  Last decade witnesses significant methodological and theoretical advances in
estimating large precision matrices. In particular, there are scientific
applications such as longitudinal data, meteorology and spectroscopy in which
the ordering of the variables can be interpreted through a bandable structure
on the Cholesky factor of the precision matrix. However, the minimax theory has
still been largely unknown, as opposed to the well established minimax results
over the corresponding bandable covariance matrices. In this paper, we focus on
two commonly used types of parameter spaces, and develop the optimal rates of
convergence under both the operator norm and the Frobenius norm. A striking
phenomenon is found: two types of parameter spaces are fundamentally different
under the operator norm but enjoy the same rate optimality under the Frobenius
norm, which is in sharp contrast to the equivalence of corresponding two types
of bandable covariance matrices under both norms. This fundamental difference
is established by carefully constructing the corresponding minimax lower
bounds. Two new estimation procedures are developed: for the operator norm, our
optimal procedure is based on a novel local cropping estimator targeting on all
principle submatrices of the precision matrix while for the Frobenius norm, our
optimal procedure relies on a delicate regression-based thresholding rule.
Lepski's method is considered to achieve optimal adaptation. We further
establish rate optimality in the nonparanormal model. Numerical studies are
carried out to confirm our theoretical findings.
