  The Blumenthal-Getoor (BG) index characterizes the jump measure of an
infinitely active L\'evy process. It determines sample path properties and
affects the behavior of various econometric procedures. If the process contains
a diffusion term, existing estimators of the BG index based on high-frequency
observations only achieve rates of convergence which are suboptimal by a
polynomial factor. In this paper, a novel estimator for the BG index and the
successive BG indices is presented, attaining the optimal rate of convergence.
If an additional proportionality factor needs to be inferred, the proposed
estimator is rate-optimal up to logarithmic factors. Furthermore, our method
yields a new efficient volatility estimator which accounts for jumps of
infinite variation. All parameters are estimated jointly by the generalized
method of moments. A simulation study compares the finite sample behavior of
the proposed estimators with competing methods from the financial econometrics
literature.
