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.     

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     

Factorized quasi-Newton methods for nonlinear least squares problems

This paper provides a modification to the Gauss—Newton method for nonlinear least squares problems. The new method is based on structured quasi-Newton methods which yield a good approximation to the second derivative matrix of the objective function. In particular, we propose BFGS-like and DFP-like updates in a factorized form which give descent search directions for the objective function. We prove local and q-superlinear convergence of our methods, and give results of computational experiments for the BFGS-like and DFP-like updates.This work was supported in part by the Grant-in-Aid for Encouragement of Young Scientists of the Japanese Ministry of Education: (A)61740133 and (A)62740137.     

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.     

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.     

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.     

Statistical estimation in partial linear models with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. A model calibration approach and a weighting approach are developed to define the estimators of the parametric and nonparametric parts in the partial linear model, respectively. It is shown that the estimators for the parametric part are asymptotically normal and the estimators of g(·) converge to g(·) with an optimal convergent rate. Also, a comparison between the proposed estimators and the complete case estimator is made. A simulation study is conducted to compare the finite sample behaviors of these estimators based on bias and standard error.     

Asymptotic properties of a particular nonlinear regression quantile estimation

In this paper we shall be concerned with the asymptotic properties of the regression quantile estimation in the nonlinear regression time series models. For these, first we prove the strong consistency and derive the asymptotic normality of the regression quantile estimators for a particular sinusoidal regression model with a simple harmonic component. Next, we extend the results to more complicated sinusoidal models of several harmonic components.     

On nonlinear weighted total least squares parameter estimation problem for the three-parameter Weibull density

The three-parameter Weibull density function is widely employed as a model in reliability and lifetime studies. Estimation of its parameters has been approached in the literature by various techniques, because a standard maximum likelihood estimate does not exist. In this paper we consider the nonlinear weighted total least squares fitting approach. As a main result, a theorem on the existence of the total least squares estimate is obtained, as well as its generalization in the total _lq$l_q$ norm (q?1$q?1$). Some numerical simulations to support the theoretical work are given.     

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.     

Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares

A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deficient in two ways. Considering the problem formulated as subject to the constraints , the Jacobian and/or the Jacobian , , may be ill conditioned at the solution.

We analyze the important special case when and/or do not have full rank at the solution. In order to solve such a problem, we formulate a nonlinear least norm problem. Next we describe a truncated Gauss-Newton method. We show that the local convergence rate is determined by the maximum of three independent Rayleigh quotients related to three different spaces in .

Another way of solving an ill-posed nonlinear least squares problem is to regularize the problem with some parameter that is reduced as the iterates converge to the minimum. Our approach is a Tikhonov based local linear regularization that converges to a minimum norm problem. This approach may be used both for almost and rank-deficient Jacobians.

Finally we present computational tests on constructed problems verifying the local analysis.

     

约束最小二乘估计的优良性

对于线性模型y=Xθ ε,ε服从椭球等高单峰分布,未知参数θ满足不等式约束a′θ≥0,证明了在参数估计优良性的集中概率准则下,θ的约束最小二乘估计θ~*优于最小二乘估计θ.     

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     

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...     

Regularized nonlinear least squares methods for hit position reconstruction in small gamma cameras

In order to improve the performance of a gamma camera, it’s fundamental to accurately reconstruct the photon hit position on the detector surface. In the last years, the increasing demand of small highly-efficient PET systems led to the development of new hit position estimation methods, with the purpose of improving the performances near the edges of the detector, where most of the information is typically lost. In this paper we apply iterative optimization methods, based on the regularization of the nonlinear least squares problem, to estimate the photon hit position. Numerical results show that, compared with the classic Anger algorithm, the proposed methods allow to recover more information near the edges.     

Consistency conditions on the least squares estimator in single common factor analysis model

Summary This paper is concerned with the consistency of estimators in a single common factor analysis model when the dimension of the observed vector is not fixed. In the model several conditions on the sample sizen and the dimensionp are established for the least squares estimator (L.S.E.) to be consistent. Under some assumptions,p/n→0 is a necessary and sufficient condition that the L.S.E. converges in probability to the true value. A sufficient condition for almost sure convergence is also given.     

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.     

Parameter estimation in two-type continuous-state branching processes with immigration

We study the parameter estimation of two-type continuous-state branching processes with immigration based on low frequency observations at equidistant time points. The ergodicity of the processes is proved. The estimators are based on the minimization of a sum of squared deviation about conditional expectations. We also establish the strong consistency and central limit theorems of the conditional least squares estimators and the weighted conditional least squares estimators of the drift and diffusion coefficients based on low frequency observations.     

An efficient line search for nonlinear least squares

The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength which satisfies certain standard conditions. Prototype algorithms are described which guarantee finding such a step in a finite number of operations. This is achieved by first bracketing an interval of acceptable values and then reducing this bracket uniformly by the repeated use of sectioning in a systematic way. Some new theorems about convergence and termination of the line search are presented.Use of these algorithms to solve the line search subproblem in methods for nonlinear least squares is considered. We show that substantial gains in efficiency can be made by making polynomial interpolations to the individual residual functions rather than the overall objective function. We also study modified schemes in which the Jacobian matrix is evaluated as infrequently as possible, and show that further worthwhile savings can be made. Numerical results are presented.This work was supported by the award of a Syrian Ministry of Higher Education Scholarship.