On an extremal problem for nonoverlapping domains *

The paper considers the problem of finding the range of functional I = J f (z 0), f (z 0), F ($\zeta$ 0), F ($\zeta$ 0) , defined on the class M of pairs functions (f (z), F ($\zeta$)) that are univalent in the system of the disk and the interior of the disk, using the method of internal variations. We establish that the range of this functional is bounded by the curve whose equation is written in terms of elliptic integrals, depending on the parameters of the functional I

Improved estimation in a non-Gaussian parametric regression

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The noise is specified by the Ornstein-Uhlenbeck process driven by the mixture of a Brownian motion and a compound Poisson process. Improved estimates for the unknown regression parameters, based on a special modification of the James-Stein procedure with smaller quadratic risk than the usual least squares estimates, are proposed. The developed estimation scheme is applied for the improved parameter estimation in the discrete time regression with the autoregressive noise depending on unknown nuisance parameters.Comment: arXiv admin note: substantial text overlap with arXiv:1105.503

Adaptive model selection method for a conditionally Gaussian semimartingale regression in continuous time

This paper considers the problem of robust adaptive efficient estimating of a periodic function in a continuous time regression model with the dependent noises given by a general square integrable semimartingale with a conditionally Gaussian distribution. An example of such noise is the non-Gaussian Ornstein-Uhlenbeck-Levy processes. An adaptive model selection procedure, based on the improved weighted least square estimates, is proposed. Under some conditions on the noise distribution, sharp oracle inequality for the robust risk has been proved and the robust efficiency of the model selection procedure has been established. The numerical analysis results are given.Comment: 50 pages, 2 figures. arXiv admin note: text overlap with arXiv:1710.03111, arXiv:1712.0645