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

optimal detection of sparse principal components in high dimension we perform a finite sample analysis of the detection levels for sparse principal components of a high dimensional covariance matrix our minimax optimal test is based on a sparse eigenvalue statistic alas computing this test is known to be np complete in general and we describe a computationally efficient alternative test using convex relaxations our relaxation is also proved to detect sparse principal components at near optimal detection levels and performs very well on simulated datasets ams 2000 subject classifications primary 62h25 secondary 62f04 90c22

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

asymptotics of empirical eigenstructure for high dimensional spiked covariance we derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime which takes into account the magnitude of spiked eigenvalues sample size and dimensionality this regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues sample size and dimensionality play in principal component analysis our results are a natural extension of those in paul 2007 to a more general setting and solve the rates of convergence problems in shen et al 2013 they also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis and lead to a new covariance estimator for the approximate factor model called shrinkage principal orthogonal complement thresholding s poet that corrects the biases our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies

an exact approach to the oscillator radiation process in an arbitrarily large cavity starting from a solution of the problem of a mechanical oscillator coupled to a scalar field inside a reflecting sphere of radius r we study the behaviour of the system in free space as the limit of an arbitrarily large radius in the confined solution from a mathematical point of view we show that this way of addressing the problem is not equivalent to considering the system a priori embedded in infinite space in particular the matrix elements of the transformation turning the system to the principal axis do not tend to distributions in the limit of an arbitrarily large sphere as should be the case if the two procedures were mathematically equivalent also we introduce dressed coordinates which allow an exact description of the oscillator radiation process expanding in powers of the coupling constant we recover from our exact expressions the well known decay formulae from perturbation theory

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

optimal structural design considering flexibility a topology optimization method based on the homogenization method the homogenization design method has been successfully applied to a variety of optimization problems such as the stiffness maximization problem and the eigen frequency maximization problem in this study a methodology to obtain the optimal structure design considering flexibility is developed as a new extension of the homogenization design method first flexibility is formulated using the mutual energy concept second a new multi objective function is proposed to obtain optimal solutions incorporating flexibility and stiffness next the topology optimization procedure is constructed using the homogenization method and sequential linear programming slp finally some examples are presented to confirm that the methodology presented here can provide flexible structures satisfying the problem specifications

risks of large portfolios the risk of a large portfolio is often estimated by substituting a good estimator of the volatility matrix however the accuracy of such a risk estimator is largely unknown we study factor based risk estimators under a large amount of assets and introduce a high confidence level upper bound h club to assess the estimation the h club is constructed using the confidence interval of risk estimators with either known or unknown factors we derive the limiting distribution of the estimated risks in high dimensionality we find that when the dimension is large the factor based risk estimators have the same asymptotic variance no matter whether the factors are known or not which is slightly smaller than that of the sample covariance based estimator numerically h club outperforms the traditional crude bounds and provides an insightful risk assessment in addition our simulated results quantify the relative error in the risk estimation which is usually negligible using 3 month daily data

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

projected principal component analysis in factor models this paper introduces a projected principal component analysis projected pca which employees principal component analysis to the projected smoothed data matrix onto a given linear space spanned by covariates when it applies to high dimensional factor analysis the projection removes noise components we show that the unobserved latent factors can be more accurately estimated than the conventional pca if the projection is genuine or more precisely when the factor loading matrices are related to the projected linear space when the dimensionality is large the factors can be estimated accurately even when the sample size is finite we propose a flexible semi parametric factor model which decomposes the factor loading matrix into the component that can be explained by subject specific covariates and the orthogonal residual component the covariates effects on the factor loadings are further modeled by the additive model via sieve approximations by using the newly proposed projected pca the rates of convergence of the smooth factor loading matrices are obtained which are much faster than those of the conventional factor analysis the convergence is achieved even when the sample size is finite and is particularly appealing in the high dimension low sample size situation this leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings the proposed method is illustrated by both simulated data and the returns of the components of the s p 500 index

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

finite element hourglassing control a new hourglassing control technique for quadrilateral and hexahedral elements used in dynamic finite element codes is presented based on expansion of the strees in a taylor series and retention of additional terms beyond the usual constant stress term this technique has the advantage that actual rather than artificial material properties are used simplifications that render this technique competitive with artificial viscosity and stiffness procedures in terms of additional storage and computation requirements are described

large covariance estimation through elliptical factor models we propose a general principal orthogonal complement thresholding poet framework for large scale covariance matrix estimation based on the approximate factor model a set of high level sufficient conditions for the procedure to achieve optimal rates of convergence under different matrix norms is established to better understand how poet works such a framework allows us to recover existing results for sub gaussian data in a more transparent way that only depends on the concentration properties of the sample covariance matrix as a new theoretical contribution for the first time such a framework allows us to exploit conditional sparsity covariance structure for the heavy tailed data in particular for the elliptical distribution we propose a robust estimator based on the marginal and spatial kendall s tau to satisfy these conditions in addition we study conditional graphical model under the same framework the technical tools developed in this paper are of general interest to high dimensional principal component analysis thorough numerical results are also provided to back up the developed theory

robust covariance estimation for approximate factor models in this paper we study robust covariance estimation under the approximate factor model with observed factors we propose a novel framework to first estimate the initial joint covariance matrix of the observed data and the factors and then use it to recover the covariance matrix of the observed data we prove that once the initial matrix estimator is good enough to maintain the element wise optimal rate the whole procedure will generate an estimated covariance with desired properties for data with only bounded fourth moment we propose to use adaptive huber loss minimization to give the initial joint covariance estimation this approach is applicable to a much wider class of distributions beyond sub gaussian and elliptical distributions we also present an asymptotic result for adaptive huber s m estimator with a diverging parameter the conclusions are demonstrated by extensive simulations and real data analysis

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

an integration scheme for prandtl reuss elastoplastic constitutive equations we investigate the generalized midpoint rule for the time integration of elastoplastic constitutive equations for pressure independent yield criteria the incremental equations are divided into one scalar hydrostatic pressure dilation rate equation and a stress deviator strain rate deviator tensorial equation this formulation is applied to two classical problems bulging of a tube under internal pressure and tension test on a notched specimen

treatment of internal constraints by mixed finite element methods unification of concepts a general method to generate assumed stress and strain fields within the context of mixed finite element methods is presented the assumed fields are constructed in such a way that internal constraints are satisfied a priori consequently the locking behaviour commonly observed in finite element solutions of problems with internal constraints is avoided to this end the assumed stress and strain fields are constructed to satisfy a priori the homogeneous part of the equilibrium equations thus avoiding fraeijs de veubeke s limitation principle results obtained using the proposed methodology on a nearly incompressible plane strain problem and thin plate application using a shear deformable theory are indicated

large deformation triangular and tetrahedral element formulations for unstructured meshes abstract the increasing use of automatic mesh generators and remeshers has spurred the need for accurate and efficient triangular and tetrahedral elements in unstructured meshes in large deformation analysis of solids accuracy and efficiency require the treatment of volumetric locking and element distortions respectively in this paper a mixed formulations is proposed to treat these issues and applied to triangular and tetrahedral elements the formulations enforces in the weak sense the expression for the volumetric strain in terms of the deformation gradient the mixed formulation is combined with an irreducible formulation to provide a balance between the treatment of volumetric locking and the maintenance of an adequate critical timestep three techniques are proposed to determine such a balance at the local element level example computations in two and three dimensions of an iron cylinder impacting a rigid surface are used to demonstrate the formulations and to compare them to the irreducible displacement based formulation

noisy matrix completion understanding statistical guarantees for convex relaxation via nonconvex optimization this paper studies noisy low rank matrix completion given partial and noisy entries of a large low rank matrix the goal is to estimate the underlying matrix faithfully and efficiently arguably o

on c0 beam elements with shear and their corresponding penalty function formulation abstract on a model of clamped beam problem we study a finite element method with approximate constraints and the corresponding penalty function approach by a small perturbation analysis we show that the rotation is independent from the penalization this method allows us to use simple c0 elements hence we can compute approximate solutions to obstacle problems the one dimensional nature of all the results is underlined and the extension of this fem to flexible pipe lines is indicated

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

on a systematic development of trilinear three dimensional solid elements based on simo s enhanced strain formulation a systematic development of eight node trilinear three dimensional solid elements based on simo s theoretical framework for the enhanced assumed strain formulation is presented for linear elasticity the starting point is a regular brick followed by parallelepipeds and then arbitrarily distorted elements the advantages and deficits of various enhanced strain formulations are extensively discussed using the regular brick for arbitrarily distorted elements a simple modification of the element stiffness matrix is proposed to achieve a more reliable and very efficient element on some numerical examples the advantages and limits of the family of enhanced strain elements and the improved elements are discussed

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

discretization by finite elements of a model parameter dependent problem the discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness which appears as a parameter in the problem on the accuracy it is shown that the approximation achieved by a standard finite element method degenerates for thin beams in contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter the most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by gauss quadratures

minimax bounds for sparse pca with noisy high dimensional data we study the problem of estimating the leading eigenvectors of a high dimensional population covariance matrix based on independent gaussian observations we establish a lower bound on the minimax risk of estimators under the l2 loss in the joint limit as dimension and sample size increase to infinity under various models of sparsity for the population eigenvectors the lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors we also propose a new method for estimating the eigenvectors by a two stage coordinate selection scheme

the b bar method and the limitation principles abstract the generalized elastic material provides a reference model to cast in a unitary framework many structural models which are based on nonlinear monotone multivalued relations such as viscoelasticity plasticity and unilateral models the modified forms of the hu washizu and hellinger reissner principles and the displacement based variational formulation are recovered for the generalized elastic material starting from a functional in the complete set of state variables the related limitation principles are derived and their specialization to elasticity and elastoplasticity with mixed hardening are provided it is shown that the interpolating fields for the pressure and the volumetric strain usually adopted in the b bar method lead to a limitation principle accordingly the same elastic and elastoplastic solutions can be obtained by means of an approximate mixed displacement pressure variational principle a second application is concerned with the conditions ensuring the coincidence of the solutions between an approximate two field mixed formulation and the displacement based method numerical examples are provided to show the coincidence of the solutions obtained from different mixed finite element formulations in elasticity or elastoplasticity under the validity of the limitation principles

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

dressed coordinates the path integral approach the recently introduced dressed coordinates are studied in the path integral approach these coordinates are defined in the context of a harmonic oscillator linearly coupled to massless scalar field and it is shown that in this model the dressed coordinates appear as a coordinate transformation preserving the path integral functional measure the analysis also generalizes the sum rules established in a previous work

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

robust estimation of high dimensional covariance and precision matrices high dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components covariance and precision matrices provide a useful summary of such structure yet the performance of popular matrix estimators typically hinges upon a sub gaussianity assumption this paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions the proposed estimators under a bounded fourth moment assumption achieve the same minimax convergence rates as do existing methods under a sub gaussianity assumption consistency of the proposed estimators is also established under the weak assumption of bounded 2 e moments for e 0 2 the associated convergence rates depend on e

optimal estimation and rank detection for sparse spiked covariance matrices this paper considers a sparse spiked covariance matrix model in the high dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection the optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices we also establish the minimax rate under the spectral norm for estimating the principal subspace the primary object of interest in principal component analysis in addition the optimal rate for the rank detection boundary is obtained this result also resolves the gap in a recent paper by berthet and rigollet ann stat 41 4 1780 1815 2013 where the special case of rank one is considered

non local and semi local optimal weighting functions for symmetric problems involving a small parameter the method of optimal weighting functions for symmetric problems is described in a general form it is based on a petrov galerkin formulation in which the best approximation property and other mathematical features are achieved for a chosen norm different from the original energy norm of the problem the nonlocality of the weighting functions is shown to have only a minor effect on the efficiency of the method although a localization scheme is also suggested the method is applied to a one and two dimensional singular perturbation problems as well as to a cylindrical shell problem

sparse principal component analysis and iterative thresholding principal component analysis pca is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix however it behaves poorly when the number of features p is comparable to or even much larger than the sample size n in this paper we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse under a spiked covariance model we find that the new approach recovers the principal subspace and leading eigenvectors consistently and even optimally in a range of high dimensional sparse settings simulated examples also demonstrate its competitive performance

minimax sparse principal subspace estimation in high dimensions we study sparse principal components analysis in high dimensions where p the number of variables can be much larger than n the number of observations and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix we prove optimal non asymptotic lower and upper bounds on the minimax subspace estimation error under two different but related notions of lq subspace sparsity for 0 q 1 our upper bounds apply to general classes of covariance matrices and they show that lq constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions 1 introduction principal components analysis pca was introduced in the early 20th century pearson 1901 hotelling 1933 and is arguably the most well known and widely used technique for dimension reduction it is part of the mainstream statistical repertoire and is routinely used in numerous and diverse areas of application however contemporary applications often involve much higher dimensional data than envisioned by the early developers of pca in such high dimensional situations where the number of variables p is of the same order or much larger than the number of observations n serious difficulties emerge standard pca can produce inconsistent estimates of the principal directions of variation and lead to unreliable conclusions johnstone and lu 2009 paul 2007 nadler 2008 the principal directions of variation correspond to the eigenvectors of the covariance matrix and in high dimensions consistent estimation of the eigenvectors is generally not possible without additional assumptions about the covariance matrix or its eigenstructure much of the recent development in pca has focused on methodology that applies the concept of sparsity to eigenvector estimation some examples include jolliffe trendafilov and uddin

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

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

b and f projection methods for nearly incompressible linear and non linear elasticity and plasticity using higher order nurbs elements abstract this paper presents projection methods to treat the incompressibility constraint in small and large deformation elasticity and plasticity within the framework of isogeometric analysis after reviewing some fundamentals of isogeometric analysis we investigate the use of higher order non uniform rational b splines nurbs within the b projection method the higher continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results a new non linear f projection method based on a modified minimum potential energy principle and inspired by the b method is proposed for the large deformation case it leads to a symmetric formulation for which the consistent linearized operator for fully non linear elasticity is derived and used in a newton raphson iterative procedure the performance of the methods is assessed on several numerical examples and results obtained are shown to compare favorably with other published techniques

asymptotics of the principal components estimator of large factor models with weakly influential factors abstract this paper introduces a drifting parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components pc estimator in situations when the factors explanatory power does not strongly dominate the explanatory power of the cross sectionally and temporally correlated idiosyncratic terms under our asymptotics the pc estimator is inconsistent we find explicit formulae for the amount of the inconsistency and propose an estimator of the number of factors for which the pc estimator works reasonably well for the special case when the idiosyncratic terms are cross sectionally but not temporally correlated or vice versa we show that the coefficients in the ols regressions of the pc estimates of factors loadings on the true factors true loadings are asymptotically normal and find explicit formulae for the corresponding asymptotic covariance matrix we explain how to estimate the parameters of the derived asymptotic distributions our monte carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n and t we apply our theoretical results to test a hypothesis about the factor content of the us stock return data

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

analysis of some mixed finite element methods related to reduced integration we prove error estimates for the following two mixed finite element methods related to reduced integration a method for stokes problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress the main idea of the proof in the case of stokes problem is to combine a weak babuska brezzi type stability estimate for the pressure with a superapproximability property for the velocities a similar technique is used in the case of the plate problem