[Paper Review] Well Posedness of the Problem of Estimation Fractional Derivative for a Distribution Function

E. Ostrovsky, L. Sirota|arXiv (Cornell University)|Dec 21, 2014

Mathematical Approximation and Integration13 references3 citations

TL;DR

This paper establishes the well-posedness of nonparametric estimation for fractional derivatives of distribution and spectral functions when the derivative order is less than 0.5. It proves unbiasedness, asymptotic normality, optimal convergence rates, and derives central limit theorems in Lebesgue-Riesz spaces for confidence regions and non-asymptotic deviation bounds.

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.

Motivation & Objective

• To investigate the well-posedness of nonparametric estimation for fractional derivatives of unknown distribution and spectral functions.
• To establish conditions under which the estimation problem is well-posed, specifically when the derivative order is less than 0.5.
• To derive estimates with unbiasedness and optimal convergence speed for fractional derivatives.
• To construct confidence regions using central limit theorems in Lebesgue-Riesz function spaces.
• To provide non-asymptotic deviation bounds for the proposed estimates in the same functional spaces.

Proposed method

• The study employs nonparametric estimation techniques tailored for fractional derivatives of distribution and spectral functions.
• It analyzes the problem in Lebesgue-Riesz spaces to define functional norms for estimation accuracy and confidence regions.
• The authors prove asymptotic normality of the estimates under the condition that the derivative order is less than 0.5.
• Central limit theorems are established in Lebesgue-Riesz spaces to support the construction of confidence regions.
• Non-asymptotic deviation bounds are derived using functional space analysis and concentration inequalities.
• Optimal convergence rates are achieved through careful design of the estimation procedure in the specified function spaces.

Experimental results

Research questions

• RQ1 Is the nonparametric estimation problem for fractional derivatives of distribution functions well-posed when the derivative order is less than 0.5?
• RQ2 Do the proposed estimates exhibit unbiasedness and optimal convergence rates?
• RQ3 Can a central limit theorem be established in Lebesgue-Riesz spaces for the estimation of fractional derivatives?
• RQ4 What are the non-asymptotic deviation bounds of the estimates in the relevant functional spaces?
• RQ5 How can confidence regions be constructed for the estimated fractional derivatives using functional norms?

Key findings

• The estimation problem for fractional derivatives of distribution and spectral functions is well-posed when the derivative order is less than 0.5.
• The proposed estimates are unbiased and achieve the optimal rate of convergence for the given problem.
• Asymptotic normality of the estimates is established, enabling the construction of confidence regions.
• A central limit theorem is proven in Lebesgue-Riesz spaces, supporting statistical inference for the estimates.
• Non-asymptotic deviation bounds are derived, providing finite-sample performance guarantees in the same functional spaces.
• The functional norms used in the analysis are tied to Lebesgue-Riesz spaces, ensuring robustness in estimation accuracy and confidence region construction.