On an estimator of an unknown parameter for a first-order autoregressive procedure \((|\theta|>1)\). (English. Ukrainian original) Zbl 0995.62084

Bondarev, B. V.; Selyakov, K. B.

Theory Probab. Math. Stat. 64, 9-17 (2002); translation from Teor. Jmovirn. Mat. Stat. 64, 10-17 (2001).

Let \(H\) be an abstract Hilbert space. Consider the first-order autoregressive model \(x_{n+1}=\theta x_{n}+\varepsilon_{n+1}\), where \(\theta\) is an unknown parameter and innovations \(\{\varepsilon_{n}\}\) are independent, identically distributed \(H\)-valued random elements which have zero mean and satisfy the Cramér condition. Under the assumption \[ 1+\lambda_0\leq |\theta|\leq L<\infty, \] \(\lambda_0\) and \(L\) being known positive numbers, the authors prove an exponential type upper estimate for the probability \(P\{|\theta|^n|\theta_n-\theta|>R\}\), where \(\theta_n\) is the least squares estimate for \(\theta\). Results of such type lead to a method of constructing confidence intervals for the unknown value of the parameter \(\theta\).

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F10	Point estimation
62F25	Parametric tolerance and confidence regions
60F10	Large deviations
46N30	Applications of functional analysis in probability theory and statistics

Keywords:

autoregression models; Hilbert spaces; least squares estimation; large deviations