Asymptotic theory of least squares estimator in a nonregular nonlinear regression model

The asymptotic properties of the least squares estimator are derived for a nonregular nonlinear model via the study of weak convergence of the least squares process. This approach was adapted earlier by the author in the smooth case. The model discussed here is not amenable to analysis via the normal equations and Taylor expansions used by earlier authors.     

Implicitly-weighted total least squares

In a total least squares (TLS) problem, we estimate an optimal set of model parameters X, so that (A-ΔA)X=B-ΔB, where A is the model matrix, B is the observed data, and ΔA and ΔB are corresponding corrections. When B is a single vector, Rao (1997) and Paige and Strakoš (2002) suggested formulating standard least squares problems, for which ΔA=0, and data least squares problems, for which ΔB=0, as weighted and scaled TLS problems. In this work we define an implicitly-weighted TLS formulation (ITLS) that reparameterizes these formulations to make computation easier. We derive asymptotic properties of the estimates as the number of rows in the problem approaches infinity, handling the rank-deficient case as well. We discuss the role of the ratio between the variances of errors in A and B in choosing an appropriate parameter in ITLS. We also propose methods for computing the family of solutions efficiently and for choosing the appropriate solution if the ratio of variances is unknown. We provide experimental results on the usefulness of the ITLS family of solutions.     

A study over the general formula of regression sum of squares in multiple linear regression

In this study, in addition to the formula of regression sum of squares (SSR) in linear regression, a general formula of SSR in multiple linear regression is given. The derivations of the formula presented are given step by step. This new formula is proposed for estimation of the SSR in multiple linear regression. By using this formula, the researcher can find easily SSR and so the researcher can compose easily the table of variance analysis to interpret the regression made.     

A Berry-Esseen bound for least squares error variance estimators of regression parameters

We construct a precise Berry-Esseen bound for the least squares error variance estimators of regression parameters. Our bound depends explicitly on the sequence of design variables and is of the order O(N ^−1/2) if this sequence is “regular” enough. Supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 1–8, January–March, 1999.     

泛最小二乘法的改进及其容许性

考虑线性回归模型,当设计阵呈病态或秩亏时,我们用泛最小二乘法给出参数的估计,并证明其容许性;然后针对泛最小二乘估计对最小二乘估计过度压缩的缺点加以改进,使之更合理,有效.     

Minimax estimation and the least squares method

This paper is devoted to the problem of minimax estimation of parameters in linear regression models with uncertain second order statistics. The solution to the problem is shown to be the least squares estimator corresponding to the least favourable matrix of the second moments. This allows us to construct a new algorithm for minimax estimation closely connected with the least squares method. As an example, we consider the problem of polynomial regression introduced by A. N. Kolmogorov     

Asymptotic distribution of the weighted least squares estimator

This paper derives the asymptotic distribution of the weighted least squares estimator (WLSE) in a heteroscedastic linear regression model. A consistent estimator of the asymptotic covariance matrix of the WLSE is also obtained. The results are obtained under weak conditions on the design matrix and some moment conditions on the error distributions. It is shown that most of the error distributions encountered in practice satisfy these moment conditions. Some examples of the asymptotic covariance matrices are also given.     

Strong consistency of least squares estimates in multiple regression II

The strong consistency of least squares estimates in multiple regression models is established under minimal assumptions on the design and weak dependence and moment restrictions on the errors.     

Deep partial least squares for instrumental variable regression

In this paper, we propose deep partial least squares for the estimation of high-dimensional nonlinear instrumental variable regression. As a precursor to a flexible deep neural network architecture, our methodology uses partial least squares for dimension reduction and feature selection from the set of instruments and covariates. A central theoretical result, due to Brillinger (2012) Selected Works of Daving Brillinger. 589-606, shows that the feature selection provided by partial least squares is consistent and the weights are estimated up to a proportionality constant. We illustrate our methodology with synthetic datasets with a sparse and correlated network structure and draw applications to the effect of childbearing on the mother's labor supply based on classic data of Chernozhukov et al. Ann Rev Econ. (2015b):649–688. The results on synthetic data as well as applications show that the deep partial least squares method significantly outperforms other related methods. Finally, we conclude with directions for future research.     

The rate of convergence for the least squares estimator in nonlinear regression model with dependent errors

We study the parameter estimation in a nonlinear regression model with a general error's structure,strong consistency and strong consistency rate of the least squares estimator are obtained.     

基于最小二乘近似的模型平均方法

本文提出基于最小二乘近似的模型平均方法.该方法可用于线性模型、广义线性模型和分位数回归等各种常用模型.特别地,经典的Mallows模型平均方法是该方法的特例.现存的模型平均文献中,渐近分布的证明一般需要局部误设定假设,所得的极限分布的形式也比较复杂.本文将在不使用局部误设定假设的情形下证明该方法的渐近正态性.另外,本文...     

The rate of convergence of the least squares estimator in a non-linear regression model with dependent errors

The rate of convergence of the least squares estimator in a non-linear regression model with errors forming either a φ-mixing or strong mixing process is obtained. Strong consistency of the least squares estimator is obtained as a corollary.     

Parallel variable distribution for total least squares

A novel parallel method for determining an approximate total least squares (TLS) solution is introduced. Based on domain decomposition, the global TLS problem is partitioned into several dependent TLS subproblems. A convergent algorithm using the parallel variable distribution technique (SIAM J. Optim. 1994; 4 :815–832) is presented. Numerical results support the development and analysis of the algorithms. Copyright © 2005 John Wiley & Sons, Ltd.     

On the robustness of least squares procedures in regression models

The criterion robustness of the standard likelihood ratio test (LRT) under the multivariate normal regression model and also the inference robustness of the same test under the univariate set up are established for certain nonnormal distributions of errors. Restricting attention to the normal distribution of errors in the context of univariate regression models, conditions on the design matrix are established under which the usual LRT of a linear hypothesis (under homoscedasticity of errors) remains valid if the errors have an intraclass covariance structure. The conditions hold in the case of some standard designs. The relevance of C. R. Rao's (1967 In Proceedings Fifth Berkeley Symposium on Math. Stat. and Prob., Vol. 1, pp. 355–372) and G. Zyskind's (1967, Ann. Math. Statist.38 1092–1110) conditions in this context is discussed.     

Second-order nonlinear least squares estimation

The ordinary least squares estimation is based on minimization of the squared distance of the response variable to its conditional mean given the predictor variable. We extend this method by including in the criterion function the distance of the squared response variable to its second conditional moment. It is shown that this “second-order” least squares estimator is asymptotically more efficient than the ordinary least squares estimator if the third moment of the random error is nonzero, and both estimators have the same asymptotic covariance matrix if the error distribution is symmetric. Simulation studies show that the variance reduction of the new estimator can be as high as 50% for sample sizes lower than 100. As a by-product, the joint asymptotic covariance matrix of the ordinary least squares estimators for the regression parameter and for the random error variance is also derived, which is only available in the literature for very special cases, e.g. that random error has a normal distribution. The results apply to both linear and nonlinear regression models, where the random error distributions are not necessarily known.     

Convergence systems and strong consistency of least squares estimates in regression models

A recent theorem of T. L. Hai, H. Robbins, and C. Z. Wei (J. Multivariate Anal.9 (1979), 343–362) is extended to a more general form which unifies previous results in the literature on the strong consistency of least squares estimates in multiple regression models with nonrandom regressors. In particular the issue of strong consistency of the least squares estimate in the Gauss-Markov model, in the i.i.d. model with infinite second moment, and in general time series models is examined. In this connection, some basic properties of convergence systems are also obtained and are applied to the strong consistency problem.     

Circle fitting by linear and nonlinear least squares

The problem of determining the circle of best fit to a set of points in the plane (or the obvious generalization ton-dimensions) is easily formulated as a nonlinear total least-squares problem which may be solved using a Gauss-Newton minimization algorithm. This straight-forward approach is shown to be inefficient and extremely sensitive to the presence of outliers. An alternative formulation allows the problem to be reduced to a linear least squares problem which is trivially solved. The recommended approach is shown to have the added advantage of being much less sensitive to outliers than the nonlinear least squares approach.This work was completed while the author was visiting the Numerical Optimisation Centre, Hatfield Polytechnic and benefitted from the encouragement and helpful suggestions of Dr. M. C. Bartholomew-Biggs and Professor L. C. W. Dixon.     

Optimal learning rates for least squares regularized regression with unbounded sampling

A standard assumption in theoretical study of learning algorithms for regression is uniform boundedness of output sample values. This excludes the common case with Gaussian noise. In this paper we investigate the learning algorithm for regression generated by the least squares regularization scheme in reproducing kernel Hilbert spaces without the assumption of uniform boundedness for sampling. By imposing some incremental conditions on moments of the output variable, we derive learning rates in terms of regularity of the regression function and capacity of the hypothesis space. The novelty of our analysis is a new covering number argument for bounding the sample error.