Well Posedness of the Problem of Estimation Fractional Derivative for a Distribution Function

Abstract

We study the problem of nonparametric estimation of the fractional derivative of unknown distribution function and of spectral function and show that these problems are well posed when the order of derivative is less than 0.5. We prove also the unbiaseness and asymptotical 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 Lebesgue-Riesz space for offered estimates, and deduce also the non-asymptotical deviation of our estimates in these spaces.