  We derive a nonparametric estimator of the jump-activity index $\beta$ of a
"locally-stable" pure-jump It\^{o} 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 L\'{e}vy 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.
