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

nonlinear shrinkage estimation of large integrated covariance matrices integrated covariance matrices arise in intra day models of asset returns which allow volatility to change across the trading day when the number of assets is large the natural estimator of such a matrix suffers from bias contributed from extreme eigenvalues we introduce a novel nonlinear shrinkage estimator for the integrated covariance matrix which shrinks the extreme eigenvalues of a realized covariance matrix back to an acceptable level and enjoys a certain asymptotic efficiency when the number of assets is of the same order as the number of data points novel maximum exposure and actual risk bounds are derived when our estimator is used in constructing the minimum variance portfolio compared to other methods our estimator performs favorably in both simulations and a real data analysis

nonparametric eigenvalue regularized precision or covariance matrix estimator 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

covariance matrix selection and estimation via penalised normal likelihood summary we propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix we reparameterise a covariance matrix through the modified cholesky decomposition of its inverse or the one step ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models the cholesky factor containing these regression coefficients is likely to have many off diagonal elements that are zero or close to zero penalised normal likelihoods in this situation with l1 and l2 penalities are shown to be closely related to tibshirani s 1996 lasso approach and to ridge regression adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the cholesky factor while because of its singularity the l1 penalty will set some elements to zero and produce interpretable models an algorithm is developed for computing the estimator and selecting the tuning parameter the proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix

condition number regularized covariance estimation estimation of high dimensional covariance matrices is known to be a difficult problem has many applications and is of current interest to the larger statistics community in many applications including so called the large p small n setting the estimate of the covariance matrix is required to be not only invertible but also well conditioned although many regularization schemes attempt to do this none of them address the ill conditioning problem directly in this paper we propose a maximum likelihood approach with the direct goal of obtaining a well conditioned estimator no sparsity assumption on either the covariance matrix or its inverse are are imposed thus making our procedure more widely applicable we demonstrate that the proposed regularization scheme is computationally efficient yields a type of steinian shrinkage estimator and has a natural bayesian interpretation we investigate the theoretical properties of the regularized covariance estimator comprehensively including its regularization path and proceed to develop an approach that adaptively determines the level of regularization that is required finally we demonstrate the performance of the regularized estimator in decision theoretic comparisons and in the financial portfolio optimization setting the proposed approach has desirable properties and can serve as a competitive procedure especially when the sample size is small and when a well conditioned estimator is required

finiteness of stationary configurations of the four vortex problem we show that the number of relative equilibria equilibria and rigidly translating configurations in the problem ot tour point vortices is hnite the proof is based on symbolic and exact integer computations which are carried out by computer we also provide upper bounds for these classes of stationary configurations

integrable and chaotic motions of four vortices i the case of identical vortices it is shown that the three vortex problem in two dimensional hydrodynamics is integrable whereas the motion of four identical vortices is not a sequence of canonical transformations is obtained that reduces the n degree of freedom hamiltonian which describes the interaction of n identical vortices to one with n 2 degrees of freedom for n 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions for n 4 the reduction procedure leads to an effective hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators resonant interaction terms in this hamiltonian suggest non integrable behaviour and this is verified by numerical experiments explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible the relevance of the results obtained to fundamental problems in hydrodynamics such as the question of integrability of euler s equation in two dimensions is discussed the paper also contains a general exposition of the hamiltonian and poisson bracket formalism for point vortices

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

concurrence vectors in arbitrary multipartite quantum systems for a given pure state of a multipartite system the concurrence vector is defined by employing the defining representation of generators of the corresponding rotation groups the norm of a concurrence vector is considered as a measure of entanglement for the multipartite pure state the concurrence vector is regarded as the direct sum of concurrence subvectors in the sense that each subvector is associated with a pair of particles it is proposed to use the norm of each subvector as the contribution of the corresponding pair in entanglement of the system

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

design free estimation of variance matrices abstract this paper introduces a new method for estimating variance matrices starting from the orthogonal decomposition of the sample variance matrix we exploit the fact that orthogonal matrices are never ill conditioned and therefore focus on improving the estimation of the eigenvalues we estimate the eigenvectors from just a fraction of the data then use them to transform the data into approximately orthogonal series that deliver a well conditioned estimator by construction even when there are fewer observations than dimensions we also show that our estimator has lower error norms than the traditional one our estimator is design free we make no assumptions on the distribution of the random sample or on any parametric structure the variance matrix may have simulations confirm our theoretical results and they also show that our simple estimator does very well in comparison with other existing 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

nonconcave penalized likelihood with a diverging number of parameters a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by fan and li to simultaneously estimate parameters and select important variables they demonstrated that this class of procedures has an oracle property when the number of parameters is finite however in most model selection problems the number of parameters should be large and grow with the sample size in this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators furthermore the consistency of the sandwich formula of the covariance matrix is demonstrated nonconcave penalized likelihood ratio statistics are discussed and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions the asymptotic results are augmented by a simulation study and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary

a nonparametric eigenvalue regularized integrated covariance matrix estimator for asset return data abstract in high frequency data analysis the extreme eigenvalues of a realized covariance matrix are biased when its dimension p is large relative to the sample size n furthermore with non synchronous trading and contamination of microstructure noise we propose a nonparametrically eigenvalue regularized integrated covariance matrix estimator nerive which does not assume specific structures for the underlying integrated covariance matrix we show that nerive is positive definite in probability with extreme eigenvalues shrunk nonlinearly under the high dimensional framework p n c 0 we also prove that in portfolio allocation the minimum variance optimal weight vector constructed using nerive has maximum exposure and actual risk upper bounds of order p 1 2 incidentally the same maximum exposure bound is also satisfied by the theoretical minimum variance portfolio weights all these results hold true also under a jump diffusion model for the log price processes with jumps removed using the wavelet method proposed in fan and wang 2007 they are further extended to accommodate the existence of pervasive factors such as a market factor under the setting p 3 2 n c 0 the practical performance of nerive is illustrated by comparing to the usual two scale realized covariance matrix as well as some other nonparametric alternatives using different simulation settings and a real data set

nonlinear shrinkage estimation of large dimensional covariance matrices many statistical applications require an estimate of a covariance matrix and or its inverse when the matrix dimension is large compared to the sample size which happens frequently the sample covariance matrix is known to perform poorly and may suffer from ill conditioning there already exists an extensive literature concerning improved estimators in such situations in the absence of further knowledge about the structure of the true covariance matrix the most successful approach so far arguably has been shrinkage estimation shrinking the sample covariance matrix to a multiple of the identity by taking a weighted average of the two turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean while retaining the sample eigenvectors our paper extends this approach by considering nonlinear transformations of the sample eigenvalues we show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work as demonstrated in extensive monte carlo simulations the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage

stationary configurations of point vortices the motion of point vortices in a plane of fluid is an old problem of fluid mechanics which was given a hamiltonian formulation by kirchhoff stationary configurations are those which remain self similar throughout the motion results of two types are presented configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry which establish necessary and sufficient conditions for existence relative equilibria rigidly rotating configurations which lie on a line are studied using a topological construction applicable to other power law systems upper and lower bounds for such configurations are found for vortices with mixed circulations arrangements of three vortices which collide in finite time are well known one dimensional families of such configurations are shown to exist for more than three vortices stationary configurations of four vortices are examined in detail the motion of point vortices in a plane of fluid is an old problem of fluid mechanics which was given a hamiltonian formulation by kirchhoff stationary configurations are those which remain self similar throughout the motion and are of considerable physical interest it is known that a configuration of vortices in equilibrium must have total vortex angular momentum 0 a converse is proved namely that for almost every choice of circulations with zero vortex angular momentum there are exactly n 2 equilibrium configurations a similar statement is proved for rigidly translating configurations with total circulation zero the proofs involve ideas from algebraic geometry relative equilibria rigidly rotating configurations were studied by palmore in the case of positive circulations upper and lower bounds for the number of collinear relative equilibria for arbitrary circulations are obtained by means of a topological construction which is applicable to other power law systems arrangements of three vortices which collide in finite time are well known one dimensional families of such configurations are shown to exist for n vortices n 3 stationary configurations of four vortices are examined in detail chapter 0 introduction the motion of point vortices in the plane is an old problem in fluid mechanics it was first given a hamiltonian formulation by kirchhoff 1876 who proved that n vortices with distinct positions xi yi i 1 n in the x y plane and received by the editors july 31 1985 and in revised form june 13 1986 1980 mathematics subject classification 1985 revision primary 76c05 secondary 70h05 t1987 american mathematical society 0002 9947 87 1 00 25 per page

minimal polynomial systems for point vortex equilibria point vortices in a two dimensional fluid may be arranged so that all resulting vortex velocities are identical these are equivalent to force free arrangements of 2d coulomb charges in a homogeneous field for generic choice of vortex circulations the number of stationary configurations is finite but a few infinite solution families of arbitrary positive dimension are known to exist this paper extends the classes of point vortex systems known to have a finite or infinite number of stationary configurations respectively the main tool is a new minimal system of polynomial equations characterizing these configurations the polynomial system is extended to include symmetric vortex configurations such as vortex streets and rings as well as circulation symmetries vortices with identical circulations by expressing the systems as a multilinear differential equation one dimensional solution sets can be constructed for a wide range of systems including vortex streets and rings

high dimensional covariance matrix estimation in approximate factor models the variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems classical methods of estimating the covariance matrices are based on the strict factor models assuming independent idiosyncratic components this assumption however is restrictive in practical applications by assuming sparse error covariance matrix we allow the presence of the cross sectional correlation even after taking out common factors and it enables us to combine the merits of both methods we estimate the sparse covariance using the adaptive thresholding technique as in cai and liu j amer statist assoc 106 2011 672 684 taking into account the fact that direct observations of the idiosyncratic components are unavailable the impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied

a sieve m theorem for bundled parameters in semiparametric models with application to the efficient estimation in a linear model for censored data in many semiparametric models that are parameterized by two types of parameters a euclidean parameter of interest and an infinite dimensional nuisance parameter the two parameters are bundled together i e the nuisance parameter is an unknown function that contains the parameter of interest as part of its argument for example in a linear regression model for censored survival data the unspecified error distribution function involves the regression coefficients motivated by developing an efficient estimating method for the regression parameters we propose a general sieve m theorem for bundled parameters and apply the theorem to deriving the asymptotic theory for the sieve maximum likelihood estimation in the linear regression model for censored survival data the numerical implementation of the proposed estimating method can be achieved through the conventional gradient based search algorithms such as the newton raphson algorithm we show that the proposed estimator is consistent and asymptotically normal and achieves the semiparametric efficiency bound simulation studies demonstrate that the proposed method performs well in practical settings and yields more efficient estimates than existing estimating equation based methods illustration with a real data example is also provided

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

factor modeling for high dimensional time series inference for the number of factors this paper deals with the factor modeling for high dimensional time series based on a dimension reduction viewpoint under stationary settings the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix and is therefore applicable when the dimension of time series is on the order of a few thousands asymptotic properties of the proposed method are investigated under two settings i the sample size goes to infinity while the dimension of time series is fixed and ii both the sample size and the dimension of time series go to infinity together in particular our estimators for zero eigenvalues enjoy faster convergence or slower divergence rates hence making the estimation for the number of factors easier in particular when the sample size and the dimension of time series go to infinity together the estimators for the eigenvalues are no longer consistent however our estimator for the number of the factors which is based on the ratios of the estimated eigenvalues still works fine furthermore this estimation shows the so called blessing of dimensionality property in the sense that the performance of the estimation may improve when the dimension of time series increases a two step procedure is investigated when the factors are of different degrees of strength numerical illustration with both simulated and real data is also reported

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

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