Conditional least squares estimation in nonstationary nonlinear stochastic regression models

Details Conditional least squares estimation in nonstationary nonlinear stochastic regression models Conditional least squares estimation in nonstationary nonlinear stochastic regression models

2010-01-13

arxiv.org/abs/1001.2102v1

Annals of Statistics 2010, Vol. 38, No. 1, 566-597

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/09-AOS733 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/09-AOS733

Let $\{Z_n\}$ be a real nonstationary stochastic process such that $E(Z_n|{\mathcaligr F}_{n-1})\stackrel{\mathrm{a.s.}}{<}\infty$ and $E(Z^2_n|{\mathcaligr F}_{n-1})\stackrel{\mathrm{a.s.}}{<}\infty$, where $\{{\mathcaligr F}_n\}$ 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 $q\leq\infty$, we study the asymptotic properties of $\hat{\theta}_n:=\arg\min_{\theta}\sum_{k=1}^n(Z_k-g_k({\theta,\hat{\nu}}))^2\lambda_k^{-1}$, where $\lambda_k$ is ${\mathcaligr F}_{k-1}$-measurable, $\hat{\nu}=\{\hat{\nu}_k\}$ is a sequence of estimations of $\nu_0$, $g_n(\theta,\hat{\nu})$ is Lipschitz in $\theta$ and $g^{(2)}_n(\theta_0,\hat{\nu})-g^{(2)}_n(\theta,\hat{\nu})$ is asymptotically negligible relative to $g^{(1)}_n(\theta_0)-g^{(1)}_n(\theta)$. We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of $\{\hat{\theta}_n\}$ 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 $\hat{\theta}_n$. We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.

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