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2 High-dimensional covariance matrix estimators 8 2.1 Literature review............................. 8 2.2 Theoretical outline............................ 12

is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs. Neighborhood selection estimates the conditional independence restrictions separately for each node in the graph and is hence equivalent to variable selection for Gaussian linear models. We

Estimating a covariance matrix is an important task in applications where the number of vari-ables is larger than the number of observations. In the literature, shrinkage approaches for estimat-ing a high-dimensional covariance matrix are employed to circumvent the limitations of the sample

This paper presents a parameter covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic. This estimator does not depend on a formal model of the structure of the heteroskedasticity. By comparing the elements of the new estimator

component analysis, prediction, discriminant analysis among others. In this paper we consider both low- and high-dimensional covariance matrix estimation problems and present a review for asymptotic properties of sample covariances and covariance matrix estimates. In particular, we shall provide

of basis functions. Using a matrix regression model, we propose a new methodology for high-dimensional covariance matrix estimation based on empirical contrast regularization by a group Lasso penalty. Using such a penalty, the method selects a sparse set of basis functions in the dictionary used

covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method

A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this "variance equation " completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary

into account the auto-correlation of the data) works well when the number of states is large enough. Two corrections based on asymptotic approximation of the variance-covariance matrix work well for moderate numbers of states and one correction that collapses the time series information into a “pre” and “post