Estimating Large Precision Matrices via Modified Cholesky Decomposition

Abstract: We introduce the $k$-banded Cholesky prior for estimating a high-dimensional bandable precision matrix via the modified Cholesky decomposition. The bandable assumption is imposed on the Cholesky factor of the decomposition. We obtained the P-loss convergence rate under the spectral norm and the matrix $\ell_{\infty}$ norm and the minimax lower bounds. Since the P-loss convergence rate (Lee and Lee (2017)) is stronger than the posterior convergence rate, the rates obtained are also posterior convergence rates. Furthermore, when the true precision matrix is a $k_0$-banded matrix with some finite $k_0$, the obtained P-loss convergence rates coincide with the minimax rates. The established convergence rates are slightly slower than the minimax lower bounds, but these are the fastest rates for bandable precision matrices among the existing Bayesian approaches. A simulation study is conducted to compare the performance to the other competitive estimators in various scenarios.