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We derive a nonparametric estimator of the jump-activity index $\beta$ of a"locally-stable" pure-jump It\^{o} semimartingale from discrete observations ofthe process on a fixed time interval with mesh of the observation gridshrinking to zero. The estimator is based on the empirical characteristicfunction of the increments of the process scaled by local power variationsformed from blocks of increments spanning shrinking time intervals precedingthe increments to be scaled. The scaling serves two purposes: (1) it controlsfor the time variation in the jump compensator around zero, and (2) it ensuresself-normalization, that is, that the limit of the characteristicfunction-based estimator converges to a nondegenerate limit which depends onlyon $\beta$. The proposed estimator leads to nontrivial efficiency gains overexisting estimators based on power variations. In the L\'{e}vy case, theasymptotic variance decreases multiple times for higher values of $\beta$. Thelimiting asymptotic variance of the proposed estimator, unlike that of theexisting power variation based estimators, is constant. This leads to furtherefficiency gains in the case when the characteristics of the semimartingale arestochastic. Finally, in the limiting case of $\beta=2$, which corresponds tojump-diffusion, our estimator of $\beta$ can achieve a faster rate thanexisting estimators.

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