A nonparametric empirical Bayes approach to covariance matrix estimation

We propose an empirical Bayes method to estimate high-dimensional covariance matrices. Our procedure centers on vectorizing the covariance matrix and treating matrix estimation as a vector estimation problem. Drawing from the compound decision theory literature, we introduce a new class of decision rules that generalizes several existing procedures. We then use a nonparametric empirical Bayes g-modeling approach to estimate the oracle optimal rule in that class. This allows us to let the data itself determine how best to shrink the estimator, rather than shrinking in a pre-determined direction such as toward a diagonal matrix. Simulation results and a gene expression network analysis shows that our approach can outperform a number of state-of-the-art proposals in a wide range of settings, sometimes substantially.Comment: 20 pages, 4 figure

Coupling of pion condensate, chiral condensate and Polyakov loop in an extended NJL model

The Nambu Jona-Lasinio model with a Polyakov loop is extended to finite isospin chemical potential case, which is characterized by simultaneous coupling of pion condensate, chiral condensate and Polyakov loop. The pion condensate, chiral condensate and the Polyakov loop as functions of temperature and isospin chemical potential are investigated by minimizing the thermodynamic potential of the system. The resulting $(T,\mu_I)$ phase diagram is studied with emphasis on the critical point and Polyakov loop dynamics. The tricritical point for the pion superfluidity phase transition is confirmed and the phase transition for isospin symmetry restoration in high isospin chemical potential region perfectly coincides with the crossover phase transition for Polyakov loop. These results are in agreement with the Lattice QCD data.Comment: 15pages, 8 figure