Conditional least squares estimation in nonstationary nonlinear stochastic regression models
The Annals of Statistics · 2010 · Volume 38 · Issue 1
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.
Keywords
60F15, 60G46, 62F12, 62J02, 62M05, 62M09, 62M10, 62P05, 62P10, 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
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