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

TL;DR

This paper investigates the nonparametric estimation of the fractional derivative of the spectral function in Gaussian stationary processes, establishing theoretical properties and convergence rates for the estimators.

Contribution

It introduces a new estimation method for fractional derivatives of spectral functions, proving asymptotic unbiasedness, normality, and optimal convergence rates under certain conditions.

Findings

Estimates are asymptotically unbiased and normally distributed.

The estimation problem is well posed for derivatives of order less than 0.5.

Central Limit Theorem established for the estimators in functional space.

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.