An asymptotic expansion connected with the empirical regression function. (Russian) Zbl 0593.62060

Bardadym, T. A.; Iwanov, A. V.

Teor. Veroyatn. Mat. Stat. 30, 8-15 (1984).

Let \({\mathcal B}^ n\) be a Borel \(\sigma\)-algebra in \(R^ n\) and \(\Theta \subset R^ p\) an open convex set. The authors consider the sequence of statistical experiments \(\{R^ n,{\mathcal B}^ n,P^ n_{\theta}\); \(\theta\in \Theta \}\) generated by independent observations \(X_ 1,X_ 2,...,X_ n\), where \[ X_ j=g(j,\theta)+\epsilon_ j,\quad j=1,...,n \] g(j,\(\theta)\), \(\theta \in \Theta^ c\) are nonrandom functions, \(\epsilon_ i\) are independently identically distributed random variables, \(E\epsilon_ 1=0\), \(E\epsilon^ 2_ 1=\sigma^ 2\). The deviation of \(g(j,\theta_ n)\) from g(j,\(\theta)\) \((\theta_ n\) is a least squares estimator) is measured by \[ \Psi_ n(\theta_ n,\theta)=\sum^{n}_{j=1}[g(j,\theta_ n)-g(j,\theta)]^ 2. \] Under some regularity conditions the authors prove that \[ \sup_{C\subseteq \Theta}\sup_{u\geq 0}| P^ n_{\theta}\{\Psi_ n(\theta_ n,\theta)<2\sigma^ 2u\}-\quad \int^{u}_{0}f_{\chi^ 2_ p}(x)(1+\sum^{[(k-2)/2]}_{\nu =1}n^{-\nu}Q_{\quad \nu n}(\theta,x))dx| = \]
\[ o(n^{-(k-2)/2}) \] where C is compact in \(\Theta\), \(f_{\chi^ 2_ p}(x)\) is a density of \(\chi^ 2_ p\), \(Q_{\nu n}\) are polynomials of power \(3\nu\) with coefficients uniformly bounded in \(\theta\in C\) and n.

Reviewer: V.Konakov

MSC:

62J02

General nonlinear regression

Keywords:

empirical regression function; asymptotic expansion; least squares estimator