Estimation of Large Precision Matrices Through Block Penalization
| dc.creator | Lam, Clifford | |
| dc.date | 2008-05-26 | |
| dc.date.accessioned | 2026-07-07T09:40:50Z | |
| dc.date.available | 2026-07-07T09:40:50Z | |
| dc.description | This paper focuses on exploring the sparsity of the inverse covariance matrix $\bSigma^{-1}$, or the precision matrix. We form blocks of parameters based on each off-diagonal band of the Cholesky factor from its modified Cholesky decomposition, and penalize each block of parameters using the $L_2$-norm instead of individual elements. We develop a one-step estimator, and prove an oracle property which consists of a notion of block sign-consistency and asymptotic normality. In particular, provided the initial estimator of the Cholesky factor is good enough and the true Cholesky has finite number of non-zero off-diagonal bands, oracle property holds for the one-step estimator even if $p_n \gg n$, and can even be as large as $\log p_n = o(n)$, where the data $\y$ has mean zero and tail probability $P(|y_j| > x) \leq K\exp(-Cx^d)$, $d > 0$, and $p_n$ is the number of variables. We also prove an operator norm convergence result, showing the cost of dimensionality is just $\log p_n$. The advantage of this method over banding by Bickel and Levina (2008) or nested LASSO by Levina \emph{et al.} (2007) is that it allows for elimination of weaker signals that precede stronger ones in the Cholesky factor. A method for obtaining an initial estimator for the Cholesky factor is discussed, and a gradient projection algorithm is developed for calculating the one-step estimate. Simulation results are in favor of the newly proposed method and a set of real data is analyzed using the new procedure and the banding method. | |
| dc.description | 42 pages article | |
| dc.identifier | https://arxiv.org/abs/0805.3798 | |
| dc.identifier | http://arxiv.org/abs/0805.3798 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/161614 | |
| dc.subject | Methodology | |
| dc.subject | Statistics Theory | |
| dc.title | Estimation of Large Precision Matrices Through Block Penalization | |
| dc.type | text |