central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

ll filters a new class of order statistic filters the ll filters are introduced to generate the order statistic filters l filters and the nonrecursive linear or finite duration impulse response fir filters such estimators are particularly effective filtering signals that do not necessarily follow gaussian distributions they can be designed to restore one dimensional or multidimensional signals corrupted by noise of impulsive type such filters are appealing since they are suitable for being made robust against the presence of spurious outliers in the data

roy s largest root test under rank one alternatives roy s largest root is a common test statistic in a variety of hy pothesis testing problems despite its popularity obtaining accurate tractable approximations to its distribution under the alternative has been a long standing open problem in multivariate statistics in this paper assuming gaussian observations and a rank one alternative also known as concen trated non centrality we derive simple yet accurate approximations for the distribution of roy s largest root test for five of the most common settings these include signal detection in noise multivariate analysis of variance and canonical correlation analysis our main result is that in all five cases roy s test can be approximated using simple combinations of standard uni variate distributions such as central and non central 2 and f our results allow approximate power calculations for roy s test as well as estimates of sample size required to detect given rank one effects by this test both of which are important quantities in hypothesis driven research

testing independence among a large number of high dimensional random vectors capturing dependence among a large number of high dimensional random vectors is a very important and challenging problem by arranging n random vectors of length p in the form of a matrix we develop a linear spectral statistic of the constructed matrix to test whether the n random vectors are independent or not specifically the proposed statistic can also be applied to n random vectors each of whose elements can be written as either a linear stationary process or a linear combination of independent random variables the asymptotic distribution of the proposed test statistic is established for the case of as n to avoid estimating the spectrum of each random vector a modified test statistic which is based on splitting the original n vectors into two equal parts and eliminating the term that contains the inner structure of each random vector or time series is constructed the facts that the limiting distribution is normal and there is no need to know the inner structure of each investigated rando

hecke algebras of classical groups over p adic fields and supercuspidal representations let k be a p adic field with involution 0 and let k 0 be its fixed subfield of k let v be a finite dimensional vector space defined over k equipped with e hermitian form v k 0 is large

central limit theorems for classical likelihood ratio tests for high dimensional normal distributions for random samples of size n obtained from p variate normal distributions we consider the classical likelihood ratio tests lrt for their means and covariance matrices in the high dimensional setting these test statistics have been extensively studied in multivariate analysis and their limiting distributions under the null hypothesis were proved to be chi square distributions as n goes to infinity and p remains fixed in this paper we consider the highdimensional case where both p and n go to infinity with p n y 0 1 we prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and variances we perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chisquare approximations for analyzing high dimensional data

inference on multiple correlation coefficients with moderately high dimensional data summary whenthemultiplecorrelationcoefficientisusedtomeasurehowstronglyagivenvariablecanbelinearly associated with a set of covariates it suffers from an upward bias that cannot be ignored in the presence of a moderately high dimensional covariate under an independent component model we derive an asymptotic approximation to the distribution of the squared multiple correlation coefficient that depends on a simple correction factor we show that this approximation enables us to construct reliable confidence intervals on the population coefficient even when the ratio of the dimension to the sample size is close to unity and the variables are non gaussian

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

asymptotic power of sphericity tests for high dimensional data this paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity we establish the convergence under the null hypothesis and the alternative of the log ratio of the joint densities of the sample covariance eigenvalues to a gaussian process indexed by the norm of the perturbation when the perturbation norm is larger than the phase transition threshold studied in baik et al 2005 the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain when the norm is below the threshold the process is non degenerate so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous using the asymptotic theory of statistical experiments we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region in particular we show that the asymptotic power of the tracy widom type tests is trivial whereas that of the eigenvalue based likelihood ratio test is strictly larger than the size and close to the power envelope

weighted statistic in detecting faint and sparse alternatives for high dimensional covariance matrices abstractthis article considers testing equality of two population covariance matrices when the data dimension p diverges with the sample size n p n c 0 we propose a weighted test statistic that is data driven and powerful in both faint alternatives many small disturbances and sparse alternatives several large disturbances its asymptotic null distribution is derived by large random matrix theory without assuming the existence of a limiting cumulative distribution function of the population covariance matrix the simulation results confirm that our statistic is powerful against all alternatives while other tests given in the literature fail in at least one situation supplementary materials for this article are available online

thresholding implemented in the frequency domain image processing procedures are usually carried out in the spatial domain where the images are acquired and presented utilized the linear nature of the fourier transform allows only those operations that are linear to be mapped into the frequency domain in contrast nonlinear operations and manipulations cannot be realized directly in the frequency domain one of these nonlinear operations is thresholding when operating in the spatial domain to segment image contents into object and background thresholding is simple and efficient however it has no obvious representation in the frequency domain and cannot be carried out there in a straightforward fashion in this paper a means to relax the rigid linear limitation of the fourier transform was investigated a novel approach was established to achieve spatial thresholding using only frequency domain operations the spatial grayscale or scalar data set two dimensional 2 d image or three dimensional 3 d volume was expanded into a binary volume in hyperspace having one more dimension than the original data set the extended dimension is the gray level of the original data manipulating only on that dimension produces the effect of thresholding

construction of tame supercuspidal representations the notion of depth is defined by moy prasad mp2 the notion of a generic character will be defined in 9 when g gln or g is the multiplicative group of a central division algebra of dimension n with n p 1 our generic characters are just the generic characters in my where the definition is due to kutzko moreover in these cases our construction literally specializes to howe s construction as formulated in my and it is known that the construction yields all supercuspidal representations my hm notice that the initial datum in our construction consists of very simple objects linear characters and supercuspidal representations of depth zero the latter are well understood by the work of moy prasad and morris see 3 by the work of howe irreducible supercuspidal representations of gln f with n p 1 are parametrized by certain characters of e as e varies through

on estimation of the noise variance in high dimensional probabilistic principal component analysis in this paper we develop new statistical theory for probabilistic principal component analysis models in high dimensions the focus is the estimation of the noise variance which is an important and unresolved issue when the number of variables is large in comparison with the sample size we first unveil the reasons of a widely observed downward bias of the maximum likelihood estimator of the variance when the data dimension is high we then propose a bias corrected estimator using random matrix theory and establish its asymptotic normality the superiority of the new bias corrected estimator over existing alternatives is first checked by monte carlo experiments with various combinations of p n dimension and sample size in order to demonstrate further potential benefits from the results of the paper to general probability pca analysis we provide evidence of net improvements in two popular procedures ulfarsson and solo 2008 bai and ng 2002 for determining the number of principal components when the respective variance estimator proposed by these authors is replaced by the bias corrected estimator the new estimator is also used to derive new asymptotics for the related goodness of fit statistic under the high dimensional scheme

pythagorean generalization of testing the equality of two symmetric positive definite matrices abstract we provide a new test for equality of two symmetric positive definite matrices that leads to a convenient mechanism for testing specification using the information matrix equality or the sandwich asymptotic covariance matrix of the gmm estimator the test relies on a new characterization of equality between two k dimensional symmetric positive definite matrices a and b the traces of a b 1 and b a 1 are equal to k if and only if a b using this simple criterion we introduce a class of omnibus test statistics for equality and examine their null and local alternative approximations under some mild regularity conditions a preferred test in the class with good omni directional power is recommended for practical work monte carlo experiments are conducted to explore performance characteristics under the null and local as well as fixed alternatives the test is applicable in many settings including gmm estimation svar models and high dimensional variance matrix settings

distribution free tests of independence in high dimensions we consider the testing of mutual independence among all entries in a formula see text dimensional random vector based on formula see text independent observations we study two families of distribution free test statistics which include kendall s tau and spearman s rho as important examples we show that under the null hypothesis the test statistics of these two families converge weakly to gumbel distributions and we propose tests that control the type i error in the high dimensional setting where formula see text we further show that the two tests are rate optimal in terms of power against sparse alternatives and that they outperform competitors in simulations especially when formula see text is large

nonlinear filtering by threshold decomposition a new threshold decomposition architecture is introduced to implement stack filters the architecture is also generalized to a new class of nonlinear filters known as threshold decomposition td filters which are shown to be equivalent to the class of l1 filters under certain conditions another new class of filters known as linear and order statistic los filters result from the intersection of the class of td and l1 filters performance comparisons among several filters are then presented it was found that td is compatible with l1 los and linear filters in suppressing gaussian noise and is superior in suppressing salt and pepper noise los filters however provide a better compromise in performance and complexity

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

minimum mean absolute error estimation over the class of generalized stack filters a class of sliding window operators called generalized stack filters is developed this class of filters which includes all rank order filters stack filters and digital morphological filters is the set of all filters possessing the threshold decomposition architecture and a consistency property called the stacking property conditions under which these filters possess the weak superposition property known as threshold decomposition are determined an algorithm is provided for determining a generalized stack filter which minimizes the mean absolute error mae between the output of the filter and a desired input signal given noisy observations of that signal the algorithm is a linear program whose complexity depends on the window width of the filter and the number of threshold levels observed by each of the filters in the superposition architecture the results show that choosing the generalized stack filter which minimizes the mae is equivalent to massively parallel threshold crossing decisions making when the decision are consistent with each other

adaptive stack filtering under the mean absolute error criterion an adaptive filter algorithm is developed for the class of stack filters which is a class of nonlinear filters obeying a weak superposition property the adaptation algorithm can be interpreted as a learning algorithm for a group of decision making units the decisions of which are subject to a set of constraints called the stacking constraints under a rather weak statistical assumption on the training inputs the decision strategy adopted by the group which evolves according to the proposed learning algorithm is shown to converge to an optimal strategy in the sense that it corresponds to an optimal stack filter under the mean absolute error criterion this adaptive algorithm requires only increment decrement and comparison operations and only local interconnections between the learning units implementation of the algorithm in hardware is therefore very feasible an example is provided to show how the adaptive stack filtering algorithm can be used in an application in image processing

structure and properties of generalized adaptive neural filters for signal enhancement this article addresses the structure and properties of a new class of nonlinear adaptive filters called generalized adaptive neural filters ganfs various properties such as an upper bound of the mean absolute error of the filters are analytically derived experimental results are presented to demonstrate the performance of the filters for signal and image enhancement it is shown that ganfs not only extend the class of stack filters but also have better performance in noise suppression

corrections to lrt on large dimensional covariance matrix by rmt in this paper we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from gaussian populations when the dimension p is large compared to the sample size n next using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random f matrices we propose necessary corrections for these lr tests to cope with high dimensional effects the asymptotic distributions of these corrected tests under the null are given simulations demonstrate that the corrected lr tests yield a realized size close to nominal level for both moderate p around 20 and high dimension while the traditional lr tests with 2 approximation fails another contribution from the paper is that for testing the equality between two covariance matrices the proposed correction applies equally for non gaussian populations yielding a valid pseudo likelihood ratio test

substitution principle for clt of linear spectral statistics of high dimensional sample covariance matrices with applications to hypothesis testing abstract sample covariance matrices are widely used in multivariate statistical analysis the central limit theorems clt s for linear spectral statistics of high dimensional non centralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high dimensional statistical problems however known population mean vectors are assumed for non centralized sample covariance matrices some of which even assume gaussian like moment conditions in fact there are still another two most frequently used sample covariance matrices the me moment estimator constructed by subtracting the sample mean vector from each sample vector and the unbiased sample covariance matrix by changing the denominator n as n n 1 in the me without depending on unknown population mean vectors in this paper we not only establish the new clt s for non centralized sample covariance matrices when the gaussian like moment conditions do not hold but also characterize the non negligible differences among the clt s for the three classes of high dimensional sample covariance matrices by establishing a substitution principle by substituting the adjusted sample size n n 1 for the actual sample size n in the centering term of the new clt s we obtain the clt of the unbiased sample covariance matrices moreover it is found that the difference between the clt s for the me and unbiased sample covariance matrix is non negligible in the centering term although the only difference between two sample covariance matrices is a normalization by n and n 1 respectively the new results are applied to two testing problems for high dimensional covariance matrices

on structure testing for component covariance matrices of a high dimensional mixture summary by studying the family of p dimensional scale mixtures the paper shows for the first time a non trivial example where the eigenvalue distribution of the corresponding sample covariance matrix does not converge to the celebrated marcenko pastur law a different and new limit is found and characterized the reasons for failure of the marcenko pastur limit in this situation are found to be a strong dependence between the p co ordinates of the mixture next we address the problem of testing whether the mixture has a spherical covariance matrix to analyse the traditional john s type test we establish a novel and general central limit theorem for linear statistics of eigenvalues of the sample covariance matrix it is shown that john s test and its recent high dimensional extensions both fail for high dimensional mixtures precisely because of the different spectral limit above as a remedy a new test procedure is constructed afterwards for the sphericity hypothesis this test is then applied to identify the covariance structure in model based clustering it is shown that the test has much higher power than the widely used integrated classification likelihood and bayesian information criteria in detecting non spherical component covariance matrices of a high dimensional mixture

a construction of the supercuspidal representations of _ adic let f be a nondiscrete locally compact non archimedean field in this paper we construct all irreducible supercuspidal representations of g gl n f for each such representation which we may as well assume is unitary we give a subgroup j of g that is compact mod the center z of g and a finite dimensional representation of j such that inducing to g gives the proof that all supercuspidals have been constructed appeals to a theorem the matching theorem that has been proved by global methods

correlation tests and linear spectral statistics of the sample correlation matrix testing the independence of the entries of multidimensional gaussian observations is a very important problem in statistics with a number of applications in signal processing radar cognitive radio seismography and multiple other fields typically the problem is formulated as a binary hypothesis test whereby the presence of correlation is declared when the value of a certain statistic is higher than a certain predetermined threshold most of the statistics for correlation tests are constructed from the sample correlation matrix also known as sample coherence matrix in signal processing which is defined as a power normalized version of the sample covariance matrix in this paper correlation tests constructed from linear spectral statistics lss of the sample correlation matrix are analyzed under the asymptotic framework where both sample size and observation dimension become large but comparable in magnitude a central limit theorem clt is established on this class of statistics which is valid for generally correlated gaussian observations results show that lss asymptotically fluctuate as gaussian random variables under both the hypotheses with an asymptotic mean and variance that can be established for each particular test in particular this general clt can be used to establish the asymptotic behavior of two of the most important correlation test statistics namely the generalized likelihood ratio test and the frobenius norm test under both null and alternative hypotheses as a by product it is established that lss of sample covariance and sample correlation matrices have exactly the same first order behavior but quite different asymptotic fluctuations in the second order regime in both the cases the lss asymptotically behave as gaussian random variables although with quite different asymptotic means and variances

hypothesis testing on linear structures of high dimensional covariance matrix this paper is concerned with test of significance on high dimensional covariance structures and aims to develop a unified framework for testing commonly used linear covariance structures we first construct a consistent estimator for parameters involved in the linear covariance structure and then develop two tests for the linear covariance structures based on entropy loss and quadratic loss used for covariance matrix estimation to study the asymptotic properties of the proposed tests we study related high dimensional random matrix theory and establish several highly useful asymptotic results with the aid of these asymptotic results we derive the limiting distributions of these two tests under the null and alternative hypotheses we further show that the quadratic loss based test is asymptotically unbiased we conduct monte carlo simulation study to examine the finite sample performance of the two tests our simulation results show that the limiting null distributions approximate their null distributions quite well and the corresponding asymptotic critical values keep type i error rate very well our numerical comparison implies that the proposed tests outperform existing ones in terms of controlling type i error rate and power our simulation indicates that the test based on quadratic loss seems to have better power than the test based on entropy loss

unrefined minimal k types for p adic groups via their restriction to compact open subgroups was begun by mautner shalika and tanaka for groups of type ai in contrast to real reductive groups where the representation theory of a maximal compact subgroup k is given in terms of slight modifications to cartan s theory of the highest weight the representation theory of a compact parahoric subgroup is quite complicated there is still no comprehensive theory for classifying the irreducible representa tions in the case of