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February 2010 Conditional least squares estimation in nonstationary nonlinear stochastic regression models
Christine Jacob
Ann. Statist. 38(1): 566-597 (February 2010). DOI: 10.1214/09-AOS733


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.


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Christine Jacob. "Conditional least squares estimation in nonstationary nonlinear stochastic regression models." Ann. Statist. 38 (1) 566 - 597, February 2010.


Published: February 2010
First available in Project Euclid: 31 December 2009

zbMATH: 1181.62133
MathSciNet: MR2590051
Digital Object Identifier: 10.1214/09-AOS733

Primary: 62F12 , 62J02 , 62M05 , 62M09 , 62M10 , 62P05 , 62P10
Secondary: 60F15 , 60G46

Keywords: asymptotic distribution , branching process , conditional least squares estimator , consistency , Heteroscedasticity , martingale difference , nonstationary process , polymerase chain reaction , quasi-likelihood estimator , Stochastic nonlinear regression , submartingale , time series

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 1 • February 2010
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