Abstract
Assuming a banded structure is one of the common practice in the estimation of high-dimensional precision matrices. In this case, estimating the bandwidth of the precision matrix is a crucial initial step for subsequent analysis. Although there exist some consistent frequentist tests for the bandwidth parameter, bandwidth selection consistency for precision matrices has not been established in a Bayesian framework. In this paper, we propose a prior distribution tailored to the bandwidth estimation of high-dimensional precision matrices. The banded structure is imposed via the Cholesky factor from the modified Cholesky decomposition. We establish strong model selection consistency for the bandwidth as well as consistency of the Bayes factor. The convergence rates for Bayes factors under both the null and alternative hypotheses are derived which yield similar order of rates. As a by-product, we also propose an estimation procedure for the Cholesky factors yielding an almost optimal order of convergence rates. Two-sample bandwidth test is also considered, and it turns out that our method is able to consistently detect the equality of bandwidths between two precision matrices. The simulation study confirms that our method in general outperforms the existing frequentist and Bayesian methods.
Highlights
Estimating a large covariance or precision matrix is a challenging task in both frequentist and Bayesian frameworks
We focus on banded precision matrices, where the banded structure is encoded via the Cholesky factor of the precision matrix
When there is a natural ordering in the data set, estimating the bandwidth of the precision matrix is important for detecting the dependence structure
Summary
Estimating a large covariance or precision matrix is a challenging task in both frequentist and Bayesian frameworks. In the Bayesian literature, Banerjee and Ghosal (2014) studied the estimation of bandable precision matrices which include the banded precision matrix as a special case They derived the posterior convergence rate of the precision matrix under the GWishart prior (Roverato, 2000). Lee and Lee (2018a) considered a similar class to that of Banerjee and Ghosal (2014), but assumed bandable Cholesky factors instead of bandable precision matrices They showed the posterior convergence rates of the precision matrix as well as the minimax lower bounds. The supplementary material (Lee and Lin, 2019) includes a result on the nearly optimal estimation of the Cholesky factors, and proofs of main results
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