regression on manifolds estimation of the exterior derivative collinearity and near collinearity of predictors cause difficulties when doing regression in these cases variable selection becomes untenable because of mathematical issues concerning the existence and numerical stability of the regression coefficients and interpretation of the coefficients is ambiguous because gradients are not defined using a differential geometric interpretation in which the regression coefficients are interpreted as estimates of the exterior derivative of a function we develop a new method to do regression in the presence of collinearities our regularization scheme can improve estimation error and it can be easily modified to include lasso type regularization these estimators also have simple extensions to the large p small n context

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

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

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

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

adapting to unknown sparsity by controlling the false discovery rate we attempt to recover an n dimensional vector observed in white noise where n is large and the vector is known to be sparse but the degree of sparsity is unknown we consider three different ways of defining sparsity of a vector using the fraction of nonzero terms imposing power law decay bounds on the ordered entries and controlling the lp norm for p small we obtain a procedure which is asymptotically minimax for l r loss simultaneously throughout a range of such sparsity classes the optimal procedure is a data adaptive thresholding scheme driven by control of the false discovery rate fdr fdr control is a relatively recent innovation in simultaneous testing ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections in our treatment the fdr control parameter qn also plays a determining role in asymptotic minimaxity if q lim qn 0 1 2 and also qn log n we get sharp asymptotic minimaxity simultaneously over a wide range of sparse parameter spaces and loss functions on the other hand q lim qn 1 2 1 forces the risk to exceed the minimax risk by a factor growing with q to our knowledge this relation between ideas in simultaneous inference and asymptotic decision theory is new our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors these new rules impose complexity penalization of the form 2 log potential model size actual model size we exhibit a close connection with fdr controlling procedures under stringent control of the false discovery rate

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

a direct formulation for sparse pca using semidefinite programming given a covariance matrix we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination this problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis pca and has wide applications ranging from biology to finance we use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix where cardinality is constrained and derive a semidefinite programming based relaxation for our problem we also discuss nesterov s smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse pca problem the method has complexity o n 4 sqrt log n epsilon where n is the size of the underlying covariance matrix and epsilon is the desired absolute accuracy on the optimal value of the problem

doubly functional graphical models in high dimensions we consider estimating a functional graphical model from multivariate functional observations in functional data analysis the classical assumption is that each function has been measured over a densely sampled grid however in practice the functions have often been observed with measurement error at a relatively small number of points we propose a class of doubly functional graphical models to capture the evolving conditional dependence relationship among a large number of sparsely or densely sampled functions our approach first implements a nonparametric smoother to perform functional principal components analysis for each curve then estimates a functional covariance matrix and finally computes sparse precision matrices which in turn provide the doubly functional graphical model we derive some novel concentration bounds uniform convergence rates and model selection properties of our estimator for both sparsely and densely sampled functional data in the high dimensional large p small n regime we demonstrate via simulations that the proposed method significantly outperforms possible competitors our proposed method is applied to a brain imaging dataset

hub discovery in partial correlation graphs one of the most important problems in large scale inference problems is the identification of variables that are highly dependent on several other variables when dependence is measured by partial correlations these variables identify those rows of the partial correlation matrix that have several entries with large magnitudes i e hubs in the associated partial correlation graph this paper develops theory and algorithms for discovering such hubs from a few observations of these variables we introduce a hub screening framework in which the user specifies both a minimum partial correlation rho and a minimum degree delta to screen the vertices the choice of rho and delta can be guided by our mathematical expressions for the phase transition correlation threshold rho c governing the average number of discoveries they can also be guided by our asymptotic expressions for familywise discovery rates under the assumption of large number p of variables fixed number n of multivariate samples and weak dependence under the null hypothesis that the dispersion covariance matrix is sparse these limiting expressions can be used to enforce familywise error constraints and to rank the discoveries in order of increasing statistical significance for n ll p the computational complexity of the proposed partial correlation screening method is low and is therefore highly scalable thus it can be applied to significantly larger problems than previous approaches the theory is applied to discovering hubs in a high dimensional gene microarray dataset

sparse linear discriminant analysis by thresholding for high dimensional data in many social economical biological and medical studies one objective is to classify a subject into one of several classes based on a set of variables observed from the subject because the probability distribution of the variables is usually unknown the rule of classification is constructed using a training sample the well known linear discriminant analysis lda works well for the situation where the number of variables used for classification is much smaller than the training sample size because of the advance in technologies modern statistical studies often face classification problems with the number of variables much larger than the sample size and the lda may perform poorly we explore when and why the lda has poor performance and propose a sparse lda that is asymptotically optimal under some sparsity conditions on the unknown parameters for illustration of application we discuss an example of classifying human cancer into two classes of leukemia based on a set of 7 129 genes and a training sample of size 72 a simulation is also conducted to check the performance of the proposed method

partial estimation of covariance matrices a classical approach to accurately estimating the covariance matrix of a p variate normal distribution is to draw a sample of size n p and form a sample covariance matrix however many modern applications operate with much smaller sample sizes thus calling for estimation guarantees in the regime n ll p we show that a sample of size n o m log6p is sufficient to accurately estimate in operator norm an arbitrary symmetric part of consisting of m n nonzero entries per row this follows from a general result on estimating hadamard products m where m is an arbitrary symmetric matrix

threshold selection for covariance estimation thresholding is a regularization method commonly used for covariance estimation which provides consistent estimators if the population covariance satisfies certain sparsity condition bickel and levina 2008a cai and liu 2011 however the performance of the thresholding estimators heavily depends on the threshold level by minimizing the frobenius risk of the adaptive thresholding estimator for covariances we conduct a theoretical study for the optimal threshold level and obtain its analytical expression a consistent estimator based on this expression is proposed for the optimal threshold level which is easy to implement in practice and efficient in computation numerical simulations and a case study on gene expression data are conducted to illustrate the proposed method

on linearly constrained minimum variance beamforming beamforming is a widely used technique for source localization in signal processing and neuroimaging a number of vector beamformers have been introduced to localize neuronal activity by using magnetoencephalography meg data in the literature however the existing theoretical analyses on these beamformers have been limited to simple cases where no more than two sources are allowed in the associated model and the theoretical sensor covariance is also assumed known the information about the effects of the meg spatial and temporal dimensions on the consistency of vector beamforming is incomplete in the present study we consider a class of vector beamformers defined by thresholding the sensor covariance matrix which include the standard vector beamformer as a special case a general asymptotic theory is developed for these vector beamformers which shows the extent of effects to which the meg spatial and temporal dimensions on estimating the neuronal activity index the performances of the proposed beamformers are assessed by simulation studies superior performances of the proposed beamformers are obtained when the signal to noise ratio is low we apply the proposed procedure to real meg data sets derived from five sessions of a human face perception experiment finding several highly active areas in the brain a good agreement between these findings and the known neurophysiology of the meg response to human face perception is shown

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

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

a penalized empirical likelihood method in high dimensions we formulate a penalized empirical likelihood pel method for inference on the population mean when the dimension of the observations become unbounded with the sample size we derive the asymptotic distribution of the pel ratio statistic we show that the limit distribution of the proposed pel ratio statistic can vary widely depending on the correlation structure of the components of the observations that we classify as i non ergodic ii long range dependent and iii short range dependent the limit laws differ from the usual chi squared limit of the empirical likelihood ratio statistic in the finite dimensional case we propose a subsampling approximation for calibrating the pel ratio test statistic and establish its validity finite sample properties of the method are investigated through a simulation study joint work with deep mukhopadhyay

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

banded regularization of autocovariance matrices in application to parameter estimation and forecasting of time series summary the paper addresses a large p small n problem in a time series framework and considers properties of banded regularization of an empirical autocovariance matrix of a time series process utilizing the banded autocovariance matrix enables us to fit a much longer auto regressive ar p model to the observed data than typically suggested by the akaike information criterion while controlling how many parameters are to be estimated precisely and the level of accuracy we present results on asymptotic consistency of banded autocovariance matrices under the frobenius norm and provide a theoretical justification on optimal band selection by using cross validation remarkably the cross validation loss function for banded prediction is related to the conditional mean square prediction error and thus may be viewed as an alternative model selection criterion the procedure proposed is illustrated by simulations and application to predicting the sea surface temperature index in the nino 3 4 region

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

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

shrinkage estimation for multivariate hidden markov models abstractmotivated from a changing market environment over time we consider high dimensional data such as financial returns generated by a hidden markov model that allows for switching between different regimes or states to get more stable estimates of the covariance matrices of the different states potentially driven by a number of observations that are small compared to the dimension we modify the expectation maximization em algorithm so that it yields the shrinkage estimators for the covariance matrices the final algorithm turns out to reproduce better estimates not only for the covariance matrices but also for the transition matrix it results into a more stable and reliable filter that allows for reconstructing the values of the hidden markov chain in addition to a simulation study performed in this article we also present a series of theoretical results that include dimensionality asymptotics and provide the motivation for certain techniques used in the algorithm supplementary materials fo

regularized estimation of linear functionals of precision matrices for high dimensional time series this paper studies a dantzig selector type regularized estimator for linear functionals of high dimensional linear processes explicit rates of convergence of the proposed estimator are obtained and they cover the broad regime from independent identically distributed samples to long range dependent time series and from sub gaussian innovations to those with mild polynomial moments it is shown that the convergence rates depend on the degree of temporal dependence and the moment conditions of the underlying linear processes the dantzig selector estimator is applied to the sparse markowitz portfolio allocation and the optimal linear prediction for time series in which the ratio consistency when compared with an oracle estimator is established the effect of dependence and innovation moment conditions is further illustrated in the simulation study finally the regularized estimator is applied to classify the cognitive states on a real functional magnetic resonance imaging dataset and to portfolio optimization on a financial dataset

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

on an additive partial correlation operator and nonparametric estimation of graphical models we introduce an additive partial correlation operator as an extension of partial correlation to the nonlinear setting and use it to develop a new estimator for nonparametric graphical models our graphical models are based on additive conditional independence a statistical relation that captures the spirit of conditional independence without having to resort to high dimensional kernels for its estimation the additive partial correlation operator completely characterizes additive conditional independence and has the additional advantage of putting marginal variation on appropriate scales when evaluating interdependence which leads to more accurate statistical inference we establish the consistency of the proposed estimator through simulation experiments and analysis of the dream4 challenge dataset we demonstrate that our method performs better than existing methods in cases where the gaussian or copula gaussian assumption does not hold and that a more appropriate scaling for our method further enhances its performance

high dimensional variable selection via tilting the paper considers variable selection in linear regression models where the number of covariates is possibly much larger than the number of observations high dimensionality of the data brings in many complications such as possibly spurious high correlations between the variables which result in marginal correlation being unreliable as a measure of association between the variables and the response we propose a new way of measuring the contribution of each variable to the response which takes into account high correlations between the variables in a data driven way the proposed tilting procedure provides an adaptive choice between the use of marginal correlation and tilted correlation for each variable where the choice is made depending on the values of the hard thresholded sample correlation of the design matrix we study the conditions under which this measure can successfully discriminate between the relevant and the irrelevant variables and thus be used as a tool for variable selection finally an iterative variable screening algorithm is constructed to exploit the theoretical properties of tilted correlation and its good practical performance is demonstrated in a comparative simulation study

comment ridge regression and regularization of large matrices we view ridge regression through the lens of eigenvalue shrinkage and consider its influence on two modern problems in high dimensional statistical inference covariance estimation and community d

complete characterization of hadamard powers preserving loewner positivity monotonicity and convexity abstract entrywise powers of symmetric matrices preserving positivity monotonicity or convexity with respect to the loewner ordering arise in various applications and have received much attention recently in the literature following fitzgerald and horn 1977 8 it is well known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness similar phenomena have also recently been shown by hiai 2009 to occur for monotonicity and convexity in this paper we complete the characterization of all the entrywise powers below and above the critical exponents that are positive monotone or convex on the cone of positive semidefinite matrices we then extend the original problem by fully classifying the positive monotone or convex powers in a more general setting where additional rank constraints are imposed on the matrices we also classify the entrywise powers that are super sub additive with respect to the loewner ordering finally we extend all the previous characterizations to matrices with negative entries our analysis consequently allows us to answer a question raised by bhatia and elsner 2007 regarding the smallest dimension for which even extensions of the power functions do not preserve loewner positivity

detection of correlations we consider the hypothesis testing problem of deciding whether an observed high dimensional vector has independent normal components or alternatively if it has a small subset of correlated components the correlated components may have a certain combinatorial structure known to the statistician we establish upper and lower bounds for the worst case minimax risk in terms of the size of the correlated subset the level of correlation and the structure of the class of possibly correlated sets we show that some simple tests have near optimal performance in many cases while the generalized likelihood ratio test is suboptimal in some important cases

a high dimensional nonparametric multivariate test for mean vector this work is concerned with testing the population mean vector of nonnormal high dimensional multivariate data several tests for high dimensional mean vector based on modifying the classical hotelling t2 test have been proposed in the literature despite their usefulness they tend to have unsatisfactory power performance for heavy tailed multivariate data which frequently arise in genomics and quantitative finance this article proposes a novel high dimensional nonparametric test for the population mean vector for a general class of multivariate distributions with the aid of new tools in modern probability theory we proved that the limiting null distribution of the proposed test is normal under mild conditions when p is substantially larger than n we further study the local power of the proposed test and compare its relative efficiency with a modified hotelling t2 test for high dimensional data an interesting finding is that the newly proposed test can have even more substantial power gain with l

a new approach to cholesky based covariance regularization in high dimensions summary in this paper we propose a new regression interpretation of the cholesky factor of the covariance matrix as opposed to the well known regression interpretation of the cholesky factor of the inverse covariance which leads to a new class of regularized covariance estimators suitable for high dimensional problems regularizing the cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator in particular one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of bickel levina 2008b which is not guaranteed to be positive definite we also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint and compare the numerical performance of several methods in simulations and on a sonar data example

the sparse matrix transform for covariance estimation and analysis of high dimensional signals covariance estimation for high dimensional signals is a classically difficult problem in statistical signal analysis and machine learning in this paper we propose a maximum likelihood ml approach to covariance estimation which employs a novel non linear sparsity constraint more specifically the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform smt the smt is formed by a product of pairwise coordinate rotations known as givens rotations using this framework the covariance can be efficiently estimated using greedy optimization of the log likelihood function and the number of givens rotations can be efficiently computed using a cross validation procedure the resulting estimator is generally positive definite and well conditioned even when the sample size is limited experiments on a combination of simulated data standard hyperspectral data and face image sets show that the smt based covariance estimates are consistently more accurate than both traditional shrinkage estimates and recently proposed graphical lasso estimates for a variety of different classes and sample sizes an important property of the new covariance estimate is that it naturally yields a fast implementation of the estimated eigen transformation using the smt representation in fact the smt can be viewed as a generalization of the classical fast fourier transform fft in that it uses butterflies to represent an orthonormal transform however unlike the fft the smt can be used for fast eigen signal analysis of general non stationary signals

large volatility matrix inference via combining low frequency and high frequency approaches it is increasingly important in financial economics to estimate volatilities of asset returns however most of the available methods are not directly applicable when the number of assets involved is large due to the lack of accuracy in estimating high dimensional matrices therefore it is pertinent to reduce the effective size of volatility matrices in order to produce adequate estimates and forecasts furthermore since high frequency financial data for different assets are typically not recorded at the same time points conventional dimension reduction techniques are not directly applicable to overcome those difficulties we explore a novel approach that combines high frequency volatility matrix estimation together with low frequency dynamic models the proposed methodology consists of three steps i estimate daily realized covolatility matrices directly based on high frequency data ii fit a matrix factor model to the estimated daily covolatility matrices and iii fit a vector autoregressive mo

large covariance estimation for compositional data via composition adjusted thresholding high dimensional compositional data arise naturally in many applications such as metagenomic data analysis the observed data lie in a high dimensional simplex and conventional statistical methods

high dimensional minimum variance portfolio estimation under statistical factor models abstract we propose a high dimensional minimum variance portfolio estimator under statistical factor models and show that our estimated portfolio enjoys sharp risk consistency our approach relies on properly integrating l 1 constraint on portfolio weights with an appropriate covariance matrix estimator in terms of covariance matrix estimation we extend the theoretical results of poet fan et al 2013 to a setting that is coherent with principal component analysis simulation and extensive empirical studies on s p 100 index constituent stocks demonstrate favorable performance of our mvp estimator compared with benchmark portfolios

regularized parameter estimation in high dimensional gaussian mixture models finite gaussian mixture models are widely used in statistics thanks to their great flexibility however parameter estimation for gaussian mixture models with high dimensionality can be challenging because of the large number of parameters that need to be estimated in this letter we propose a penalized likelihood estimator to address this difficulty the type penalty we impose on the inverse covariance matrices encourages sparsity on its entries and therefore helps to reduce the effective dimensionality of the problem we show that the proposed estimate can be efficiently computed using an expectation maximization algorithm to illustrate the practical merits of the proposed method we consider its applications in model based clustering and mixture discriminant analysis numerical experiments with both simulated and real data show that the new method is a valuable tool for high dimensional data analysis

local ζ function techniques vs point splitting procedure a few rigorous results some general properties of local function procedures to renormalize some quantities in d dimensional euclidean quantum field theory in curved background are rigorously discussed for positive scalar operators v x in general closed d manifolds and a few comments are given for nonclosed manifolds too a general comparison is carried out with respect to the more known point splitting procedure concerning the effective lagrangian and the field fluctuations it is proven that for d 1 the local function and point splitting approaches lead essentially to the same results apart from some differences in the subtraction procedure of the hadamard divergences it is found that the function procedure picks out a particular term w 0 w y in the hadamard expansion the presence of an untrivial kernel of the operator v x may produce some differences between the two analyzed approaches finally a formal identity concerning the field fluctuations used by physicists is discussed and proven within the local function approach this is done also to reply to recent criticism against function techniques

sparse estimation of conditional graphical models with application to gene networks in many applications the graph structure in a network arises from two sources intrinsic connections and connections due to external effects we introduce a sparse estimation procedure for graphical models that is capable of isolating the intrinsic connections by removing the external effects technically this is formulated as a conditional graphical model in which the external effects are modeled as predictors and the graph is determined by the conditional precision matrix we introduce two sparse estimators of this matrix using the reproduced kernel hilbert space combined with lasso and adaptive lasso we establish the sparsity variable selection consistency oracle property and the asymptotic distributions of the proposed estimators we also develop their convergence rate when the dimension of the conditional precision matrix goes to infinity the methods are compared with sparse estimators for unconditional graphical models and with the constrained maximum likelihood estimate that assumes a know

embracing the blessing of dimensionality in factor models factor modeling is an essential tool for exploring intrinsic dependence structures among high dimensional random variables much progress has been made for estimating the covariance matrix from a high dimensional factor model however the blessing of dimensionality has not yet been fully embraced in the literature much of the available data are often ignored in constructing covariance matrix estimates if our goal is to accurately estimate a covariance matrix of a set of targeted variables shall we employ additional data which are beyond the variables of interest in the estimation in this article we provide sufficient conditions for an affirmative answer and further quantify its gain in terms of fisher information and convergence rate in fact even an oracle like result as if all the factors were known can be achieved when a sufficiently large number of variables is used the idea of using data as much as possible brings computational challenges a divide and conquer algorithm is thus proposed to alleviate the computational burden and also shown not to sacrifice any statistical accuracy in comparison with a pooled analysis simulation studies further confirm our advocacy for the use of full data and demonstrate the effectiveness of the above algorithm our proposal is applied to a microarray data example that shows empirical benefits of using more data supplementary materials for this article are available online

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

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

augmented factor models with applications to validating market risk factors and forecasting bond risk premia abstract we study factor models augmented by observed covariates that have explanatory powers on the unknown factors in financial factor models the unknown factors can be reasonably well explained by a few observable proxies such as the fama french factors in diffusion index forecasts identified factors are strongly related to several directly measurable economic variables such as consumption wealth variable financial ratios and term spread with those covariates both the factors and loadings are identifiable up to a rotation matrix even only with a finite dimension to incorporate the explanatory power of these covariates we propose a smoothed or projected principal component analysis pca i regress the data onto the observed covariates and ii take the principal components of the fitted data to estimate the loadings and factors this allows us to more accurately estimate the percentage of both explained and unexplained components in factors and thus to assess the explanatory power of covariates we show that both the estimated factors and loadings can be estimated with improved rates of convergence compared to the benchmark method the degree of improvement depends on the strength of the signals representing the explanatory power of the covariates on the factors the proposed estimator is robust to possibly heavy tailed distributions we apply the model to forecast us bond risk premia and find that the observed macroeconomic characteristics contain strong explanatory powers of the factors the gain of forecast is more substantial when the characteristics are incorporated to estimate the common factors than directly used for forecasts

testing for trends in high dimensional time series the article considers statistical inference for trends of high dimensional time series based on a modified l2 distance between parametric and nonparametric trend estimators we propose a de diagon

estimation of a covariance matrix under stein s loss on decrit la technique generale de stein pour ameliorer l estimateur sans biais invariant le meilleur et l estimateur minimax de la matrice de covariance normale

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

a dynamic structure for high dimensional covariance matrices and its application in portfolio allocation abstractestimation of high dimensional covariance matrices is an interesting and important research topic in this article we propose a dynamic structure and develop an estimation procedure for high dimensional covariance matrices asymptotic properties are derived to justify the estimation procedure and simulation studies are conducted to demonstrate its performance when the sample size is finite by exploring a financial application an empirical study shows that portfolio allocation based on dynamic high dimensional covariance matrices can significantly outperform the market from 1995 to 2014 our proposed method also outperforms portfolio allocation based on the sample covariance matrix the covariance matrix based on factor models and the shrinkage estimator of covariance matrix supplementary materials for this article are available online

tracy widom limit for the largest eigenvalue of a large class of complex sample covariance matrices we consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity more precisely let x be an n x p matrix and let its rows be i i d complex normal vectors with mean 0 and covariance p we show that for a large class of covariance matrices p the largest eigenvalue of x x is asymptotically distributed after recentering and rescaling as the tracy widom distribution that appears in the study of the gaussian unitary ensemble we give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance n and p the main theorem applies to a number of covariance models found in applications for example well behaved toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms under some conditions on the mass of the atoms are among the models the theorem can handle generalizations of the theorem to certain spiked versions of our models and a s results about the largest eigenvalue are given we also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics

source localization with meg data a beamforming approach based on covariance thresholding summary reconstructing neural activities using non invasive sensor arrays outside the brain is an ill posed inverse problem since the observed sensor measurements could result from an infinite number of possible neuronal sources the sensor covariance based beamformer mapping represents a popular and simple solution to the above problem in this article we propose a family of beamformers by using covariance thresholding a general theory is developed on how their spatial and temporal dimensions determine their performance conditions are provided for the convergence rate of the associated beamformer estimation the implications of the theory are illustrated by simulations and a real data analysis

covariance regularized regression and classification for high dimensional problems in recent years many methods have been developed for regression in high dimensional settings we propose covariance regularized regression a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction an estimate of the inverse covariance matrix is obtained by maximizing its log likelihood under a multivariate normal model subject to a constraint on its elements this estimate is then used to estimate coefficients for the regression of the response onto the features we show that ridge regression the lasso and the elastic net are special cases of covariance regularized regression and we demonstrate that certain previously unexplored forms of covariance regularized regression can outperform existing methods in a range of situations the covariance regularized regression framework is extended to generalized linear models and linear discriminant analysis and is used to analyze gene expression data sets with multiple class and survival outcomes

optimal sparse volatility matrix estimation for high dimensional ito processes with measurement errors stochastic processes are often used to model complex scientiflc problems in flelds ranging from biology and flnance to engineering and physical science this paper investigates rate optimal estimation of the volatility matrix of a high dimensional it o process observed with measurement errors at discrete time points the minimax rate of convergence is established for estimating sparse volatility matrices by combining the multi scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate the minimax lower bound is derived by considering a subclass of it o processes for which the minimax lower bound is obtained through a novel equivalent model of covariance matrix estimation for independent but non identically distributed observations and through a delicate construction of the least favorable parameters in addition a simulation study was conducted to test the flnite sample performance of the optimal estimator and the simulation results were found to support the established asymptotic theory

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

bootstrapping factor models with cross sectional dependence abstract we consider bootstrap methods for factor augmented regressions with cross sectional dependence among idiosyncratic errors this is important to capture the bias of the ols estimator derived recently by goncalves and perron 2014 we first show that a common approach of resampling cross sectional vectors over time is invalid in this context because it induces a zero bias we then propose the cross sectional dependent csd bootstrap where bootstrap samples are obtained by taking a random vector and multiplying it by the square root of a consistent estimator of the covariance matrix of the idiosyncratic errors we show that if the covariance matrix estimator is consistent in the spectral norm then the csd bootstrap is consistent and we verify this condition for the thresholding estimator of bickel and levina 2008 finally we apply our new bootstrap procedure to forecasting inflation using convenience yields as recently explored by gospodinov and ng 2013

certifiably optimal sparse inverse covariance estimation we consider the maximum likelihood estimation of sparse inverse covariance matrices we demonstrate that current heuristic approaches primarily encourage robustness instead of the desired sparsity we give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality the approach uses techniques from mixed integer optimization and convex optimization and provides a high quality solution with a guarantee on its suboptimality even if the algorithm is terminated early using a variety of synthetic and real datasets we demonstrate that our approach can solve problems where the dimension of the inverse covariance matrix is up to 1000 s we also demonstrate that our approach produces significantly sparser solutions than glasso and other popular learning procedures makes less false discoveries while still maintaining state of the art accuracy

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

spectrum estimation for large dimensional covariance matrices using random matrix theory estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics the eigenvalues of covariance matrices play a key role in many widely used techniques in particular in principal component analysis pca in many modern data analysis problems statisticians are faced with large datasets where the sample size n is of the same order of magnitude as the number of variables p random matrix theory predicts that in this context the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance we propose to use a fundamental result in random matrix theory the marcenko pastur equation to better estimate the eigenvalues of large dimensional covariance matrices the marcenko pastur equation holds in very wide generality and under weak assumptions the estimator we obtain can be thought of as shrinking in a nonlinear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues inspired by ideas of random matrix theory we also suggest a change of point of view when thinking about estimation of high dimensional vectors we do not try to estimate directly the vectors but rather a probability measure that describes them we think this is a theoretically more fruitful way to think about these problems our estimator gives fast and good or very good results in extended simulations our algorithmic approach is based on convex optimization we also show that the proposed estimator is consistent

on an additive semigraphoid model for statistical networks with application to pathway analysis we introduce a nonparametric method for estimating non gaussian graphical models based on a new statistical relation called additive conditional independence which is a three way relation among random vectors that resembles the logical structure of conditional independence additive conditional independence allows us to use one dimensional kernel regardless of the dimension of the graph which not only avoids the curse of dimensionality but also simplifies computation it also gives rise to a parallel structure to the gaussian graphical model that replaces the precision matrix by an additive precision operator the estimators derived from additive conditional independence cover the recently introduced nonparanormal graphical model as a special case but outperform it when the gaussian copula assumption is violated we compare the new method with existing ones by simulations and in genetic pathway analysis supplementary materials for this article are available online

estimating sufficient reductions of the predictors in abundant high dimensional regressions we study the asymptotic behavior of a class of methods for sufficient dimension reduction in high dimension regressions as the sample size and number of predictors grow in various alignments it is demonstrated that these methods are consistent in a variety of settings particularly in abundant regressions where most predictors contribute some information on the response and oracle rates are possible simulation results are presented to support the theoretical conclusion

differentially private high dimensional sparse covariance matrix estimation abstract in this paper we study the problem of estimating the covariance matrix under differential privacy where the underlying covariance matrix is assumed to be sparse and of high dimensions we propose a new method called dp thresholding to achieve a non trivial l 2 norm based error bound whose dependence on the dimension drops to logarithmic instead of polynomial it is significantly better than the existing ones which add noise directly to the empirical covariance matrix we also extend the l 2 norm based error bound to a general l w norm based one for any 1 w and show that they share the same upper bound asymptotically our approach can be easily extended to local differential privacy experiments on the synthetic datasets show results that are consistent with theoretical claims

vast volatility matrix estimation for high frequency financial data high frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro structure noise and realized volatility based methods are often used to estimate integrated volatility for problems involving a large number of assets the estimation objects we face are volatility matrices of large size the existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large in fact they are inconsistent when both the number p of the assets and the average sample size n of the price data on the p assets go to infinity this paper proposes a new type of estimators for the integrated volatility matrix and establishes asymptotic theory for the proposed estimators in the framework that allows both n and p to approach to infinity the theory shows that the proposed estimators achieve high convergence rates under a sparsity assumption on the integrated volatility matrix the numerical studies demonstrate that the proposed estimators perform well for large p and complex price and volatility models the proposed method is applied to real high frequency financial data

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 dimensional influence measure influence diagnosis is important since presence of influential observations could lead to distorted analysis and misleading interpretations for high dimensional data it is particularly so as the increased dimensionality and complexity may amplify both the chance of an observation being influential and its potential impact on the analysis in this article we propose a novel high dimensional influence measure for regressions with the number of predictors far exceeding the sample size our proposal can be viewed as a high dimensional counterpart to the classical cook s distance however whereas the cook s distance quantifies the individual observation s influence on the least squares regression coefficient estimate our new diagnosis measure captures the influence on the marginal correlations which in turn exerts serious influence on downstream analysis including coefficient estimation variable selection and screening moreover we establish the asymptotic distribution of the proposed influence measure by letting the predictor dimension go to infinity availability of this asymptotic distribution leads to a principled rule to determine the critical value for influential observation detection both simulations and real data analysis demonstrate usefulness of the new influence diagnosis measure

estimating the number of sources in magnetoencephalography using spiked population eigenvalues abstractmagnetoencephalography meg is an advanced imaging technique used to measure the magnetic fields outside the human head produced by the electrical activity inside the brain various source localization methods in meg require the knowledge of the underlying active sources which are identified by a priori common methods used to estimate the number of sources include principal component analysis or information criterion methods both of which make use of the eigenvalue distribution of the data thus avoiding solving the time consuming inverse problem unfortunately all these methods are very sensitive to the signal to noise ratio snr as examining the sample extreme eigenvalues does not necessarily reflect the perturbation of the population ones to uncover the unknown sources from the very noisy meg data we introduce a framework referred to as the intrinsic dimensionality id of the optimal transformation for the snr rescaling functional it is defined as the number of the spiked population

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

high dimensional volatility matrix estimation via waveletsand thresholding we propose a locally stationary linear model for the evolution of high dimensional financial returns where the time varying volatility matrix is modelled as a piecewise constant function of time we introduce a new wavelet based technique for estimating the volatility matrix which 10 combines four ingredients a haar wavelet decomposition variance stabilization of the haar coefficients via the fisz transform prior to thresholding a bias correction and extra time domain thresholding soft or hard under the assumption of sparsity we demonstrate the interval wise consistency of the proposed estimators of the volatility matrix and its inverse in the operator norm with rates which adapt to the features of the target matrix we also propose a version of 15 the estimators based on the polarization identity which permits a more precise derivation of the thresholds we discuss the practicalities of the algorithm including parameter selection and how to perform it online a simulation study shows the benefits of the method which is illustrated using a stock index portfolio

some theory for generalized boosting algorithms we give a review of various aspects of boosting clarifying the issues through a few simple results and relate our work and that of others to the minimax paradigm of statistics we consider the population version of the boosting algorithm and prove its convergence to the bayes classifier as a corollary of a general result about gauss southwell optimization in hilbert space we then investigate the algorithmic convergence of the sample version and give bounds to the time until perfect separation of the sample we conclude by some results on the statistical optimality of the l2 boosting

wkb analysis of mathcal p mathcal t symmetric sturm liouville problems most studies of symmetric quantum mechanical hamiltonians have considered the schr dinger eigenvalue problem on an infinite domain this paper examines the consequences of imposing the boundary conditions on a finite domain as is the case with regular hermitian sturm liouville problems the eigenvalues of the symmetric sturm liouville problem grow like n2 for large n however the novelty is that a eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another for the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane wkb analysis is used to calculate the asymptotic behaviours of the real eigenvalues and the locations of the critical points the method turns out to be surprisingly accurate even at low energies this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators