JUMP ACTIVITY ESTIMATION FOR PURE-JUMP SEMIMARTINGALES VIA SELF-NORMALIZED STATISTICS

• 成果类型: Article
• 署名作者: Todorov, Viktor
• 署名单位: Northwestern University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1327
• 发表日期: 2015
• 页码: 1831-1864
• 关键词: limit-theorems; Levy processes; activity index; inference; ORDER

摘要

We derive a nonparametric estimator of the jump-activity index beta 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 beta. 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 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.