Statistical Inference for Quasi-Infinitely Divisible Distributions via Fourier Methods

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.