## The Annals of Statistics

### Jump activity estimation for pure-jump semimartingales via self-normalized statistics

Viktor Todorov

#### Abstract

We derive a nonparametric estimator of the jump-activity index $\beta$ of a “locally-stable” pure-jump Itô semimartingale from discrete observations of the process on a fixed time interval with mesh of the observation grid shrinking to zero. The estimator is based on the empirical characteristic function of the increments of the process scaled by local power variations formed from blocks of increments spanning shrinking time intervals preceding the increments to be scaled. The scaling serves two purposes: (1) it controls for the time variation in the jump compensator around zero, and (2) it ensures self-normalization, that is, that the limit of the characteristic function-based estimator converges to a nondegenerate limit which depends only on $\beta$. The proposed estimator leads to nontrivial efficiency gains over existing estimators based on power variations. In the Lévy case, the asymptotic variance decreases multiple times for higher values of $\beta$. The limiting asymptotic variance of the proposed estimator, unlike that of the existing power variation based estimators, is constant. This leads to further efficiency gains in the case when the characteristics of the semimartingale are stochastic. Finally, in the limiting case of $\beta=2$, which corresponds to jump-diffusion, our estimator of $\beta$ can achieve a faster rate than existing estimators.

#### Article information

Source
Ann. Statist., Volume 43, Number 4 (2015), 1831-1864.

Dates
Revised: December 2014
First available in Project Euclid: 17 June 2015

https://projecteuclid.org/euclid.aos/1434546224

Digital Object Identifier
doi:10.1214/15-AOS1327

Mathematical Reviews number (MathSciNet)
MR3357880

Zentralblatt MATH identifier
1317.62022

#### Citation

Todorov, Viktor. Jump activity estimation for pure-jump semimartingales via self-normalized statistics. Ann. Statist. 43 (2015), no. 4, 1831--1864. doi:10.1214/15-AOS1327. https://projecteuclid.org/euclid.aos/1434546224

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