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 $\psi_i$.
Publié le : 1994-12-14
Classification:
Stochastic regressors,
nonlinear autoregressive models,
control systems,
optimal experimental design,
strong consistency,
asymptotic normality,
martingales in Hilbert spaces,
62J02,
62M10,
62F12,
60F15
@article{1176325764,
author = {Lai, Tze Leung},
title = {Asymptotic Properties of Nonlinear Least Squares Estimates in Stochastic Regression Models},
journal = {Ann. Statist.},
volume = {22},
number = {1},
year = {1994},
pages = { 1917-1930},
language = {en},
url = {http://dml.mathdoc.fr/item/1176325764}
}
Lai, Tze Leung. Asymptotic Properties of Nonlinear Least Squares Estimates in Stochastic Regression Models. Ann. Statist., Tome 22 (1994) no. 1, pp. 1917-1930. http://gdmltest.u-ga.fr/item/1176325764/