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Estimation for the nonlinear functional relationship. (English) Zbl 0669.62046
Let $$\{b_ n\}^{\infty}_{n=1}$$ and $$\{a_ n\}^{\infty}_{n=1}$$ be sequences of positive real numbers such that $$a_ n\cdot b_ n=n$$. Let $f(z^ 0_ t,\beta^ 0)=0,\quad t=1,2,...,b_ n$ be a functional relationship, where $$z^ 0_ t$$ are unobservable fixed vectors belonging to a parameter space $$\Gamma \subset R^ p$$, $$\beta^ 0\in \Omega \subset R^ k$$ is a $$1\times k$$ vector of unknown parameters, $f(z,\beta):\quad \Gamma \times \Omega \to R^ 1.$ The observations are the p-dimensional vectors $Z_{nt}=z^ 0_ t+\epsilon_{nt},\quad t=1,...,b_ n,$ where $$\epsilon_{nt}$$ are i.i.d. r.v. with mean zero and covariance matrix $$\Sigma_ n=a_ n^{- 1}\Phi$$, where $$\Phi >0$$ is a fixed matrix. The maximum likelihood estimators $${\hat \beta}$$ and $$\hat z_ t$$ are the values of $$\beta$$ in $$\Omega$$ and $$z_ t$$ in $$\Gamma$$ that minimize $\sum^{b_ n}_{t=1}(Z_{nt}-z_ t)\Sigma_ n^{-1}(Z_{nt}-z_ t)'$ subject to $$f(z_ t,\beta)=0$$, $$t=1,...,b_ n$$. Conditions for consistency and asymptotic normality of $${\hat \beta}$$ and $$\hat z_ t$$ are investigated. Modifications of the maximum likelihood estimators (bias-adjusted estimators) are also considered.
Reviewer: N.Leonenko

##### MSC:
 62J02 General nonlinear regression 62F12 Asymptotic properties of parametric estimators 62H12 Estimation in multivariate analysis
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