Hypothesis Testing for Band Size Detection of High Dimensional Banded Precision Matrices

Abstract

Many statistical analysis procedures require a good estimator for the high dimensional covariance matrix or its inverse, the precision matrix. When the precision matrix is banded, the Cholesky-based method can often yield a good estimator for the precision matrix. One important aspect of this method is to determine the band size of the precision matrix. In practice, cross validation is commonly used. However, we show that cross validation is not only computationally intensive, it can be very unstable. In this paper, a new hypothesis testing procedure is proposed to determine the precision matrix band size in the high dimensional case. Our proposed test statistic is shown to be asymptotically normal under the null hypothesis. Theoretical power of the test is studied as well. Numerical examples demonstrate the effectiveness of our testing procedure.

Keywords: Band size; Banded matrix; Cholesky decomposition; Covariance matrix; High-dimensional hypothesis test; Multiple comparison; Precision Matrix

JEL Classification: C10, C13