## The Annals of Statistics

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

Christine Jacob

#### Abstract

Let {Zn} be a real nonstationary stochastic process such that $E(Z_{n}|{\mathcal{F}}_{n-1})\stackrel{\mathrm{a.s.}}<\infty$ and $E(Z_{n}^{2}|{\mathcal{F}}_{n-1})\stackrel{\mathrm{a.s.}}<\infty$, where $\{{\mathcal{F}}_{n}\}$ is an increasing sequence of σ-algebras. Assuming that $E(Z_{n}|{\mathcal{F}}_{n-1})=g_{n}(\theta_{0},\nu_{0})=g_{n}^{(1)}(\theta_{0})+g_{n}^{(2)}(\theta _{0},\nu_{0})$, θ0∈ℝp, p<∞, ν0∈ℝq and q≤∞, we study the asymptotic properties of $\widehat{\theta}_{n}:=\arg\min_{\theta}\sum_{k=1}^{n}(Z_{k}-g_{k}({\theta,\widehat{\nu}}))^{2}\lambda _{k}^{-1}$, where λk is ${\mathcal{F}}_{k-1}$-measurable, ̂ν={̂νk} is a sequence of estimations of ν0, gn(θ, ̂ν) is Lipschitz in θ and gn(2)(θ0, ̂ν)−gn(2)(θ, ̂ν) is asymptotically negligible relative to gn(1)(θ0)−gn(1)(θ). We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of {̂θn} 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 ̂θn. We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.

#### Article information

Source
Ann. Statist., Volume 38, Number 1 (2010), 566-597.

Dates
First available in Project Euclid: 31 December 2009

https://projecteuclid.org/euclid.aos/1262271624

Digital Object Identifier
doi:10.1214/09-AOS733

Mathematical Reviews number (MathSciNet)
MR2590051

Zentralblatt MATH identifier
1181.62133

#### Citation

Jacob, Christine. Conditional least squares estimation in nonstationary nonlinear stochastic regression models. Ann. Statist. 38 (2010), no. 1, 566--597. doi:10.1214/09-AOS733. https://projecteuclid.org/euclid.aos/1262271624

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