covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

multitask quantile regression under the transnormal model abstractwe consider estimating multitask quantile regression under the transnormal model with focus on high dimensional setting we derive a surprisingly simple closed form solution through rank based covariance regularization in particular we propose the rank based l1 penalization with positive definite constraints for estimating sparse covariance matrices and the rank based banded cholesky decomposition regularization for estimating banded precision matrices by taking advantage of the alternating direction method of multipliers nearest correlation matrix projection is introduced that inherits sampling properties of the unprojected one our work combines strengths of quantile regression and rank based covariance regularization to simultaneously deal with nonlinearity and nonnormality for high dimensional regression furthermore the proposed method strikes a good balance between robustness and efficiency achieves the oracle like convergence rate and provides the provable prediction interval und

shrinking characteristics of precision matrix estimators we propose a framework to shrink a user specified characteristic of a precision matrix estimator that is needed to fit a predictive model estimators in our framework minimize the gaussian negative loglikelihood plus an l 1 penalty on a linear or affine function evaluated at the optimization variable corresponding to the precision matrix we establish convergence rate bounds for these estimators and propose an alternating direction method of multipliers algorithm for their computation our simulation studies show that our estimators can perform better than competitors when they are used to fit predictive models in particular we illustrate cases where our precision matrix estimators perform worse at estimating the population precision matrix but better at prediction

admm for high dimensional sparse penalized quantile regression abstractsparse penalized quantile regression is a useful tool for variable selection robust estimation and heteroscedasticity detection in high dimensional data analysis the computational issue of the sparse penalized quantile regression has not yet been fully resolved in the literature due to non smoothness of the quantile regression loss function we introduce fast alternating direction method of multipliers admm algorithms for computing the sparse penalized quantile regression the convergence properties of the proposed algorithms are established numerical examples demonstrate the competitive performance of our algorithm it significantly outperforms several other fast solvers for high dimensional penalized quantile regression supplementary materials for this article are available online

generalized kahler potentials from supergravity we consider supersymmetric mathcal n 2 solutions with non vanishing ns three form building on worldsheet results we reduce the problem to a single generalized monge ampere equation on the generalized kahler potential k recently interpreted geometrically by lindstrom rocek von unge and zabzine one input in the procedure is a holomorphic function w that can be thought of as the effective superpotential for a d3 brane probe the procedure is hence likely to be useful for finding gravity duals to field theories with non vanishing abelian superpotential such as leigh strassler theories we indeed show that a purely ns precursor of the lunin maldacena dual to the deformed mathcal n 4 super yang mills falls in our class

an adaptive two sample test for high dimensional means several two sample tests for high dimensional data have been proposed recently but they are powerful only against certain limited alternative hypotheses in practice since the true alternative hypothesis is unknown it is unclear how to choose a powerful test we propose an adaptive test that maintains high power across a wide range of situations and study its asymptotic properties its finite sample performance is compared with existing tests we apply it and other tests to detect possible associations between bipolar disease and a large number of single nucleotide polymorphisms on each chromosome based on a genome wide association study dataset numerical studies demonstrate the superior performance and high power of the proposed test across a wide spectrum of applications

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

superstrings and topological strings at large n we embed the large n chern simons topological string duality in ordinary superstrings this corresponds to a large n duality between generalized gauge systems with n 1 supersymmetry in four dimensions and superstrings propagating on noncompact calabi yau manifolds with certain fluxes turned on we also show that in a particular limit of the n 1 gauge theory system certain superpotential terms in the n 1 system including deformations if spacetime is noncommutative are captured to all orders in 1 n by the amplitudes of noncritical bosonic strings propagating on a circle with self dual radius we also consider d brane anti d brane system wrapped over vanishing cycles of compact calabi yau manifolds and argue that at large n they induce a shift in the background to a topologically distinct calabi yau which we identify as the ground state system of the brane anti brane system

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

a multiple testing approach to the regularisation of large sample correlation matrices abstract this paper proposes a regularisation method for the estimation of large covariance matrices that uses insights from the multiple testing m t literature the approach tests the statistical significance of individual pair wise correlations and sets to zero those elements that are not statistically significant taking account of the multiple testing nature of the problem the effective p values of the tests are set as a decreasing function of n the cross section dimension the rate of which is governed by the nature of dependence of the underlying observations and the relative expansion rates of n and t the time dimension in this respect the method specifies the appropriate thresholding parameter to be used under gaussian and non gaussian settings the m t estimator of the sample correlation matrix is shown to be consistent in the spectral and frobenius norms and in terms of support recovery so long as the true covariance matrix is sparse the performance of the proposed m t estimator is compared to a number of other estimators in the literature using monte carlo experiments it is shown that the m t estimator performs well and tends to outperform the other estimators particularly when n is larger than t

a well conditioned and sparse estimation of covariance and inverse covariance matrices using a joint penalty we develop a method for estimating well conditioned and sparse covariance and inverse covariance matrices from a sample of vectors drawn from a sub gaussian distribution in high dimensional setting the proposed estimators are obtained by minimizing the quadratic loss function and joint penalty of l1 norm and variance of its eigenvalues in contrast to some of the existing methods of covariance and inverse covariance matrix estimation where often the interest is to estimate a sparse matrix the proposed method is flexible in estimating both a sparse and well conditioned covariance matrix simultaneously the proposed estimators are optimal in the sense that they achieve the mini max rate of estimation in operator norm for the underlying class of covariance and inverse covariance matrices we give a very fast algorithm for computation of these covariance and inverse covariance matrices which is easily scalable to large scale data analysis problems the simulation study for varying sample sizes and variables shows that the proposed estimators performs better than several other estimators for various choices of structured covariance and inverse covariance matrices we also use our proposed estimator for tumor tissues classification using gene expression data and compare its performance with some other classification methods

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

generalized complex manifolds and supersymmetry we find a worldsheet realization of generalized complex geometry a notion introduced recently by hitchin which interpolates between complex and symplectic manifolds the two dimensional model we construct is a supersymmetric relative of the poisson sigma model used in the context of deformation quantization

the impact of dynamic events on the number of errors in networks in order to achieve routing in a graph nodes need to store routing information in the case of shortest path routing for a given destination every node has to store an advice that is an outgoing link toward a neighbor if this neighbor does not belong to a shortest path then the advice is considered as an error and the node giving this advice will be qualified a liar this article focuses on the impact of graph dynamics on the advice set for a given destination more precisely we show that for a weighted graph g of diameter d with n nodes and m edges the expected number of errors after m edge deletions is bounded by o n m d m we also show that this bound is tight when m o n moreover for m node deletions the expected number of errors is o m d finally we show that after a single edge addition the expected number of liars can be n for some families of graphs

oracles for distances avoiding a failed node or link we consider the problem of preprocessing an edge weighted directed graph g to answer queries that ask for the length and first hop of a shortest path from any given vertex x to any given vertex y avoiding any given vertex or edge as a natural application this problem models routing in networks subject to node or link failures we describe a deterministic oracle with constant query time for this problem that uses o n 2 log n space where n is the number of vertices in g the construction time for our oracle is o mn 2 n 3 log n however if one is willing to settle for theta n 2 5 space we can improve the preprocessing time to o mn 1 5 n 2 5 log n while maintaining the constant query time our algorithms can find the shortest path avoiding a failed node or link in time proportional to the length of the path

mirror symmetric su 3 structure manifolds with ns fluxes when string theory is compactified on a six dimensional manifold with a nontrivial ns flux turned on mirror symmetry exchanges the flux with a purely geometrical composite ns form associated with lack of integrability of the complex structure on the mirror side considering a general class of t3 fibered geometries admitting su 3 structure we find an exchange of pure spinors e ij and in dual geometries under fiberwise t duality and study the transformations of the ns flux and the components of intrinsic torsion a complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with ns flux

fault tolerant subgraph for single source reachability general and optimal let g be a directed graph with n vertices m edges and a designated source vertex s we address the problem of single source reachability ssr from s in the presence of failures of vertices edges we show that for every k geq1 there is a subgraph h of g with at most 2 kn edges that preserves the reachability from s even after the failure of any k edges formally given a set f of k edges a vertex v in v g is reachable from s in g setminus f if and only if v is reachable from s in h setminus f we call h a k fault tolerant reachability subgraph textsc k ftrs we also prove a matching lower bound of omega 2 kn edges for such subgraphs that holds for all n k with 2 k leq n our results extend to vertex failures without any extra overhead the construction of k ftrs is interesting from several different perspectives from the graph theory perspective it reveals a separation between ssr and single source shortest paths sssp in directed grap

adaptive thresholding for sparse covariance matrix estimation 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 ar

positive definite l1 penalized estimation of large covariance matrices the thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices but it often has negative eigenvalues when used in real data analysis to fix this drawback of thresholding estimation we develop a positive definite l1 penalized covariance estimator for estimating sparse large covariance matrices we derive an efficient alternating direction method to solve the challenging optimization problem and establish its convergence properties under weak regularity conditions nonasymptotic statistical theory is also established for the proposed estimator the competitive finite sample performance of our proposal is demonstrated by both simulation and real applications

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

a new approach to dynamic all pairs shortest paths we study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems our approach yields a fully dynamic algorithm for general directed graphs with non negative real valued edge weights that supports any sequence of operations in o n2log3n amortized time per update and unit worst case time per distance query where n is the number of vertices we can also report shortest paths in optimal worst case time these bounds improve substantially over previous results and solve a long standing open problem our algorithm is deterministic uses simple data structures and appears to be very fast in practice

sparse covariance matrix estimation by dca based algorithms this letter proposes a novel approach using the formula see text norm regularization for the sparse covariance matrix estimation scme problem the objective function of scme problem is composed of a nonconvex part and the formula see text term which is discontinuous and difficult to tackle appropriate dc difference of convex functions approximations of formula see text norm are used that result in approximation scme problems that are still nonconvex dc programming and dca dc algorithm powerful tools in nonconvex programming framework are investigated two dc formulations are proposed and corresponding dca schemes developed two applications of the scme problem that are considered are classification via sparse quadratic discriminant analysis and portfolio optimization a careful empirical experiment is performed through simulated and real data sets to study the performance of the proposed algorithms numerical results showed their efficiency and their superiority compared with seven state of the art methods

alternating direction methods for latent variable gaussian graphical model selection chandrasekaran parrilo and willsky 2012 proposed a convex optimization problem for graphical model selection in the presence of unobserved variables this convex optimization problem aims to estimate an inverse covariance matrix that can be decomposed into a sparse matrix minus a low rank matrix from sample data solving this convex optimization problem is very challenging especially for large problems in this letter we propose two alternating direction methods for solving this problem the first method is to apply the classic alternating direction method of multipliers to solve the problem as a consensus problem the second method is a proximal gradient based alternating direction method of multipliers our methods take advantage of the special structure of the problem and thus can solve large problems very efficiently a global convergence result is established for the proposed methods numerical results on both synthetic data and gene expression data show that our methods usually solve problems with 1 million variables in 1 to 2 minutes and are usually 5 to 35i times faster than a state of the art newton cg proximal point algorithm

dynamic covariance models abstractan important problem in contemporary statistics is to understand the relationship among a large number of variables based on a dataset usually with p the number of the variables much larger than n the sample size recent efforts have focused on modeling static covariance matrices where pairwise covariances are considered invariant in many real systems however these pairwise relations often change to characterize the changing correlations in a high dimensional system we study a class of dynamic covariance models dcms assumed to be sparse and investigate for the first time a unified theory for understanding their nonasymptotic error rates and model selection properties in particular in the challenging high dimensional regime we highlight a new uniform consistency theory in which the sample size can be seen as n4 5 when the bandwidth parameter is chosen as h n 1 5 for accounting for the dynamics we show that this result holds uniformly over a range of the variable used for modeling t

positive definite estimators of large covariance matrices using convex optimization we construct a sparse estimator of the covariance matrix that is positive definite and performs well in high dimensional settings a lasso type penalty is used to encourage sparsity and a logarithmic barrier function is used to enforce positive definiteness consistency and convergence rate bounds are established as both the number of variables and sample size diverge an efficient computational algorithm is developed and the merits of the approach are illustrated with simulations and a speech signal classification example copyright 2012 oxford university press

efficient estimation of approximate factor models via penalized maximum likelihood we study the estimation of a high dimensional approximate factor model in the presence of both cross sectional dependence and heteroskedasticity the classical method of principal components analysis pca does not efficiently estimate the factor loadings or common factors because it essentially treats the idiosyncratic error to be homoskedastic and cross sectionally uncorrelated for the efficient estimation it is essential to estimate a large error covariance matrix we assume the model to be conditionally sparse and propose two approaches to estimating the common factors and factor loadings both are based on maximizing a gaussian quasi likelihood and involve regularizing a large covariance sparse matrix in the first approach the factor loadings and the error covariance are estimated separately while in the second approach they are estimated jointly extensive asymptotic analysis has been carried out in particular we develop the inferential theory for the two step estimation because the proposed approaches take into account the large error covariance matrix they produce more efficient estimators than the classical pca methods or methods based on a strict factor model

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

high dimensionality effects in the markowitz problem and other quadratic programs with linear constraints risk underestimation we study the properties of solutions of quadratic programs with linear equality constraints whose parameters are estimated from data in the high dimensional setting where p the number of variables in the problem is of the same order of magnitude as n the number of observations used to estimate the parameters the markowitz problem in finance is a subcase of our study assuming normality and independence of the observations we relate the ecient frontier computed empirically to the true ecient frontier our computations show that there is a separation of the errors induced by estimating the mean of the observations and estimating the covariance matrix in particular the price paid for estimating the covariance matrix is an underestimation of the variance by a factor roughly equal to 1 p n therefore the risk of the optimal population solution is underestimated when we estimate it by solving a similar quadratic program with estimated parameters we also characterize the statistical behavior of linear functionals of the empirical optimal vector and show that they are biased estimators of the corresponding population quantities we investigate the robustness of our gaussian results by extending the study to certain elliptical models and models where our n observations are correlated in time we show a lack of robustness of the gaussian results but are still able to get results concerning rst order properties of the quantities of interest even in the case of relatively heavy tailed data we require two moments risk underestimation is still present in the elliptical case and more pronounced that in the gaussian case we discuss properties of the non parametric and parametric bootstrap in this context we show several results including the interesting fact that standard applications of the bootstrap generally yields inconsistent estimates of bias finally we propose some strategies to correct these problems and practically validate them in some simulations in all the paper we will assume that p n and n p tend to innity and p n

geometric u folds in four dimensions we describe a general construction of geometric u folds compatible with the global formulation of four dimensional extended supergravity on a differentiable spin manifold the topology of geometric u folds depends on certain fiber bundles which encode how supergravity fields are globally glued together smooth non trivial u folds of this type can exist only in theories where both the scalar and space time manifolds have non trivial fundamental group and in addition the configuration of scalar fields of the solution is homotopically non trivial nonetheless certain geometric u folds extend to simply connected backgrounds containing localized sources consistency with string theory requires smooth geometric u folds to be glued using subgroups of the effective discrete u duality group implying that the fundamental group of the scalar manifold of such solutions must be a subgroup of the latter we construct simple examples of geometric u folds in a generalization of the axiondilaton model of n 2 supergravity coupled to a single vector multiplet whose scalar manifold is a generally non compact riemann surface of genus at least two endowed with its uniformizing metric we also discuss the relation between geometric u folds and a moduli space of flat connections defined on the scalar manifold which involves certain character varieties not studied in the literature

covariance regression analysis abstractthis article introduces covariance regression analysis for a p dimensional response vector the proposed method explores the regression relationship between the p dimensional covariance matrix and auxiliary information we study three types of estimators maximum likelihood ordinary least squares and feasible generalized least squares estimators then we demonstrate that these regression estimators are consistent and asymptotically normal furthermore we obtain the high dimensional and large sample properties of the corresponding covariance matrix estimators simulation experiments are presented to demonstrate the performance of both regression and covariance matrix estimates an example is analyzed from the chinese stock market to illustrate the usefulness of the proposed covariance regression model supplementary materials for this article are available online