Conditional least squares estimation in nonstationary nonlinear stochastic regression models

Abstract

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 $E(Z_n|{\mathcaligr F}_{n-1})=g_n(θ_0,ν_0)=g^{(1)}_n(θ_0)+g^{(2)}_n(θ_0,ν_0)$, ${\theta }_0\in {\mathbb{R}}^p$, $p<\infty$, ${\nu }_0\in {\mathbb{R}}^q$ and $q\le \infty$, 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 $g_n^{(2)}({\theta }_0,\hat{\nu })-g_n^{(2)}(\theta ,\hat{\nu })$ 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 $\{{\hat{\theta }}_n\}$ 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.

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

References in corpus (3)

• Conditional least squares estimation in nonstationary nonlinear stochastic regression models
• Identification of multitype branching processes
• Estimation in a class of nonlinear heteroscedastic time series models

Cited by in corpus (3)

• Conditional least squares estimation in nonstationary nonlinear stochastic regression models
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