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Asymptotic theory of nonlinear regression. (English) Zbl 0874.62070

Mathematics and its Applications (Dordrecht). 389. Dordrecht: Kluwer Academic Publishers. vi, 327 p. (1997).
Consider the statistical experiments \(E^n=\{R^n,{\mathbf B}^n,P^n_\theta,\theta\in\Theta\}\) generated by a nonlinear regression model, i.e., observations of the form \[ X_j=g(j,\theta)+\varepsilon_j\quad (j=1,\dots,n), \] where the \(\varepsilon_j\)’s are independent random variables with common distribution functions not depending on \(\theta\). The least squares estimator of an unknown parameter \(\theta\in\Theta\) obtained from the observations \(X=(X_1,\dots,X_n)\) is any vector \(\widetilde{\theta}_n=\widetilde{\theta}_n(X)\) having the property: \[ L(\widetilde{\theta}_n)=\inf_{\tau\in\Theta^c}\sum^n_{j=1} [X_j-g(j,\tau)]^2. \] This book is devoted to the study of the asymptotic theory concerning least squares estimators which has been investigated by the author for a long time. Among others it contains the following: asymptotic normality of distributions of least squares estimators and asymptotic expansions related to functionals of the least squares estimators. In addition, the geometric properties of asymptotic expansions are considered.
From this book we can obtain some aspects of up-to-date mathematical results in the asymptotic theory on nonlinear regression.

MSC:

62J02 General nonlinear regression
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62F12 Asymptotic properties of parametric estimators
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