An asymptotic expansion of the distribution of a functional of the least square estimator. (English. Russian original) Zbl 0834.62016

Bardadym, T. A.; Ivanov, A. V.

Theory Probab. Math. Stat. 47, 1-8 (1993); translation from Teor. Jmovirn. Mat. Stat. 47, 3-10 (1992).

Summary: The authors investigate the nonlinear regression model \(x_j= g(j, \theta)+ \varepsilon_j\), \(j=1, \dots, n\), \(\theta\in \Theta \subset \mathbb{R}^q\), \(q\geq 1\), and present an asymptotic expansion of the distribution of the functional \((L_n (\theta)- L_n (\widehat {\theta}_n))/ \sigma^2\), where \[ L_n (\theta)= \sum_{j=1}^n [x_j- g(j, \theta) ]^2, \] \(\widehat {\theta}_n\) is the least square estimator of an unknown parameter \(\theta\in \Theta\) constructed from the observations \(x_1, \dots, x_n\), and \(\sigma^2\) is the variance of the observation error \(\varepsilon\).

MSC:

62E20	Asymptotic distribution theory in statistics
62J02	General nonlinear regression
62F12	Asymptotic properties of parametric estimators

Keywords:

asymptotic expansion; least square estimator