Optimal estimations for sparse precision matrices under non-standard moment conditions

Estimation of precision matrix plays a significant role for many applications in multivariate analysis. Cai–Liu–Zhou establish a pilot estimation for sample covariance matrix under the standard and exponential moment condition. Then the authors provide an optimal mean estimation for a family of sparse precision matrices by using the adaptive constrained [Formula: see text] minimization estimator (ACLIME), see [T. T. Cai, W. Liu and H. H. 23 Zhou, Estimating sparse precision matrix: optimal rates of convergence and adaptive 24 estimation, Ann. Statist. 44(2) (2016) 455–488]. This paper gives first a pilot estimation for sample covariance matrix under the non-standard and exponential moment condition. It turns that our pilot estimation implies Cai–Liu–Zhou’s. Examples show that our non-standard condition and Cai–Liu–Zhou’s standard one don’t contain each other. Then we show optimal probability and mean estimations for a class of precision matrices by using the constrained [Formula: see text] minimization estimator (CLIME) prososed by Cai-Liu-Luo in [A constrained l1 minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106(2011), 594–607]. Since the exponential moment condition is much restricted, Avella-Medina, Battey, Fan and Li construct the Huber pilot estimator under the moment condition of order 4, and study the optimal probability estimation for precision matrices whose eigenvalues [Formula: see text] satisfy [Formula: see text], see [M. Avella-Medina, H. S. Battey, J. Fan and Q. Li, Robust estimation of high-dimensional covariance and precision matrices, Biometrika 105(2) (2018) 271–284]. Motivated by their work, we prove that the truncated sample covariance matrix provides another pilot estimator under the non-standard moment condition of order [Formula: see text]. Moreover, both probability and mean estimations are given by using the CLIME estimator. Although our moment condition is a little stronger, we don’t assume [Formula: see text].