Efficient nonparametric inference for discretely observed compound Poisson processes

TL;DR

This paper develops efficient nonparametric methods for estimating the jump and Lévy distributions of a compound Poisson process observed at discrete times, providing asymptotic theory and inference tools.

Contribution

It introduces novel nonparametric estimators for the jump and Lévy distributions with proven asymptotic normality and efficiency, along with kernel estimators for related parameters.

Findings

Proposed estimators are asymptotically normal and efficient.

Established functional central limit theorems under mild conditions.

Developed inference tools like confidence regions and tests.

Abstract

A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and L\'evy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as confidence regions and tests are briefly discussed.