Nonparametric estimation of infinitely divisible distributions based on variational analysis on measures

TL;DR

This paper introduces a novel non-parametric estimation method for compound Poisson distributions using variational analysis, effective for both discrete and continuous infinitely divisible distributions, with demonstrated advantages over existing techniques.

Contribution

It develops a new estimator based on series decomposition and steepest descent in variational measure analysis, expanding estimation capabilities for Levy processes.

Findings

Effective for discrete compounding distributions not on a grid

Performs well with continuous distributions with smoothing

Demonstrates positive comparison with existing methods

Abstract

The paper develops new methods of non-parametric estimation a compound Poisson distribution. Such a problem arise, in particular, in the inference of a Levy process recorded at equidistant time intervals. Our key estimator is based on series decomposition of functionals of a measure and relies on the steepest descent technique recently developed in variational analysis of measures. Simulation studies demonstrate applicability domain of our methods and how they positively compare and complement the existing techniques. They are particularly suited for discrete compounding distributions, not necessarily concentrated on a grid nor on the positive or negative semi-axis. They also give good results for continuous distributions provided an appropriate smoothing is used for the obtained atomic measure.