  High frequency based estimation methods for a semiparametric pure-jump
subordinated Brownian motion exposed to a small additive microstructure noise
are developed building on the two-scales realized variations approach
originally developed by Zhang et. al. (2005) for the estimation of the
integrated variance of a continuous Ito process. The proposed estimators are
shown to be robust against the noise and, surprisingly, to attain better rates
of convergence than their precursors, method of moment estimators, even in the
absence of microstructure noise. Our main results give approximate optimal
values for the number K of regular sparse subsamples to be used, which is an
important tune-up parameter of the method. Finally, a data-driven plug-in
procedure is devised to implement the proposed estimators with the optimal
K-value. The developed estimators exhibit superior performance as illustrated
by Monte Carlo simulations and a real high-frequency data application.
