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

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