Efficient Minimax Estimation of a Class of High-Dimensional Sparse Precision Matrices

Accurate estimation of large covariance and precision matrices is an essential prerequisite of portfolio selection and financial risk management. Among factor-based covariance estimators, high-frequency data provides additional information but also introduces noise, which could degrade the estimation of betas. In this paper, we propose a new Mixed-frequency and FActor-based (MFACE) combining high-frequency (intraday) data and low-frequency (daily) betas. We apply approximate factor structure to construct the high-dimensional minimum variance portfolio. We establish the consistency and obtain the convergence rate of the covariance estimator and the corresponding precision matrix estimator. A comprehensive simulation study investigates the estimation accuracy under different combinations of intrady sampling frequency, intraday sample size and daily sample size. Out-of-sample forecasts demonstrate that our estimator explains the daily volatility of the equally-weighted portfolio pretty well. The minimum variance portfolio that embodies low-frequency betas also achieves the minimum risk at relatively low cost among several estimators.

ABSTRACTWe propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrix estimators to have equal entries and also encourage zeros in the precision matrix estimator. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze three real datasets to demonstrate our method’s applicability. Supplementary materials, including an R package implementing our method, are available online.

In this paper, we consider the estimation of a high dimensional precision matrix of Gaussian graphical model. Based on the re-parameterized likelihood, we obtain the full conditional distribution of all parameters in Cholesky factor. Furthermore, by imposing the prior information, we obtain the shrinkage Bayesian estimator of large precision matrix, and establish the asymptotic distribution of all parameters in the Cholesky factor. At last, we demonstrate our method through the simulation study and an application to telephone call center data.

Estimation of sparse covariance matrices and their inverse subject to positive definiteness constraints has drawn a lot of attention in recent years. The abundance of high-dimensional data, where the sample size (n) is less than the dimension (d), requires shrinkage estimation methods since the maximum likelihood estimator is not positive definite in this case. Furthermore, when n is larger than d but not sufficiently larger, shrinkage estimation is more stable than maximum likelihood as it reduces the condition number of the precision matrix. Frequentist methods have utilized penalized likelihood methods, whereas Bayesian approaches rely on matrix decompositions or Wishart priors for shrinkage. In this paper we propose a new method, called the Bayesian Covariance Lasso (BCLASSO), for the shrinkage estimation of a precision (covariance) matrix. We consider a class of priors for the precision matrix that leads to the popular frequentist penalties as special cases, develop a Bayes estimator for the precision matrix, and propose an efficient sampling scheme that does not precalculate boundaries for positive definiteness. The proposed method is permutation invariant and performs shrinkage and estimation simultaneously for non-full rank data. Simulations show that the proposed BCLASSO performs similarly as frequentist methods for non-full rank data.

This paper considers the ridge-type shrinkage estimation of the large dimensional precision matrix. The asymptotic optimal shrinkage coefficients and the theoretical loss are derived. Data-driven estimators for the shrinkage coefficients are also conducted based on the asymptotic results from random matrix theory. The new method is distribution-free and no assumption on the structure of the covariance matrix or the precision matrix is required. The proposed method also applies to situations where the dimension is larger than the sample size. Finally, numerical studies using simulated data and real data demonstrate that the proposed estimator performs better than the existing competitors in a wide range of settings.

The determination of the covariance matrix and its inverse, the precision matrix, is critical in the statistical analysis of cosmological measurements. The covariance matrix is typically estimated with a limited number of simulations at great computational cost before inversion into the precision matrix; therefore, it can be ill-conditioned and overly noisy when the sample size n used for estimation is not much larger than the data vector dimension. In this work, we consider a class of methods known as shrinkage estimation for the precision matrix, which combines an empirical estimate with a target that is either analytical or stochastic. These methods include linear and non-linear shrinkage applied to the covariance matrix (the latter represented by the so-called NERCOME estimator), and the direct linear shrinkage estimation of the precision matrix which we introduce in a cosmological setting. By performing Bayesian parameter inference and using metrics like matrix loss functions, the Kullback–Leibler divergence and the eigenvalue spectrum, we compare their performance against the standard sample estimator with varying sample size n. We have found the shrinkage estimators to significantly improve the posterior distribution at low n, especially for the linear shrinkage estimators either inverted from the covariance matrix or applied directly to the precision matrix, with an empirical target constructed from the sample estimate. Our results are particularly relevant to the analyses of Stage-IV spectroscopic galaxy surveys such as the Dark Energy Spectroscopic Instrument and Euclid, whose statistical power can be limited by the computational cost of obtaining an accurate precision matrix estimate.

Estimating large covariance and precision (inverse covariance) matrices has become increasingly important in high dimensional statistics because of its wide applications. The estimation problem is challenging not only theoretically due to the constraint of its positive definiteness, but also computationally because of the curse of dimensionality. Many types of estimators have been proposed such as thresholding under the sparsity assumption of the target matrix, banding and tapering the sample covariance matrix. However, these estimators are not always guaranteed to be positive-definite, especially, for finite samples, and the sparsity assumption is rather restrictive. We propose a novel two-stage adaptive method based on the Cholesky decomposition of a general covariance matrix. By banding the precision matrix in the first stage and adapting the estimates to the second stage estimation, we develop a computationally efficient and statistically accurate method for estimating high dimensional precision matrices. We demonstrate the finite-sample performance of the proposed method by simulations from autoregressive, moving average, and long-range dependent processes. We illustrate its wide applicability by analyzing financial data such S&P 500 index and IBM stock returns, and electric power consumption of individual households. The theoretical properties of the proposed method are also investigated within a large class of covariance matrices.

Large-scale precision matrix estimation is of fundamental importance yet challenging in many contemporary applications for recovering Gaussian graphical models. In this paper, we suggest a new approach of innovated scalable efficient estimation (ISEE) for estimating large precision matrix. Motivated by the innovated transformation, we convert the original problem into that of large covariance matrix estimation. The suggested method combines the strengths of recent advances in high-dimensional sparse modeling and large covariance matrix estimation. Compared to existing approaches, our method is scalable and can deal with much larger precision matrices with simple tuning. Under mild regularity conditions, we establish that this procedure can recover the underlying graphical structure with significant probability and provide efficient estimation of link strengths. Both computational and theoretical advantages of the procedure are evidenced through simulation and real data examples.

In this article, we consider the problem of estimating the precision matrix when the sample data contains cellwise contamination. For the widely employed methodologies (e.g. Graphical Lasso), using the sample covariance matrix as an input matrix potentially deteriorates the precision matrix estimation performance in the presence of outliers. We propose several robust alternatives for the covariance matrix, which are constructed by combining robust correlation estimators with robust variation measures. Through extensive numerical studies we demonstrate the robust performance of our proposed approaches compared to the standard methods based on the sample covariance matrix. Further, we apply our proposals to a real data application, aimed at studying the optimal portfolio allocations in Shanghai Stock Exchange Composite index. The results show that the proposed alternatives provide desirable out-of-sample performance.

In many engineering applications, estimation of covariance and precision matrices is of great importance, helping researchers understand the dependency and conditional dependency between variables of interest. Among various matrix estimation methods, the modified Cholesky decomposition is a commonly used technique. It has the advantage of transforming the matrix estimation task into solving a sequence of regression models. Moreover, the sparsity on the regression coefficients implies certain sparse structure on the covariance and precision matrices. In this chapter, we first overview the Cholesky-based covariance and precision matrices estimation. It is known that the Cholesky-based matrix estimation depends on a prespecified ordering of variables, which is often not available in practice. To address this issue, we then introduce several techniques to enhance the Cholesky-based estimation of covariance and precision matrices. These approaches are able to ensure the positive definiteness of the matrix estimate and applicable in general situations without specifying the ordering of variables. The advantage of Cholesky-based estimation is illustrated by numerical studies and several real-case applications.

Graphical models are well established in providing compact conditional probability descriptions of complex multivariable interactions. In the Gaussian case, graphical models are determined by zeros in the precision or concentration matrix, i.e. the inverse of the covariance matrix. Hence, there has been much recent interest in sparse precision matrices in areas such as statistics, machine learning, computer vision, pattern recognition and signal processing. In this paper we propose a simple new algorithm for constructing a sparse estimator for the precision matrix from multivariate data where the sparsity is enforced by an l 0 penalty. We compare and test the quality of our method on a synthetic graphical model.

In this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries. The estimators are fully data-driven and demonstrate excellent performance both theoretically and numerically. It is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm. In contrast, the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces. Support recovery is discussed as well. The adaptive thresholding estimators are easy to implement. The numerical performance of the estimators is studied using both simulated and real data. Simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators. The method is also illustrated in an analysis on a dataset from a small round blue-cell tumor microarray experiment. A supplement to this article presenting additional technical proofs is available online.

A sparse precision matrix can be directly translated into a sparse Gaussian graphical model under the assumption that the data follow a joint normal distribution. This neat property makes high-dimensional precision matrix estimation very appealing in many applications. However, in practice we often face nonnormal data, and variable transformation is often used to achieve normality. In this paper we consider the nonparanormal model that assumes that the variables follow a joint normal distribution after a set of unknown monotone transformations. The nonparanormal model is much more flexible than the normal model while retaining the good interpretability of the latter in that each zero entry in the sparse precision matrix of the nonparanormal model corresponds to a pair of conditionally independent variables. In this paper we show that the nonparanormal graphical model can be efficiently estimated by using a rank-based estimation scheme which does not require estimating these unknown transformation functions. In particular, we study the rank-based graphical lasso, the rank-based neighborhood Dantzig selector and the rank-based CLIME. We establish their theoretical properties in the setting where the dimension is nearly exponentially large relative to the sample size. It is shown that the proposed rank-based estimators work as well as their oracle counterparts defined with the oracle data. Furthermore, the theory motivates us to consider the adaptive version of the rank-based neighborhood Dantzig selector and the rank-based CLIME that are shown to enjoy graphical model selection consistency without assuming the irrepresentable condition for the oracle and rank-based graphical lasso. Simulated and real data are used to demonstrate the finite performance of the rank-based estimators.

We introduce nonparametric regularization of the eigenvalues of a sample covariance matrix through splitting of the data (NERCOME), and prove that NERCOME enjoys asymptotic optimal nonlinear shrinkage of eigenvalues with respect to the Frobenius norm. One advantage of NERCOME is its computational speed when the dimension is not too large. We prove that NERCOME is positive definite almost surely, as long as the true covariance matrix is so, even when the dimension is larger than the sample size. With respect to the Stein’s loss function, the inverse of our estimator is asymptotically the optimal precision matrix estimator. Asymptotic efficiency loss is defined through comparison with an ideal estimator, which assumed the knowledge of the true covariance matrix. We show that the asymptotic efficiency loss of NERCOME is almost surely 0 with a suitable split location of the data. We also show that all the aforementioned optimality holds for data with a factor structure. Our method avoids the need to first estimate any unknowns from a factor model, and directly gives the covariance or precision matrix estimator, which can be useful when factor analysis is not the ultimate goal. We compare the performance of our estimators with other methods through extensive simulations and real data analysis.

In this paper, we study the problem of precision matrix estimation when the dataset contains sensitive information. In the differential privacy framework, we develop a differentially private ridge estimator by perturbing the sample covariance matrix. Then we develop a differentially private graphical lasso estimator by using the alternating direction method of multipliers (ADMM) algorithm. Furthermore, we prove theoretical results showing that the differentially private ridge estimator for the precision matrix is consistent under fixed-dimension asymptotic, and establish a convergence rate of differentially private graphical lasso estimator in the Frobenius norm as both data dimension p and sample size n are allowed to grow. The empirical results that show the utility of the proposed methods are also provided.