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A truncated estimation method with guaranteed accuracy. (English) Zbl 1281.62209

Summary: This paper presents a truncated estimation method of ratio type functionals by dependent samples of finite size. This method makes it possible to obtain estimators with guaranteed accuracy in the sense of the \(L_m\)-norm, \(m\geq 2\). As an illustration, the parametric and nonparametric estimation problems on a time interval of a fixed length are considered. In particular, parameters of linear (autoregressive) and nonlinear 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 multivariate logarithmic derivative of a noise density of an autoregressive process with guaranteed accuracy are solved. In addition to non-asymptotic properties, the limit behavior of presented estimators is investigated. It is shown that all the truncated estimators have asymptotic properties of basic estimators. In particular, the asymptotic efficiency in the mean square sense of the truncated estimator of the dynamic parameter of a stable autoregressive process is established.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G08 Nonparametric regression and quantile regression
62H12 Estimation in multivariate analysis
62M05 Markov processes: estimation; hidden Markov models
62G07 Density estimation
62F12 Asymptotic properties of parametric estimators
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