Problem of Estimation of Fractional Derivative for a Spectral Function of Gaussian Stationary Processes

Abstract

We study the problem of nonparametric estimation of the fractional derivative of unknown spectral function of Gaussian stationary sequence (time series) and show that these problems is well posed with the classical speed of convergence when the order of derivative is less than 0.5. We prove also the asymptotical unbiaseness and normality of offered estimates with optimal speed of convergence. For the construction of the confidence region in some functional norm we establish the Central Limit Theorem in correspondent space of continuous functions for offered estimates.