On the estimation of the jump activity index in the case of random observation times

TL;DR

This paper introduces a nonparametric method to estimate the jump activity index of a pure-jump process from irregular high-frequency data, accounting for complex sampling schemes.

Contribution

It develops a consistent estimator for the jump activity index using empirical characteristic functions and derives its asymptotic properties under irregular sampling.

Findings

Estimator is consistent for the jump activity index.

Central limit theorem established for the estimator.

Method handles irregular observation times effectively.

Abstract

We propose a nonparametric estimator of the jump activity index $\beta$ of a pure-jump semimartingale $X$ driven by a $\beta$-stable process when the underlying observations are coming from a high-frequency setting at irregular times. The proposed estimator is based on an empirical characteristic function using rescaled increments of $X$, with a limit which depends in a complicated way on $\beta$ and the distribution of the sampling scheme. Utilising an asymptotic expansion we derive a consistent estimator for $\beta$ and prove an associated central limit theorem.