Asymptotic inference for a nonlinear regression with a long-range dependence. (English. Ukrainian original) Zbl 0988.62015

Ivanov, O. V.; Leonenko, M. M.

Theory Probab. Math. Stat. 63, 65-85 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 61-79 (2000).

Let \(\xi(t),\;t\in R^1,\) be a measurable mean-square continuous Gaussian process with \(E\xi(t)=0\) and covariance function \[ B(t)=cov(\xi(0),\xi(t))=(1+t^2)^{-\alpha/2},\;0<\alpha<1,\;t\in R^1. \] Let \(\Theta\subset R^1\) be a bounded open interval, and let \(\Theta^{c}\) be the closure of \(\Theta\).
The authors consider the regression model \(y_{\theta}(t)=g(t,\theta)+G(\xi(t)),\;t\in R^1\), where \(g(t,\theta):R^1\times\Theta^{c}\to R^1\) is a measurable function depending on the unknown parameter \(\theta\in \Theta\). The random noise \(\zeta(t)=G(\xi(t)),\;t\in R^1\), is such that \(E\zeta(0)=0,\;E\zeta^2(0)=1\). The nonlinear Borel function \(G:R^1\to R^1\) satisfies the condition \[ (2\pi)^{-1/2}\int_{-\infty}^{\infty}G^2(u)e^{-u^2/2}du<\infty. \] The authors investigate consistency and the asymptotic distribution for the least squares estimates of \(\theta\).

Reviewer: A.D.Borisenko (Kyïv)

MSC:

62F12	Asymptotic properties of parametric estimators
62J02	General nonlinear regression
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20	Asymptotic distribution theory in statistics

Keywords:

asymptotic inference; long-range dependence