On variable ordination of Cholesky‐based estimation for a sparse covariance matrix

Abstract

AbstractEstimation of a large sparse covariance matrix is of great importance for statistical analysis, especially in high‐dimensional settings. The traditional approach such as the sample covariance matrix performs poorly due to the high dimensionality. The modified Cholesky decomposition (MCD) is a commonly used method for sparse covariance matrix estimation. However, the MCD method relies on the order of variables, which often is not available or cannot be pre‐determined in practice. In this work, we solve this order issue by obtaining a set of covariance matrix estimates based on assuming different orders of variables used in the MCD. Then we consider an ensemble estimator as the “centre” of such a set of covariance matrix estimates with respect to the Frobenius norm. Our proposed method not only ensures that the estimator is positive definite, but also captures the underlying sparse structure of the covariance matrix. Under some regularity conditions, we establish both algorithmic and asymptotic convergence of the proposed method. Its merits are illustrated via simulation studies and a practical example using data from a prostate cancer study.