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Conditional least squares estimation in nonstationary nonlinear stochastic regression models. (English) Zbl 1181.62133
Summary: Let $$\{Z_n\}$$ be a real nonstationary stochastic process such that
$E(Z_n\,|\,{\mathcal F}_{n-1})\overset{\text{a.s.}}<\infty\text{ and } E(Z^2_n\,|\,{\mathcal F}_{n-1})\overset{\text{a.s.}}<\infty,$
where $$\{{\mathcal F}_n\}$$ is an increasing sequence of $$\sigma$$-algebras. Assuming that
$E(Z_n\,|\,{\mathcal F}_{n-1})=g_n(\theta_0,\nu_0)=g_n^{(1)}(\theta_0)+g_n^{(2)}(\theta_0,\nu_0),$
$\theta_0\in\mathbb R^p,\quad p<\infty,\;\nu_0\in\mathbb R^q\text{ and }q\leq\infty,$
we study the asymptotic properties of $$\widehat\theta_n:=\text{argmin}_{\theta}\sum^n_{k=1}(Z_k-g_k(\theta,\widehat\nu))^2\lambda^{-1}_k$$, where $$\lambda_k$$ is $${\mathcal F}_{k-1}$$-measurable, $$\widehat\nu=\{\widehat\nu_k\}$$ is a sequence of estimations of $$\nu_0$$, $$g_n(\theta,\widehat\nu)$$ is Lipschitz in $$\theta$$ and $$g_n^{(2)}(\theta_0,\widehat\nu)-g_n^{(2)}(\theta,\widehat\nu)$$ is asymptotically negligible relative to $$g_n^{(1)}(\theta_0)-g_n^{(1)}(\theta)$$.
We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of $$\{\widehat\theta_n\}$$ in the linear model. For that, we prove a strong law of large numbers for a class of submartingales. Again using this strong law, we derive the general conditions leading to the asymptotic distribution of $$\{\widehat\theta_n\}$$. We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60F15 Strong limit theorems 62J02 General nonlinear regression 62F12 Asymptotic properties of parametric estimators 62M05 Markov processes: estimation; hidden Markov models 62M09 Non-Markovian processes: estimation 60G46 Martingales and classical analysis 62E20 Asymptotic distribution theory in statistics
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