On an estimate of the unknown parameter (\(|\theta|>1\)) of a first order autoregression process under Gaussian perturbations. (English. Ukrainian original) Zbl 0932.62100

Bondarev, B. V.

Theory Probab. Math. Stat. 58, 1-7 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 1-7 (1998).

Let \(X\) be an abstract separable Hilbert space and let \({\mathcal B}\) be the \(\sigma\)-algebra of the Borel subsets of \(X\). Let \(x_0,x_1, \ldots,x_{n}\) be observations of the autoregressive process of the form \(x_{n}=\theta x_{n-1}+\varepsilon_{n}\), where \(x_0=\varepsilon_0\), \(\varepsilon_{n}\in (X,\mathcal B)\), \(\theta\in\Theta\) is an unknown parameter. The author investigates properties of the least squares estimates of the parameter \(\theta\), \(\theta_{n}={\sum_{k=1}^{n}(x_{k},x_{k-1})/ \sum_{k=1}^{n}(x_{k-1},x_{k-1})}.\)
Let \(\{\varepsilon_0, \varepsilon_1,\ldots,\varepsilon_{n}\}\) be a sequence of independent, identically distributed normal random variables and let \(|\theta|\geq 1+\lambda_0.\) The author finds conditions under which one can obtain exponential estimates for the probabilities \(P\{|\theta|^{n}|\theta_{n} -\theta|>R\}\) and \(P\{|\theta_{n}-\theta|>\delta_{n}\}\), where \(\delta_{n}=((1+(3/4))\lambda_0/(1+\lambda_0))^{n}.\) For example, the following estimate (\(B\) is correlation operator of \(\varepsilon_{n}\), \(\delta>0\) is any constant) is obtained: \[ P\{|\theta|^{n}|\theta_{n} -\theta|>R\} \leq 2\exp\{ -R^2\delta^2/ 2\|B\|\}+C_1\delta \sqrt{Sp B}/\|B\|_2+ \]
\[ +\exp\{ (-r_{n}/2\|B \|)(1-\sqrt{n Sp B/r_{n}})^{2}\} +C_{n} Sp^2 B/\varepsilon^{n}, \] where \(C_1\), \(C_{n}\), \(r_{n}\) are some constants.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F10	Point estimation
62J05	Linear regression; mixed models

Keywords:

Hilbert spaces; method of least squares; autoregressive processes