One investigation method of ratio type estimators

Vyacheslav A. Vasiliev

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

62G05 Estimation

62G07 Density estimation

62F99 None of the above, but in this section

93B30 System identification

93E10 Estimation and detection

Abstract
This paper presents a truncated modification of basic ratio type estimators constructed by dependent sample of finite size. This method gives a possibility to obtain estimators with guaranteed accuracy in the sense of Lm-norm, m >= 2: As an illustration, parametric and non-parametric estimation problems on a time interval of a ¯xed length are considered. In particular, parameters of linear (autoregressive) and non-linear (ARARCH) discrete-time processes are estimated. Moreover, the parameter estimation problem of non-Gaussian Ornstein-Uhlenbeck process by discrete-time observations and the estimation problem of a logarithmic derivative of a noise density of an autoregressive process with guaranteed accuracy are solved. In addition to non-asymptotic properties, the limiting behavior of presented estimators is investigated. It is shown, in particular, that all parametric truncated estimators have rates of convergence of basic estimators. Non-parametric estimator has optimal (as compared to the case of independent inputs) rate of convergence.