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

Viktor Todorov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


We derive a nonparametric estimator of the jump-activity index β of a "locally-stable" pure-jump Ito 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 β. The proposed estimator leads to nontrivial efficiency gains over existing estimators based on power variations. In the Levy case, the asymptotic variance decreases multiple times for higher values of β. 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 β = 2, which corresponds to jump-diffusion, our estimator of β can achieve a faster rate than existing estimators.

Original languageEnglish (US)
Pages (from-to)1831-1864
Number of pages34
JournalAnnals of Statistics
Issue number4
StatePublished - Aug 1 2015


  • Central limit theorem
  • High-frequency data
  • Itô semimartingale
  • Jump activity index
  • Jumps
  • Power variation
  • Stochastic volatility

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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