The properties of shift invariant operators are proved: It is shown that Q has polynomial order r iff r is the rate of convergence of . A weak saturation theorem is given. If f is replaced by in the weak saturation formula the asymptotics of the expression is calculated. Moreover, bootstrap approximation is introduced.
We prove the central limit theorem for the integrated square error of multivariate box-spline density estimators.
An approximation error and an asymptotic formula are given for shift invariant operators of polynomial order ϱ. Density estimators based on shift invariant operators are introduced and AMISE is calculated.
In this paper we consider a smoothness parameter estimation problem for a density function. The smoothness parameter of a function is defined in terms of Besov spaces. This paper is an extension of recent results (K. Dziedziul, M. Kucharska, B. Wolnik, ). The construction of the estimator is based on wavelets coefficients. Although we believe that the effective estimation of the smoothness parameter is impossible in general case, we can show that it becomes possible for some classes of the density...
We study the universal estimator for the regression problem in learning theory considered by Binev et al. This new approach allows us to improve their results.
The uniform approach to calculation of MISE for histogram and density box-spline estimators gives us a possibility to obtain estimators of derivatives of densities and the asymptotic constant.
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