Ratio estimation
truncated estimation method
dependent observations
guaran teed accuracy
finite sample size
autoregression; ARARCH model; nonGaussian OrnsteinUhlenbeck process
nonparametric logarithmic density derivative estimation
One investigation method of ratio type estimators
Vyacheslav A. Vasiliev
Vasiliev
Vyacheslav A.
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976),
Vyacheslav A. Vasiliev
Preprint series:
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976),
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 Lmnorm, m >= 2: As an illustration, parametric and
nonparametric estimation problems on a time interval of a ¯xed length are
considered. In particular, parameters of linear (autoregressive) and nonlinear
(ARARCH) discretetime processes are estimated. Moreover, the parameter estimation problem of nonGaussian OrnsteinUhlenbeck process by discretetime 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 nonasymptotic 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. Nonparametric
estimator has optimal (as compared to the case of independent inputs) rate of
convergence.
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