Asymptotics of estimators for nonparametric multivariate regression models with long memory

In this paper, a nonparametric multivariate regression model with long memory covariates and long memory errors is considered. We approximate the nonparametric multivariate regression function by the weighted additive one-dimensional functions. The local linear smoothing and least squares method are proposed for the one-dimensional regression estimation and the weight parameters estimation, respectively. The asymptotic behaviors of the proposed estimators are investigated.     

A weighted least squares method for scattered data fitting

In this paper, we present a weighted least squares method to fit scattered data with noise. Existence and uniqueness of a solution are proved and an error bound is derived. The numerical experiments illustrate that our weighted least squares method has better performance than the traditional least squares method in case of noisy data.     

Generalized least squares estimation of the functional multivariate linear errors-in-variables model

Estimators of the parameters of the functional multivariate linear errors-in-variables model are obtained by the application of generalized least squares to the sample matrix of mean squares and products. The generalized least squares estimators are shown to be consistent and asymptotically multivariate normal. Relationships between generalized least squares estimation of the functional model and of the structural model are demonstrated. It is shown that estimators constructed under the assumption of normal x are appropriate for fixed x.     

Tikhonov regularization for weighted total least squares problems

In this work, we study and analyze the regularized weighted total least squares (RWTLS) formulation. Our regularization of the weighted total least squares problem is based on the Tikhonov regularization. Numerical examples are presented to demonstrate the effectiveness of the RWTLS method.     

Detection of outliers in weighted least squares regression

In multiple linear regression model, we have presupposed assumptions (independence, normality, variance homogeneity and so on) on error term. When case weights are given because of variance heterogeneity, we can estimate efficiently regression parameter using weighted least squares estimator. Unfortunately, this estimator is sensitive to outliers like ordinary least squares estimator. Thus, in this paper, we proposed some statistics for detection of outliers in weighted least squares regression.     

Approximate decoupling of multivariate polynomials using weighted tensor decomposition

Many scientific and engineering disciplines use multivariate polynomials. Decomposing a multivariate polynomial vector function into a sandwiched structure of univariate polynomials surrounded by linear transformations can provide useful insight into the function while reducing the number of parameters. Such a decoupled representation can be realized with techniques based on tensor decomposition methods, but these techniques have only been studied in the exact case. Generalizing the existing techniques to the noisy case is an important next step for the decoupling problem. For this extension, we have considered a weight factor during the tensor decomposition process, leading to an alternating weighted least squares scheme. In addition, we applied the proposed weighted decoupling algorithm in the area of system identification, and we observed smaller model errors with the weighted decoupling algorithm than those with the unweighted decoupling algorithm.     

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??In this paper, we studied the inverse probability weighted least squares estimation of single-index model with response variable missing at random. Firstly, the B-spline technique is used to approximate the unknown single-index function, and then the objective function is established based on the inverse probability weighted least squares method. By the two-stage Newton iterative algorithm, the estimation of index parameters and the B-spline coefficients can be obtained. Finally, through many simulation examples and a real data application, it can be concluded that the method proposed in this paper performs very well for moderate sample     

An algorithm for linear least squares problems with equality and nonnegativity constraints

We present a new algorithm for solving a linear least squares problem with linear constraints. These are equality constraint equations and nonnegativity constraints on selected variables. This problem, while appearing to be quite special, is the core problem arising in the solution of the general linearly constrained linear least squares problem. The reduction process of the general problem to the core problem can be done in many ways. We discuss three such techniques.The method employed for solving the core problem is based on combining the equality constraints with differentially weighted least squares equations to form an augmented least squares system. This weighted least squares system, which is equivalent to a penalty function method, is solved with nonnegativity constraints on selected variables.Three types of examples are presented that illustrate applications of the algorithm. The first is rank deficient, constrained least squares curve fitting. The second is concerned with solving linear systems of algebraic equations with Hilbert matrices and bounds on the variables. The third illustrates a constrained curve fitting problem with inconsistent inequality constraints.     

Error Process Indexed by Bandwidth Matrices in Multivariate Local Linear Smoothing

We focus on nonparametric multivariate regression function estimation by locally weighted least squares. The asymptotic behavior for a sequence of error processes indexed by bandwidth matrices is derived. We discuss feasible data-driven consistent estimators minimizing asymptotic mean squared error or efficient estimators reducing asymptotic bias at points where opposite sign curvatures of the regression function are present in different directions.     

Local Weighted Composite Quantile Estimating for Varying Coefficient Models

A generalization of classical linear models is varying coefficient models, which offer a flexible approach to modeling nonlinearity between covariates. A method of local weighted composite quantile regression is suggested to estimate the coefficient functions. The local Bahadur representation of the local estimator is derived and the asymptotic normality of the resulting estimator is established. Comparing to the local least squares estimator, the asymptotic relative efficiency is examined for the local weighted composite quantile estimator. Both theoretical analysis and numerical simulations reveal that the local weighted composite quantile estimator can obtain more efficient than the local least squares estimator for various non-normal errors. In the normal error case, the local weighted composite quantile estimator is almost as efficient as the local least squares estimator. Monte Carlo results are consistent with our theoretical findings. An empirical application demonstrates the potential of the proposed method.     

Strong underrelaxation in Kaczmarz's method for inconsistent systems

Summary We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.     

基于Pena距离的加权最小二乘估计的影响分析

利用Pena距离对加权线性最小二乘估计的影响问题进行讨论,得到加权最小二乘估计的Pena距离的表达式,对其性质进行讨论,从而得到高杠异常点的判别方法.文中对Pena距离与Cook距离的性能进行了对比,得到在一定条件下Pena距离优于Cook距离的结论.并通过数值实验对此方法的有效性进行验证.     

变系数部分线性模型的加权随机约束s-K估计

研究随机约束条件下半参数变系数部分线性模型的参数估计问题,当回归模型线性部分变量存在多重共线性时,基于Profile最小二乘方法、s-K估计和加权混合估计构造参数向量的加权随机约束s-K估计,并在均方误差矩阵准则下给出新估计量优于s-K估计和加权混合估计的充要条件,最后通过蒙特卡洛数值模拟验证所提出估计量的有限样本性质.     

Iterative weighted partial spline least squares estimation in semiparametric modeling of longitudinal data

In this paper we consider the estimating problem of a semiparametric regression modelling whenthe data are longitudinal.An iterative weighted partial spline least squares estimator(IWPSLSE)for the para-metric component is proposed which is more efficient than the weighted partial spline least squares estimator(WPSLSE)with weights constructed by using the within-group partial spline least squares residuals in the sense     

Scattered data fitting of Hermite type by a weighted meshless method

In many practical problems, it is often desirable to interpolate not only the function values but also the values of derivatives up to certain order, as in the Hermite interpolation. The Hermite interpolation method by radial basis functions is used widely for solving scattered Hermite data approximation problems. However, sometimes it makes more sense to approximate the solution by a least squares fit. This is particularly true when the data are contaminated with noise. In this paper, a weighted meshless method is presented to solve least squares problems with noise. The weighted meshless method by Gaussian radial basis functions is proposed to fit scattered Hermite data with noise in certain local regions of the problem’s domain. Existence and uniqueness of the solution is proved. This approach has one parameter which can adjust the accuracy according to the size of the noise. Another advantage of the weighted meshless method is that it can be used for problems in high dimensions with nonregular domains. The numerical experiments show that our weighted meshless method has better performance than the traditional least squares method in the case of noisy Hermite data.     

Generalized least squares and maximum likelihood estimations of multivariate polychoric correlations

This paper investigates the generalized least squares estimation and the maximum likelihood estimation of the parameters in a multivariate polychoric correlations model, based on data from a multidimensional contingency table. Asymptotic properties of the estimators are discussed. An iterative procedure based on the Gauss-Newton algorithm is implemented to produce the generalized least squares estimates and the standard errors estimates. It is shown that via an iteratively reweighted method, the algorithm produces the maximum likelihood estimates as well. Numerical results on the finite sample behaviors of the methods are reported.     

Bridge estimators and the adaptive lasso under heteroscedasticity

In this paper we investigate penalized least squares methods in linear regression models with heteroscedastic error structure. It is demonstrated that the basic properties with respect to model selection and parameter estimation of bridge estimators, Lasso and adaptive Lasso do not change if the assumption of homoscedasticity is violated. However, these estimators do not have oracle properties in the sense of Fan and Li (2001) if the oracle is based on weighted least squares. In order to address this problem we introduce weighted penalized least squares methods and demonstrate their advantages by asymptotic theory and by means of a simulation study.     

基于偏最小二乘变换与Kriging模型的全局优化方法

针对传统Kriging模型在多变量(高维)输入全局优化中因超参数过多而引发收敛速度慢,精度低,建模效率不高问题,提出了基于偏最小二乘变换技术和Kriging模型的有效全局优化方法.首先,构造偏最小二乘高斯核函数;其次,借助差分进化算法寻找满足期望改进准则最大化条件的新样本点;然后,将不同核函数和期望改进准则组合,构建四种有效全局优化算法并进行比较;最后,数值算例结果表明,基于偏最小二乘变换的Kriging全局优化方法在解决高维全局优化问题方面相比于标准的全局优化算法在收敛精度及收敛速度方面更具优势.     

Asymptotic inference of unstable periodic ARCH processes

This paper studies asymptotic properties of the quasi maximum likelihood and weighted least squares estimates (QMLE and WLSE) of the conditional variance slope parameters of a strictly unstable ARCH model with periodically time varying coefficients (PARCH in short). The model is strictly unstable in the sense that its parameters lie outside the strict periodic stationarity domain and its boundary. Obtained from the regression form of the PARCH, the WLSE is a variant of the least squares method weighted by the square of the conditional variance evaluated at any fixed value in the parameter space. In calculating the QMLE and WLSE, the conditional variance intercepts are set to any arbitrary values not necessarily the true ones. The theoretical finding is that the QMLE and WLSE are consistent and asymptotically Gaussian with the same asymptotic variance irrespective of the fixed conditional variance intercepts and the weighting parameters. So because of its numerical complexity, the QMLE may be dropped in favor of the WLSE which enjoys closed form.     

Multivariate Locally Weighted Polynomial Fitting and Partial Derivative Estimation

Nonparametric regression estimator based on locally weighted least squares fitting has been studied by Fan and Ruppert and Wand. The latter paper also studies, in the univariate case, nonparametric derivative estimators given by a locally weighted polynomial fitting. Compared with traditional kernel estimators, these estimators are often of simpler form and possess some better properties. In this paper, we develop current work on locally weighted regression and generalize locally weighted polynomial fitting to the estimation of partial derivatives in a multivariate regression context. Specifically, for both the regression and partial derivative estimators we prove joint asymptotic normality and derive explicit asymptotic expansions for their conditional bias and conditional convariance matrix (given observations of predictor variables) in each of the two important cases of local linear fit and local quadratic fit.