Testing a linear hypothesis using Haar transform

The paper is concerned with the problem of testing a linear hypothesis about regression function. New testing procedure based on the Haar transform is proposed which is adaptive to unknown smoothness properties of the underlying function. The results show that under mild conditions on the design and smoothness of the regression function, this procedure provides with the near optimal rate of testing. (orig.)SIGLEAvailable from TIB Hannover: RR 5549 (314)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Data-driven testing the fit of linear models

The paper is concerned with the problem of testing a linear hypothesis about regression function. We propose a new testing procedure based on the Haar transform which is adaptive to unknown smoothness properties of the underlying function. The results show rate optimality of this procedure under mild conditions on the model. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(472)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Moderate deviations for integral functionals of diffusion process

Available from TIB Hannover: RR 5549(377) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

On estimating a dynamic function of stochastic system with averaging

We consider a two-scaled diffusion system, when drift and diffusion parameters of a 'slow' component are contaminated by an unobservable 'fast' one. The goal is to estimate the dynamic function which is defined by averaging the drift coefficient of the 'slow' component w.r.t. the stationary distribution of the 'fast' one. For estimation we use a locally linear smoother with a datadriven choice of bandwidth. A procedure proposed is fully adaptive and nearly optimal up to a log log factor. (orig.)Available from TIB Hannover: RR 5549(381) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Deviation probability bound for martingales with applications to statistical estimation

SIGLEAvailable from TIB Hannover: RR 5549(471)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

On estimation of non-smooth functionals

Let a function f be observed with noise. In the present paper we concern the problem of nonparametric estimation of some non-smooth functionals of f, more precisely, L_r-norm parallel f parallel _r of f. Existing in the literature results on estimation of functionals deal mostly with two extreme cases: Estimation of a smooth (differentiable in L_2) functional or estimation of a singular functional like the value of f at a certain point or the maximum of f. In the first case, the rate of estimation is typically n"-"1"/"2, n being the number of observations. In the second case, the rate of functional estimation coincides with the nonparametric rate of estimation of the whole function f in the corresponding norm. We show that the case of estimation of parallel f parallel _r is in some sense intermediate between the above extreme two. The optimal rate of estimation is worse than n"-"1"/"2 but better than the usual nonparametric rate. The results depend on the value of r. For r even integer, the rate occurs to be n"-"#beta#"/"("2"#beta#"+"1"-"1"/"r") where #beta# is the degree of smoothness. If r is not even integer, then the nonparametric rate n"-"#beta#"/"("2"#beta#"+"1") can be improved only by some logarithmic factor. (orig.)Available from TIB Hannover: RR 5549(297)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

On estimation of non-smooth functionals

Let a function f be observed with noise. In the present paper we concern the problem of nonparametric estimation of some non-smooth functionals of f, more precisely, L_r-norm parallel f parallel _r of f. Existing in the literature results on estimation of functionals deal mostly with two extreme cases: Estimation of a smooth (differentiable in L_2) functional or estimation of a singular functional like the value of f at a certain point or the maximum of f. In the first case, the rate of estimation is typically n"-"1"/"2, n being the number of observations. In the second case, the rate of functional estimation coincides with the nonparametric rate of estimation of the whole function f in the corresponding norm. We show that the case of estimation of parallel f parallel _r is in some sense intermediate between the above extreme two. The optimal rate of estimation is worse than n"-"1"/"2 but better than the usual nonparametric rate. The results depend on the value of r. For r even integer, the rate occurs to be n"-"#beta#"/"("2"#beta#"+"1"-"1"/"r") where #beta# is the degree of smoothness. If r is not even integer, then the nonparametric rate n"-"#beta#"/"("2"#beta#"+"1") can be improved only by some logarithmic factor. (orig.)Available from TIB Hannover: RR 5549(297)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Structure adaptive approach for dimension reduction

We propose a new method of effective dimension reduction for a multi-index model which is based on iterative improvement of the family of average derivative estimates. The procedure is computationally straightforward and does not require any prior information about the structure of the underlying model. We show that in the case when the effective dimension m of the index space does not exceed 3, this space can be estimated with the rate n"-"1"/"2 under rather mild assumptions on the model. (orig.)Available from TIB Hannover: RR 5549(569)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Structure adaptive approach for dimension reduction

We propose a new method of effective dimension reduction for a multi-index model which is based on iterative improvement of the family of average derivative estimates. The procedure is computationally straightforward and does not require any prior information about the structure of the underlying model. We show that in the case when the effective dimension m of the index space does not exceed 3, this space can be estimated with the rate n"-"1"/"2 under rather mild assumptions on the model. (orig.)Available from TIB Hannover: RR 5549(569)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Exact asymptotics of minimax Bahadur risk in Lipschitz regression

The estimation problem for a Lipschitz regression at a point is studied. The exact limiting performance of the Bahadur risk is found in the minimax sense, the asymptotics being presented in the explicit form in terms of the Chernoff function. (orig.)Available from TIB Hannover: RR 5549(183)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman