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December, 1994 Asymptotic Properties of Nonlinear Least Squares Estimates in Stochastic Regression Models
Tze Leung Lai
Ann. Statist. 22(4): 1917-1930 (December, 1994). DOI: 10.1214/aos/1176325764


Stochastic regression models of the form $y_i = f_i(\theta) + \varepsilon_i$, where the random disturbances $\varepsilon_i$ form a martingale difference sequence with respect to an increasing sequence of $\sigma$-fields $\{\mathcal{G}_i\}$ and $f_i$ is a random $\mathcal{G}_{i - 1}$-measurable function of an unknown parameter $\theta$, cover a broad range of nonlinear (and linear) time series and stochastic process models. Herein strong consistency and asymptotic normality of the least squares estimate of $\theta$ in these stochastic regression models are established. In the linear case $f_i(\theta) = \theta^T\psi_i$, they reduce to known results on the linear least squares estimate $(\sum^n_1\psi_i\psi^T_i)^{-1}\sum^n_1\psi_i y_i$ with stochastic $\mathcal{G}_{i - 1}$-measurable regressors $\psi_i$.


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Tze Leung Lai. "Asymptotic Properties of Nonlinear Least Squares Estimates in Stochastic Regression Models." Ann. Statist. 22 (4) 1917 - 1930, December, 1994.


Published: December, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0824.62054
MathSciNet: MR1329175
Digital Object Identifier: 10.1214/aos/1176325764

Primary: 62J02
Secondary: 60F15 , 62F12 , 62M10

Keywords: asymptotic normality , control systems , martingales in Hilbert spaces , nonlinear autoregressive models , optimal experimental design , Stochastic regressors , strong consistency

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 4 • December, 1994
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