Statistical Inference for Quasi-Infinitely Divisible Distributions via Fourier Methods

TL;DR

This paper develops a Fourier-based statistical inference method for quasi-infinitely divisible distributions, showing improved convergence rates over traditional methods and demonstrating its effectiveness through simulations.

Contribution

It introduces a novel Fourier approach for QID distributions, achieving polynomial convergence rates and providing practical algorithms with numerical validation.

Findings

Polynomial convergence rates for certain QID subclasses

Improved rates over classical infinitely divisible distribution methods

Successful numerical demonstrations with simulated data

Abstract

This study focuses on statistical inference for the class of quasi-infinitely divisible (QID) distributions, which was recently introduced by Lindner, Pan and Sato (2018). The paper presents a Fourier approach, based on the analogue of the L{\'e}vy-Khintchine theorem with a signed spectral measure. We prove that for some subclasses of QID distributions, the considered estimates have polynomial rates of convergence. This is a remarkable fact when compared to the logarithmic convergence rates of similar methods for infinitely divisible distributions, which cannot be improved in general. We demonstrate the numerical performance of the algorithm using simulated examples.