The accurate estimation of a precision matrix plays a crucial role in the current age of high-dimensional data explosion. To deal with this problem, one of the prominent and commonly used techniques is the ℓ1 norm (Lasso) penalization for a given loss function. This approach guarantees the sparsity of the precision matrix estimate for properly selected penalty parameters. However, the ℓ1 norm penalization often fails to control the bias of obtained estimator because of its overestimation behavior. In this paper, we introduce two adaptive extensions of the recently proposed ℓ1 norm penalized D-trace loss minimization method. They aim at reducing the produced bias in the estimator. Extensive numerical results, using both simulated and real datasets, show the advantage of our proposed estimators.We would like to thank the Associate Editor, Coordinating Editor and two anonymous referees for their helpful comments that led to an improvement of this article. We express our gratitude to Teng Zhang and Hui Zou for sharing their Matlab code that solves the L1 norm penalized D-trace loss minimization problem. Andrés M. Alonso gratefully acknowledges financial support from CICYT (Spain) Grants ECO2012-38442 and ECO2015-66593. Francisco J. Nogales and Vahe Avagyan were supported by the Spanish Government through project MTM2013-44902-P. This paper is based on the first author's dissertation submitted to the Universidad Carlos III de Madrid. At the time of publication, Vahe Avagyan is a Postdoctoral fellow at Ghent University
