# Rate-optimal estimation of the Blumenthal-Getoor index of a L\'evy process

## TL;DR

This paper introduces a new estimator for the Blumenthal-Getoor index of Lévy processes that achieves the optimal convergence rate, improving upon existing methods especially when a diffusion component is present.

## Contribution

The paper proposes a novel, rate-optimal estimator for the BG index and related parameters, applicable even with infinite variation jumps, using the generalized method of moments.

## Key findings

- Estimator attains the optimal convergence rate.
- Method effectively estimates parameters jointly.
- Simulation shows superior finite sample performance.

## Abstract

The Blumenthal-Getoor (BG) index characterizes the jump measure of an infinitely active L\'evy process. It determines sample path properties and affects the behavior of various econometric procedures. If the process contains a diffusion term, existing estimators of the BG index based on high-frequency observations only achieve rates of convergence which are suboptimal by a polynomial factor. In this paper, a novel estimator for the BG index and the successive BG indices is presented, attaining the optimal rate of convergence. If an additional proportionality factor needs to be inferred, the proposed estimator is rate-optimal up to logarithmic factors. Furthermore, our method yields a new efficient volatility estimator which accounts for jumps of infinite variation. All parameters are estimated jointly by the generalized method of moments. A simulation study compares the finite sample behavior of the proposed estimators with competing methods from the financial econometrics literature.