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August 2015 Jump activity estimation for pure-jump semimartingales via self-normalized statistics
Viktor Todorov
Ann. Statist. 43(4): 1831-1864 (August 2015). DOI: 10.1214/15-AOS1327

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.

Citation

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Viktor Todorov. "Jump activity estimation for pure-jump semimartingales via self-normalized statistics." Ann. Statist. 43 (4) 1831 - 1864, August 2015. https://doi.org/10.1214/15-AOS1327

Information

Received: 1 October 2014; Revised: 1 December 2014; Published: August 2015
First available in Project Euclid: 17 June 2015

zbMATH: 1317.62022
MathSciNet: MR3357880
Digital Object Identifier: 10.1214/15-AOS1327

Subjects:
Primary: 62F12, 62M05
Secondary: 60H10, 60J75

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 4 • August 2015
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