variables is inevitable. In this paper, we propose four methods to monitor this structure in Phase II. The performance of the methods is compared through simulation studies in terms of the average run length criterion. Furthermore, a method based on likelihood ratio approach is developed to determine the location of shifts and a numerical simulation is used to evaluate the performance of the proposed method. Finally, the use of the methods is illustrated by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

线性回归模型中的递推辨识及其应用

摘要：因机械系统振动响应离散差分方程可以改写为关于结构模态参数的线性回归形式,对于此线性回归形式中的结构模态参数的辨识问题可转化为系统辨识理论中线性回归模型的未知参数矢量的辨识估计问题。当系统对象在白噪声激励下,常规的最小二乘辨识法可以给出参数估计的无偏估计。当系统对象在有色噪声作用下,在常规最小二乘辨识法的基础上提出一种新的可分离迭代最小二乘辨识法。在缺乏关于噪声的统计先验信息,仅有噪声为未知但有界的情况下,提出一种带死区的最小二乘辨识法,该辨识法不仅能给出未知参数矢量的一致性无偏估计,而且还能保证参数估计值逐渐向真值靠拢,任何相邻两估计值之间的逼近程度远远小于噪声的上界。在有界噪声出现的情况下,该算法的鲁棒性可以通过在参数修正方程中引入死区的方法来增强。最后用理论上的仿真算例和弹簧-质量-阻尼系统的振动响应来验证两方法的有效性和可行性。     

Availability maximization under partial observations

In this paper, we propose a new model for availability maximization under partial observations for maintenance applications. Compared with the widely studied cost minimization models, few structural results are known about the form of the optimal control policy for availability maximization models. We consider a failing system with unobservable operational states. Only the failure state is observable. System deterioration is driven by an unobservable, continuous-time homogeneous Markov process. Multivariate condition monitoring data which is stochastically related to the unobservable state of the system is collected at equidistant sampling epochs and is used to update the posterior state distribution for decision making. Preventive maintenance can be carried out at any sampling epoch, and corrective maintenance is carried out upon system failure. The objective is to determine the form of the optimal control policy that maximizes the long-run expected average availability per unit time. We formulate the problem as an optimal stopping problem with partial information. Under standard assumptions, we prove that a control limit policy is optimal. A computational algorithm is developed, illustrated by numerical results.     

Comparison of Bayesian objective procedures for variable selection in linear regression

In the objective Bayesian approach to variable selection in regression a crucial point is the encompassing of the underlying nonnested linear models. Once the models have been encompassed, one can define objective priors for the multiple testing problem involved in the variable selection problem. There are two natural ways of encompassing: one way is to encompass all models into the model containing all possible regressors, and the other is to encompass the model containing only the intercept into any other.     

一元线性回归分析方法在压力传感器数据拟合中的应用

针对压力传感器在工程测量中需要根据计量检定数据获取压力传感器输出值和所加载标准压力值之间关系的问题,采用正态检验和方差分析的方法分析了压力传感器输出电压值和标准压力值之间的特征关系。根据相关系数值的计算结果,建立一元线性回归模型。分析了一元线性回归模型的拟合优度、模型线性关系、回归系数和模型的残差。结果表明：一元线性回归分析方法能够应用于计量检定工作中的压力传感器检定数据拟合分析,这对计量检定人员研究其他类似检定数据的拟合方法具有重要的参考价值。