Weighted average ensemble for Cholesky-based covariance matrix estimation

The modified Cholesky decomposition (MCD) is an efficient technique for estimating a covariance matrix. However, it is known that the MCD technique often requires a pre-specified variable ordering in the estimation procedure. In this work, we propose a weighted average ensemble covariance estimation for high-dimensional data based on the MCD technique. It can flexibly accommodate the high-dimensional case and ensure the positive definiteness property of the resultant estimate. Our key idea is to obtain different weights for different candidate estimates by minimizing an appropriate risk function with respect to the Frobenius norm. Different from the existing ensemble estimation based on the MCD, the proposed method provides a sparse weighting scheme such that one can distinguish which variable orderings employed in the MCD are useful for the ensemble matrix estimate. The asymptotically theoretical convergence rate of the proposed ensemble estimate is established under regularity conditions. The merits of the proposed method are examined by the simulation studies and a portfolio allocation example of real stock data.