The Annals of Statistics

Asymptotic Properties of Nonlinear Least Squares Estimates in Stochastic Regression Models

Tze Leung Lai

Full-text: Open access

Abstract

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 $\psi_i$.

Article information

Source
Ann. Statist., Volume 22, Number 4 (1994), 1917-1930.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176325764

Digital Object Identifier
doi:10.1214/aos/1176325764

Mathematical Reviews number (MathSciNet)
MR1329175

Zentralblatt MATH identifier
0824.62054

JSTOR
links.jstor.org

Subjects
Primary: 62J02: General nonlinear regression
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62F12: Asymptotic properties of estimators 60F15: Strong theorems

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

Citation

Lai, Tze Leung. Asymptotic Properties of Nonlinear Least Squares Estimates in Stochastic Regression Models. Ann. Statist. 22 (1994), no. 4, 1917--1930. doi:10.1214/aos/1176325764. https://projecteuclid.org/euclid.aos/1176325764


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