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Register a new userMinimax Estimation of Large Precision Matrices with Bandable Cholesky Factor
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Added by
Zhao Ren
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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Last decade witnesses significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the variables can be interpreted through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this paper, we focus on two commonly used types of parameter spaces, and develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found: two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm, which is in sharp contrast to the equivalence of corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. Two new estimation procedures are developed: for the operator norm, our optimal procedure is based on a novel local cropping estimator targeting on all principle submatrices of the precision matrix while for the Frobenius norm, our optimal procedure relies on a delicate regression-based thresholding rule. Lepskis method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model. Numerical studies are carried out to confirm our theoretical findings.
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We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian or heavier-tailed observations with slowly decaying temporal dependence. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. We show that several commonly used methods for independent observations can be applied to the temporally dependent data. In particular, the rates of convergence are obtained for the generalized thresholding estimation of covariance and correlation matrices, and for the constrained $ell_1$ minimization and the $ell_1$ penalized likelihood estimation of precision matrix. Properties of sparsistency and sign-consistency are also established. A gap-block cross-validation method is proposed for the tuning parameter selection, which performs well in simulations. As a motivating example, we study the brain functional connectivity using resting-state fMRI time series data with long-range temporal dependence.
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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 $P(|y_j| > x) leq Kexp(-Cx^d)$, $d > 0$, 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.
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