Conditional least squares estimation in nonstationary nonlinear stochastic regression models

Let $\{Z_n\}$ be a real nonstationary stochastic process such that $E(Z_n\mathrm{\mid }{\text{mathcaligr}F}_{n-1})\mathrm{\infty }$ and $E(Z_n^2\mathrm{\mid }{\text{mathcaligr}F}_{n-1})\mathrm{\infty }$, where $\{{\text{mathcaligr}F}_n\}$ is an increasing sequence of $\sigma$-algebras. Assuming that $E(Z_n|{\mathcaligr F}_{n-1})=g_n(\theta_0,\nu_0)=g^{(1)}_n(\theta_0)+g^{(2)}n(\theta_0,\nu_0)$,$\theta_0\in{$\mathbb{R}$}^p$,$p<\infty$,$\nu_0\in{$\mathbb{R}$}^q$and$q\leq\infty$,westudytheasymptoticpropertiesof$\hat{\theta}n:=\arg\min{\theta}\sum{k=1}^n(Z_k-g_k({\theta,\hat{\nu}}))^2\lambda_k^{-1}$, where ${\lambda }_k$ is ${\text{mathcaligr}F}_{k-1}$-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.