Weak convergence of the distribution for an estimate of an infinite- dimensional parameter to the normal one. (Russian) Zbl 0634.62010

Dorogovtsev, A. Ya.; Zerek, N.; Kukush, A. G.

Teor. Veroyatn. Mat. Stat., Kiev 37, 39-46 (1987).

The article is the continuation of the paper “Asymptotic properties of nonlinear regression estimates in Hilbert space” by the same authors in ibid. 35, 36-44 (1986); English translation in: Theory Probab. Math. Statist. 35, 37-43 (1987). The authors describe the asymptotic properties of least squares estimators (l.s.e.) in a model of nonlinear regression in Hilbert space. Let \(y_ n:=f_ n(\theta_ 0)+\xi_ n\), \(n\geq 1\). The estimated parameter \(\theta_ 0\) belongs to a convex compact set \(\Theta\) of special type in a Hilbert space B. The functions \(\{f_ n\}\) act from \(\Theta\) into a Hilbert space H, \(\{\xi_ n\}\) are i.i.d. H- valued random elements. The l.s.e. \({\hat \theta}_ N\) is build on the basis of the observations \(y_ 1,...,y_ N\) to minimize the functional \[ Q_ N(\theta):=N^{-1}\sum^{N}_{n=1}\| y_ n-f_ n(\theta)\|^ 2_ H,\quad \theta \in \Theta. \] The normalized estimators \(\sqrt{N}({\hat \theta}_ N-\theta_ 0)\), \(N\geq 1\), are proved to converge in distribution in B to a Gaussian element. The restrictions are basically analogous to the conditions in the finite- dimensional situation. The main difficulty of the investigation is that the equality grad \(Q_ N({\hat \theta}_ N)=\bar 0\) may be false because an infinite-dimensional compact set has no inner point.
The idea of the proof of convergence is the following: Inequalities for the derivative \(Q'_{N}({\hat \theta}_ N)\) in the directions connected with the deviation \({\hat \theta}_ N-\theta_ 0\) are obtained, implying the tightness of the distributions of the normalized estimators.

Reviewer: A.Ya.Dorogovtsev

MSC:

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

Keywords:

infinite-dimensional parameter; asymptotic normality; convergence in distribution; least squares estimators; Hilbert space; convex compact set; normalized estimators