Estimating large precision matrices via modified Cholesky decomposition

We introduce a k-banded Cholesky prior for estimating high-dimensional bandable precision matrices using a modified Cholesky decomposition. The bandable assumption is imposed on the Cholesky factor of the decomposition. We obtain the P-loss convergence rate under the spectral norm and the matrix l(infinity)-norm, as well as the minimax lower bounds. Because the P-loss convergence rate is stronger than the posterior convergence rate, the rates obtained are also posterior convergence rates. Furthermore, when the true precision matrix is a ko-banded matrix, for some finite ko, we obtain the minimax rate. The established convergence rates for bandable precision matrices are slightly slower than the minimax lower bounds, but are the fastest of the existing Bayesian approaches. Simulation results show that the performance of the proposed method is better than or comparable to that of competitive estimators.