Local polynomial regression smoothers with AR-error structure

Consider the fixed regression model with random observation error that follows an AR(1) correlation structure. In this paper, we study the nonparametric estimation of the regression function and its derivatives using a modified version of estimators obtained by weighted local polynomial fitting. The asymptotic properties of the proposed estimators are studied: expressions for the bias and the variance/covariance matrix of the estimators are obtained and the joint asymptotic normality is established. In a simulation study, a better behavior of the Mean Integrated Squared Error of the proposed regression estimator with respect to that of the classical local polynomial estimator is observed when the correlation of the observations is large. This work has been partially supported by grants PB98-0182-C02-01, PGIDT01PXI10505PR and MCyT Grant BFM2002-00265 (European FEDER support included).     

Recursive robust parameter estimation in automated information—Measuring systems

Advantages and shortcomings of robust estimators as compared with the classical ones are briefly described. Recursive robust estimators of parameters of a linear regression model are proposed. Results of numerical investigations of relative efficiency of recursive and nonrecursive robust estimators of parameters in a linear regression are presented. Translated from Izmeritel'naya Tekhnika, No. 4, pp. 16–19, April, 1997.     

An Explicit Solution for Generalized Ridge Regression

The general form of ridge regression proposed by Hoerl and Kennard is examined in the context of the iterative procedure they suggest for obtaining optimal estimators. It is shown that a non-iterative, closed form solution is available for this procedure. The solution is found to depend upon certain convergence/divergence conditions which relate to the ordinary least) squares estimators. Numerical examples are given.     

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.      

Uniform integrability of the OLS estimators,and the convergence of their moments

The problem of convergence of moments of a sequence of random variables to the moments of its asymptotic distribution is important in many applications. These include the determination of the optimal training sample size in the cross-validation estimation of the generalization error of computer algorithms, and in the construction of graphical methods for studying dependence patterns between two biomarkers. In this paper, we prove the uniform integrability of the ordinary least squares estimators of a linear regression model, under suitable assumptions on the design matrix and the moments of the errors. Further, we prove the convergence of the moments of the estimators to the corresponding moments of their asymptotic distribution, and study the rate of the moment convergence. The canonical central limit theorem corresponds to the simplest linear regression model. We investigate the rate of the moment convergence in canonical central limit theorem proving a sharp improvement of von Bahr’s (Ann Math Stat 36:808–818, 1965) theorem.     

Multigrid approach to predictive wave-front reconstruction in adaptive optical systems

A computationally efficient approach, based on the principles of multigrid methods, to predictive wave-front reconstruction in adaptive optical systems is described. Local predictive estimators are computed by use of recursive least squares on multiple grids. Each grid is increasingly coarse, allowing for temporal prediction of the behavior of both high- and low-spatial-frequency aberrations. Example numerical simulation results are given, showing that implementing the recursive least-squares algorithm for predictive estimation in a multigrid fashion greatly accelerates convergence to the steady-state optimal estimator condition. By implementation of the multigrid predictive reconstructor in parallel, the computational cost of implementing a predictive wave-front reconstruction scheme that uses recursive least squares for each processor at each cycle can be reduced from [symbol: see text](m2) to [symbol: see text](2m), where m is the number of actuators.     

Dating multiple change points in the correlation matrix

A nonparametric procedure for detecting and dating multiple change points in the correlation matrix of sequences of random variables is proposed. The procedure is based on a recently proposed test for changes in correlation matrices at an unknown point in time. Although the procedure requires constant expectations and variances, only mild assumptions on the serial dependence structure are assumed. The convergence rate of the change point estimators is derived and the asymptotic validity of the procedure is proved. Moreover, the performance of the proposed algorithm in finite samples is illustrated by means of a simulation study and the analysis of a real data example with financial returns. These examples show that the algorithm has large power in finite samples.     

Recursive approach for random response analysis using non-orthogonal polynomial expansion

Using non-orthogonal polynomial expansions, a recursive approach is proposed for the random response analysis of structures under static loads involving random properties of materials, external loads, and structural geometries. In the present formulation, non-orthogonal polynomial expansions are utilized to express the unknown responses of random structural systems. Combining the high-order perturbation techniques and finite element method, a series of deterministic recursive equations is set up. The solutions of the recursive equations can be explicitly expressed through the adoption of special mathematical operators. Furthermore, the Galerkin method is utilized to modify the obtained coefficients for enhancing the convergence rate of computational outputs. In the post-processing of results, the first- and second-order statistical moments can be quickly obtained using the relationship matrix between the orthogonal and the non-orthogonal polynomials. Two linear static problems and a geometrical nonlinear problem are investigated as numerical examples in order to illustrate the performance of the proposed method. Computational results show that the proposed method speeds up the convergence rate and has the same accuracy as the spectral finite element method at a much lower computational cost, also, a comparison with the stochastic reduced basis method shows that the new method is effective for dealing with complex random problems.     

Bootstrapping forecast intervals in ARCH models

In this paper we develop a bootstrap method for the construction of prediction intervals for an ARMA model when its innovations are an autoregressive conditional heteroscedastic process. We give a proof of the validity of the proposed bootstrap for this process. For this purpose we prove the convergence to zero in probability of the Mallows metric between the empirical distribution function and the theoretical distribution function of the residuals. The potential of the proposed method is assessed through a simulation study. This research was partially supported by Grant UZ-228-26 from the Spanish Ministry of Education and Grant UZ-228-25 from University of Zaragoza.     

钛合金立铣表面粗糙度预测新方法

为实现加工前对表面粗糙度的预测,建立高精度的表面粗糙度预测模型至关重要.针对钛合金立铣表面粗糙度的特点及传统预测方法的不足,提出了表面粗糙度预测新方法.分别用递推最小二乘算法、基本蚁群算法与混合蚁群算法训练模糊系统,混合蚁群算法的收敛效果优于递推最小二乘算法和基本蚁群算法.通过回归分析建立了表面粗糙度的两种经验公式.对各方法所得模型进行测试,结果表明混合蚁群算法训练模糊系统的预测效果优于其他方法,用混合蚁群算法训练的模糊系统进行表面粗糙度预测是可行的.     

New Improved Ranked Set Sampling Designs with an Application to Real Data

This article proposes two new Ranked Set Sampling (RSS) designs for estimating the population parameters: Simple Z Ranked Set Sampling (SZRSS) and Generalized Z Ranked Set Sampling (GZRSS). These designs provide unbiased estimators for the mean of symmetric distributions. It is shown that for non-uniform symmetric distributions, the estimators of the mean under the suggested designs are more efficient than those obtained by RSS, Simple Random Sampling (SRS), extreme RSS and truncation based RSS designs. Also, the proposed RSS schemes outperform other RSS schemes and provide more efficient estimates than their competitors under imperfect rankings. The suggested mean estimators under perfect and imperfect rankings are more efficient than the linear regression estimator under SRS. Our proposed RSS designs are also extended to cover the estimation of the population median. Real data is used to examine wthe usefulness and efficiency of our estimators.     

具有非参数AR(1)误差的回归模型的局部线性估计

考虑随机设计下具有一阶非参数自回归误差的线性回归模型,构造了参数和非参数函数的局部线性估计。在适当的条件下,证明了参数估计量的渐近正态性,并给出了非参数函数估计的收敛速度。模拟算例表明局部线性方法优于核方法。     

The Robust Regression Methods for Estimating of Finite Population Mean Based on SRSWOR in Case of Outliers

The ordinary least square (OLS) method is commonly used in regression analysis. But in the presence of outlier in the data, its results are unreliable. Hence, the robust regression methods have been suggested for a long time as alternatives to the OLS to solve the outliers problem. In the present study, new ratio type estimators of finite population mean are suggested using simple random sampling without replacement (SRSWOR) utilizing the supplementary information in Bowley’s coefficient of skewness with quartiles. For these proposed estimators, we have used the OLS, Huber-M, Mallows GM-estimate, Schweppe GM-estimate, and SIS GM-estimate methods for estimating the population parameters. Theoretically, the mean square error (MSE) equations of various estimators are obtained and compared with the OLS competitor. Simulations for skewed distributions as the Gamma distribution support the results, and an application of real data set containing outliers is considered for illustration.     

双线性时间序列模型的参数辨识

导出了双线性时间序列模型参数预报误差估计的递推算法，采用伴随常微分方程的稳定性分析方法分析了双线性模型的ｍ－可逆性对可预报误差估计算法收敛性的影响，并给出了相应的仿真示例。     

基于递归希尔伯特变换的振动信号解调和瞬时频率计算方法

精确地提取振动信号的瞬时幅值和瞬时频率对结构的参数识别和健康监测有重要作用。希尔伯特变换是一种常用的信号解调及瞬时频率计算方法,但在信号不满足Bedrosian乘积定理的条件时会造成较大误差。针对这一问题,提出了一种递归希尔伯特变换方法,用前一步希尔伯特变换计算出的纯调频信号作为新的信号,递归地使用希尔伯特变换以进行信号解调,理论分析表明递归希尔伯特变换能够快速地收敛。最后采用仿真信号对比了递归希尔伯特变换与单次希尔伯特变换、经验调幅调频分解及Teager能量算子法在信号解调及瞬时频率计算中的结果,结果表明了递归希尔伯特变换方法的实用性及精确性。     

Reweighted least trimmed squares: an alternative to one-step estimators

A new class of robust regression estimators is proposed that forms an alternative to traditional robust one-step estimators and that achieves the $\sqrt{n}$ rate of convergence irrespective of the initial estimator under a wide range of distributional assumptions. The proposed reweighted least trimmed squares (RLTS) estimator employs data-dependent weights determined from an initial robust fit. Just like many existing one- and two-step robust methods, the RLTS estimator preserves robust properties of the initial robust estimate. However contrary to existing methods, the first-order asymptotic behavior of RLTS is independent of the initial estimate even if errors exhibit heteroscedasticity, asymmetry, or serial correlation. Moreover, we derive the asymptotic distribution of RLTS and show that it is asymptotically efficient for normally distributed errors. A simulation study documents benefits of these theoretical properties in finite samples.     

Multicollinearity and Avoidance in Regression Analysis

For the multiple linear regression problem, a number of alternative estimators to ordinary least squares (OLS) have been proposed for situations in which multicollinearity is present among the explanatory variables. Multicollinearity may have several adverse effects on estimated coefficients in a multiple regression analysis.

This paper investigates the relative efficiency of these 12 alternative estimators from the point of view of mean squared error (MSE) by the Monte Carlo simulation, and discusses the practical implication of the use of such estimators. The results of this study are that OLS, Ridges, BYS and ITR estimators are more efficient than the others, when multicollinearity is not present. However, when multicollinearity is present, Ridges, GRB, BYS, ITR and PCA estimators are more efficient than OLS for almost all values of σ. Ridges have uniformly smaller MSEs than OLS. Relative efficiencies of these estimators vary with the value of σ. In the interval of small σ. Ridges are more efficient than the others, but, for large σ, each of GRB, BYS, ITR, PCA and LAT is more efficient.

From our experiment in which these 12 estimators are applied to the economic data of France, we find that, while OLS has the negative coefficient, some of these alternative have positive appropriate values, where regression coefficient must have the positive sign from the point of view of Economics. Therefore, we can conclude that these alternative estimators are effective for the practical regression problem with multicollinearity.

     

Fast estimation of the median covariation matrix with application to online robust principal components analysis

The geometric median covariation matrix is a robust multivariate indicator of dispersion which can be extended to infinite dimensional spaces. We define estimators, based on recursive algorithms, that can be simply updated at each new observation and are able to deal rapidly with large samples of high-dimensional data without being obliged to store all the data in memory. Asymptotic convergence properties of the recursive algorithms are studied under weak conditions in general separable Hilbert spaces. The computation of the principal components can also be performed online and this approach can be useful for online outlier detection. A simulation study clearly shows that this robust indicator is a competitive alternative to minimum covariance determinant when the dimension of the data is small and robust principal components analysis based on projection pursuit and spherical projections for high-dimension data. An illustration on a large sample and high-dimensional dataset consisting of individual TV audiences measured at a minute scale over a period of 24 h confirms the interest of considering the robust principal components analysis based on the median covariation matrix. All studied algorithms are available in the R package Gmedian on CRAN.     

Divergence-type errors of smooth Barron-type density estimators

Barron-type estimators are histogram-based distribution estimators that have been proved to have good consistency properties according to several information theoretic criteria. However they are not continuous. In this paper, we examine a new class of continuous distribution estimators obtained as a combination of Barron-type estimators with the frequency polygon. We prove the consistency of these estimators in expected information divergence and expected χ²-divergence. For one of then we evaluate the rate of convergence in expected χ²-divergence. Jan Beirlant is supported by the University Montpellier II. Igor Vajda is supported by the GACR grant 102/99/1137 and by the University Montpellier II.     

半参数回归模型的泛补偿最小二乘估计

本文首先提出泛补偿最小二乘法：接着,使用该法考虑半参数回归模型,得到了参数及非参数的估计。然后,将泛补偿最小二乘法与补偿最小二乘法进行了比较;最后用模拟的算例说明了该方法的有效性。