On the Asymptotic Distribution of the Least-Squares Estimators in Unidentifiable Models

2004 ◽  
Vol 16 (1) ◽  
pp. 99-114 ◽  
Author(s):  
Taichi Hayasaka ◽  
Masashi Kitahara ◽  
Shiro Usui
Keyword(s):  
Asymptotic Normality ◽  
Least Squares ◽  
Asymptotic Distribution ◽  
Linear Models ◽  
Stochastic Property ◽  
Learning Machines ◽  
Least Squares Estimators

In order to analyze the stochastic property of multilayered perceptrons or other learning machines, we deal with simpler models and derive the asymptotic distribution of the least-squares estimators of their parameters. In the case where a model is unidentified, we show different results from traditional linear models: the well-known property of asymptotic normality never holds for the estimates of redundant parameters.

scholarly journals Convergence of least squares estimators in the adaptive Wynn algorithm for some classes of nonlinear regression models

Metrika ◽  
2021 ◽  
Author(s):  
Fritjof Freise ◽  
Norbert Gaffke ◽  
Rainer Schwabe
Keyword(s):  
Asymptotic Normality ◽  
Least Squares ◽  
Parameter Space ◽  
Generalized Linear Models ◽  
Nonlinear Regression ◽  
Linear Models ◽  
Nonlinear Models ◽  
Random Errors ◽  
True Parameter ◽  
Least Squares Estimators

AbstractThe paper continues the authors’ work (Freise et al. The adaptive Wynn-algorithm in generalized linear models with univariate response. arXiv:1907.02708, 2019) on the adaptive Wynn algorithm in a nonlinear regression model. In the present paper the asymptotics of adaptive least squares estimators under the adaptive Wynn algorithm is studied. Strong consistency and asymptotic normality are derived for two classes of nonlinear models: firstly, for the class of models satisfying a condition of ‘saturated identifiability’, which was introduced by Pronzato (Metrika 71:219–238, 2010); secondly, a class of generalized linear models. Further essential assumptions are compactness of the experimental region and of the parameter space together with some natural continuity assumptions. For asymptotic normality some further smoothness assumptions and asymptotic homoscedasticity of random errors are needed and the true parameter point is required to be an interior point of the parameter space.

Least Squares Regression with Integrated or Dynamic Regressors under Weak Error Assumptions

1987 ◽  
Vol 3 (1) ◽  
pp. 98-116 ◽  
Author(s):  
Donald W. K. Andrews
Keyword(s):  
Linear Regression ◽  
Regression Model ◽  
Least Squares ◽  
Asymptotic Distribution ◽  
Linear Regression Model ◽  
Multiple Regression Model ◽  
Least Squares Regression ◽  
Least Squares Estimators ◽  
Integrated Or ◽  
Weak Error

This paper establishes consistency of least squares estimators in (i) a multiple regression model with integrated regressors and explosive, non-mixing errors, and (ii) a dynamic linear regression model with regressors and errors that may have infinite variances. In the former context, the asymptotic distribution of the least squares estimator also is obtained, in certain cases.

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