Resource title

Estimation of large precision matrices through block penalization

Resource image

image for OpenScout resource :: Estimation of large precision matrices through block penalization

Resource 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.

Resource author

Resource publisher

Resource publish date

Resource language


Resource content type


Resource resource URL

Resource license