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

TL;DR

This paper investigates the well-posedness and statistical properties of nonparametric estimators for fractional derivatives of distribution functions, establishing their asymptotic normality and confidence regions for derivatives of order less than 0.5.

Contribution

It demonstrates the well-posedness of fractional derivative estimation problems and proves asymptotic properties of the estimators, including unbiasedness, normality, and deviation bounds.

Findings

Estimation problems are well posed for derivatives of order less than 0.5.

Proves asymptotic normality and unbiasedness of estimators.

Establishes confidence regions using the Central Limit Theorem in Lebesgue-Riesz spaces.

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.