a test for a specific principal component of a correlation matrix abstract in the application of principal components analysis it is common to replace an observed sample principal component vector by another vector closely resembling the sample vector but which is easier to use or interpret a useful test of hypothesis in this case is one that specifies the true ith principal component in this article we obtain an asymptotically chi squared procedure suitable for testing such a hypothesis when the principal components analysis is performed on a correlation matrix the procedure easily extends to a principal components analysis based on m estimates of scatter

structural modeling and psychometrika an historical perspective on growth and achievements the field of linear structural equation modeling with continuous variables is reviewed trends in psychometric theory and data analysis across the five decades of publication ofpsychometrika are discussed especially the clarification of concepts of population and sample explication of the parametric structure of models delineation of concepts of exploratory and confirmatory data analysis expansion of statistical theory in psychometrics estimation via optimization of an explicit objective function and implementation of general function minimization methods developments in the ideas of factor analysis latent variables as well as structural and causal modeling are noted some major conceptual achievements involving general covariance structure representations multiple population models and moment structures are reviewed the major statistical achievements of normal theory generalized least squares estimation elliptical and distribution free estimation and higher moment estimation are discussed computer programs that implement some of the theoretical developments are described

non negative solutions for a class of non positone problems in the recent past many results have been established on non negative solutions to boundary value problems of the form where 0 f 0 0 positone problems in this paper we consider the impact on the non negative solutions when f 0 f u to be convex to guarantee uniqueness of positive solutions and f u to be appropriately concave for multiple positive solutions this is in contrast to the case of positone problems where the roles of convexity and concavity were interchanged to obtain similar results we further establish the existence of non negative solutions with interior zeros which did not exist in positone problems

asymptotic properties of robust complex covariance matrix estimates in many statistical signal processing applications the estimation of nuisance parameters and parameters of interest is strongly linked to the resulting performance generally these applications deal with complex data this paper focuses on covariance matrix estimation problems in non gaussian environments and particularly the m estimators in the context of elliptical distributions first this paper extends to the complex case the results of tyler in d tyler robustness and efficiency properties of scatter matrices biometrika vol 70 no 2 p 411 1983 more precisely the asymptotic distribution of these estimators as well as the asymptotic distribution of any homogeneous function of degree 0 of the m estimates are derived on the other hand we show the improvement of such results on two applications directions of arrival doa estimation using the multiple signal classification music algorithm and adaptive radar detection based on the adaptive normalized matched filter anmf test

complex elliptically symmetric distributions survey new results and applications complex elliptically symmetric ces distributions have been widely used in various engineering applications for which non gaussian models are needed in this overview circular ces distributions are surveyed some new results are derived and their applications e g in radar and array signal processing are discussed and illustrated with theoretical examples simulations and analysis of real radar data the maximum likelihood ml estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness and for convergence of the iterative fixed point algorithm are established specific ml estimators for several ces distributions that are widely used in the signal processing literature are discussed in depth including the complex t distribution k distribution the generalized gaussian distribution and the closely related angular central gaussian distribution a generalization of ml estimators the m estimators of the scatter matrix are also discussed and asymptotic analysis is provided applications of ces distributions and the adaptive signal processors based on ml and m estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for direction of arrival doa estimation and beamforming furthermore experimental validation of the usefulness of ces distributions for modelling real radar data is given

the asymptotic distribution of principal component roots under local alternatives to multiple roots the asymptotic distribution for the principal component roots under local alternatives to multiple population roots is derived the asymptotic theory assumes the estimate of the population covariance or scatter matrix to be asymptotically normal and to possess certain invariance properties these assumptions are satisfied for the affine invariant m estimates of scatter for an elliptical distribution the local alternative framework is used in deriving a local power function for the test for subsphericity

existence of positive solutions for a class of semipositone problem in whole ℝn in this paper we show the existence of solution for the following class of semipositone problem p where n 3 a 0 h n 0 and f 0 0 are continuous functions with f having a subcritical growth the main tool used is the variational method together with estimates that involve the riesz potential

the asymptotic covariance matrix of sample correlation coefficients under general conditions abstract a general matrix expression for the asymptotic covariance matrix of correlation coefficients is derived it is applicable when the data are drawn from any distribution with finite fourth order moments the result is specialized to the cases where the data have a distribution from the elliptical class and where the sample covariance matrix has a noncentral wishart distribution

a class of asymptotic tests for principal component vectors in this paper the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered a class of invariant asymptotic tests based on the sample covariance matrix is derived tests in this class are shown to be consistent and their local power functions are given the arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of the sample covariance matrix the results are shown to generalize when the procedures are based on any affine invariant m estimate of scatter and when the population is elliptical

on some pattern reduction matrices which appear in statistics we review and extend some recent results concerning the structure of patternreduction matrices which effect the reduction of the vet of a patterned matrix to the vector consisting only of the functionally independent elements of the matrix the results are applied to the calculation of certain jacobians and to the construction of ellipsoidal confidence regions for covariance matrices on the basis of maximum likelihood or robust m estimators

robustness and efficiency properties of scatter matrices the robustness and efficiency properties of likelihood ratio tests for functions of the population covariance matrix are studied an alternative class of tests based upon affineinvariant m estimates of scatter is proposed whenever the function of the covariance matrix is invariant under a common scale change for such inferences it is shown that a single scalar valued index of efficiency is sufficient

analysis of covariance structures under elliptical distributions abstract this article examines the adjustment of normal theory methods for the analysis of covariance structures to make them applicable under the class of elliptical distributions it is shown that if the model satisfies a mild scale invariance condition and the data have an elliptical distribution the asymptotic covariance matrix of sample covariances has a structure that results in the retention of many of the asymptotic properties of normal theory methods if a scale adjustment is applied the likelihood ratio tests of fit have the usual asymptotic chi squared distributions difference tests retain their property of asymptotic independence and maximum likelihood estimators retain their relative asymptotic efficiency within the class of estimators based on the sample covariance matrix an adjustment to the asymptotic covariance matrix of normal theory maximum likelihood estimators for elliptical distributions is provided this adjustment is particularly simple in models for patterned covariance or co

spectral models for covariance matrices summary a new model for the simultaneous eigenstructure of multiple covariance matrices is proposed the model is much more flexible than existing models and subsumes most of them as special cases a fisher scoring algorithm for computing maximum likelihood estimates of the parameters under normality is given asymptotic distributions of the estimators are derived under normality as well as under arbitrary distributions having finite fourth order cumulants special attention is given to elliptically contoured distributions likelihood ratio tests are described and sufficient conditions are given under which the test statistics are asymptotically distributed as chi squared random variables procedures are derived for evaluating bartlett corrections under normality some conjectures made by flury 1988 are verified others are refuted a small simulation study of the adequacy of the bartlett correction is described and the new procedures are illustrated on two datasets

greatest lower bound to the elliptical theory kurtosis parameter on montre que le parametre d aplatissement pour une distribution elliptique continue a p variables et matrice de covariance definie positive reelle doit etre strictement plus grand que 2 p 2

invariant co ordinate selection summary a general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented the method is based on the eigenvalue eigenvector decomposition of one scatter matrix relative to another in particular it is shown that the eigenvectors can be used to generate an affine invariant co ordinate system for the multivariate data consequently we view this method as a method for invariant co ordinate selection by plotting the data with respect to this new invariant co ordinate system various data structures can be revealed for example under certain independent components models it is shown that the invariant co ordinates correspond to the independent components another example pertains to mixtures of elliptical distributions in this case it is shown that a subset of the invariant co ordinates corresponds to fisher s linear discriminant subspace even though the class identifications of the data points are unknown some illustrative examples are given

eca high dimensional elliptical component analysis in non gaussian distributions abstractwe present a robust alternative to principal component analysis pca called elliptical component analysis eca for analyzing high dimensional elliptically distributed data eca estimates the eigenspace of the covariance matrix of the elliptical data to cope with heavy tailed elliptical distributions a multivariate rank statistic is exploited at the model level we consider two settings either that the leading eigenvectors of the covariance matrix are nonsparse or that they are sparse methodologically we propose eca procedures for both nonsparse and sparse settings theoretically we provide both nonasymptotic and asymptotic analyses quantifying the theoretical performances of eca in the nonsparse setting we show that eca s performance is highly related to the effective rank of the covariance matrix in the sparse setting the results are twofold i we show that the sparse eca estimator based on a combinatoric program attains the optimal rate of convergence ii based on some recent d

multiplicativity factors for orlicz space function norms abstract let be a function norm defined by a young function with respect to a measure space t m and let l be the orlicz space determined by if l is an algebra then a constant 0 is called a multiplicativity factor for if fg f g for all f g l the main objective of this paper is to give conditions under which l is indeed an algebra and to obtain in this case the best least multiplicativity factor for the first of our principal results is that l is an algebra if and only if m inf inf m a 0 a 0 or x sup x 0 x our second main result states that if l is an algebra and t m is free of infinite atoms then the best multiplicativity factor for is 1 1 minf if minf 0 and x if minf 0

robust estimators for nondecomposable elliptical graphical models robust estimators of the restricted covariance matrices associated with elliptical graphical models are studied general asymptotic results which apply to both decomposable and nondecomposable graphical models are presented for robust plug in type estimators these extend results previously established only for the decomposable case furthermore a class of graphical m estimators for the restricted covariance matrices is introduced and compared with the corresponding plug in m estimators the two approaches are shown to be asymptotically equivalent under random sampling from an elliptical distribution a simulation study demonstrates the superiority of the graphical m estimators for small samples

semiparametrically efficient rank based inference for shape ii optimal r estimation of shape a class of r estimators based on the concepts of multivariate signed ranks and the optimal rank based tests developed in hallin and paindaveine 2006a is proposed for the estimation of the shape matrix of an elliptical distribution these r estimators are root n consistent under any radial density g without any moment assumptions and semiparametrically efficient at some prespecified density f when based on normal scores they are uniformly more efficient thanthe traditional normal theory estimator based on empirical covariance matrices the asymptotic normality of which moreover requires finite moments of order four irrespective of the actual underlying elliptical density they rely on an original rank based version of le cam s one step methodology which avoids the unpleasant nonparametric estimation of cross information quantities that is generally required in the context of r estimation although they are not strictly equivariant they are shown to be equivariant in a weak asymptotic sense simulations confirm their feasability and excellent finite sample performances

statistical inference based on pseudo maximum likelihood estimators in elliptical populations in this article we develop statistical inference based on the maximum likelihood method in elliptical populations with an unknown density function the method assuming the multivariate normal distribution using the sample mean and the sample covariance matrix is basically correct even for elliptical populations under a certain kurtosis adjustment but is not statistically efficient especially when the kurtosis of the population distribution has higher than moderate values on the other hand several methods of statistical inference assuming a particular family e g multivariate t distribution of elliptical distributions have been recommended as a robust procedure against outliers or distributions with heavy tails such inference also will be important to maintain a high efficiency of statistical inference in elliptical populations in practice however it is very difficult to choose an appropriate family of elliptical distributions and one may misspecify the family furthermore extra parameters

depth weighted scatter estimators general depth weighted scatter estimators are introduced and investigated for general depth functions we find out that these affine equivariant scatter estimators are fisher consistent and unbiased for a wide range of multivariate distributions and show that the sample scatter estimators are strong and n consistent and asymptotically normal and the influence functions of the estimators exist and are bounded in general we then concentrate on a specific case of the general depth weighted scatter estimators the projection depth weighted scatter estimators which include as a special case the well known stahel donoho scatter estimator whose limiting distribution has long been open until this paper large sample behavior including consistency and asymptotic normality and efficiency and finite sample behavior including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators are thoroughly investigated the influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined unlike typical high breakdown competitors the projection depth weighted scatter estimators can integrate high breakdown point and high efficiency while enjoying a bounded influence function and a moderate maximum bias curve comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted scatter estimators behave very well overall and consequently represent very favorable choices of affine equivariant multivariate scatter estimators

elliptical graphical modelling we propose elliptical graphical models based on conditional uncorrelatedness as a gen eralization of gaussian graphical models by letting the population distribution be elliptical instead of normal allowing the tting of data with arbitrarily heavy tails we study the class of propor tionally ane equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling leading to a new class of partial correlation estimators and analogues of the classical deviance test general expressions for the asymptotic variance of partial correlation esti mators unconstrained and under decomposable models are given and the asymptotic chi square approximation of the pseudo deviance test statistic is proved the feasibility of our approach is demonstrated by a simulation study using among others tyler s scatter estimator which is distribution free within the elliptical model our approach provides a robustication of gauss ian graphical modelling the latter is likelihood based and known to be very sensitive to model misspecication and outlying observations 1 introduction and notation the statistical theory of undirected graphical models for continuous variables is usually based on the assumption of multivariate normality in practice data may deviate from the normal model in various ways outliers and heavy tails pose a problem of particular gravity they frequently occur and the normal likelihood methods such as the sample covariance matrix are very susceptible to them our objective is to deal with heavy tailed data and to safeguard graphical modelling against the impact of faulty outliers we restrict our attention to the case where we have only continuous variables and only undirected edges joint multivariate normality is often assumed in this situation and the statistical methodology is called gaussian graphical modelling we propose the class of elliptical distributions as a more general model and call our approach elliptical graphical modelling the lack of robustness of gaussian graphical modelling has been noted by several authors four proposals of robust approaches to gaussian graphical modelling are known to us becker 2005 and gottard and pacillo 2010 suggest replacing the sample covariance matrix by the reweighted minimum covariance determinant estimator miyamura and kano 2006 propose an alternative m type estimation and finegold drton arxiv 1009 3669 consider robustied versions of the graphical lasso by friedman et al 2008 this article delivers a systematic treatment of the plug in approach used in the rst two ref erences we show that the sample covariance matrix may be replaced by any ane equivariant

on the asymptotic distributions of two statistics for two level covariance structure models within the class of elliptical distributions since data in social and behavioral sciences are often hierarchically organized special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures most of these developments are based on a multivariate normality distribution assumption which may not be realistic for practical data it is of interest to know whether normal theory based inference can still be valid with violations of the distribution condition various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions this paper shows that similar results still hold for 2 level covariance structure models specifically when both the level 1 within cluster and level 2 between cluster random components follow the same elliptical distribution the rescaled statistic recently developed by yuan and bentler asymptotically follows a chi square distribution when level 1 and level 2 have different elliptical distributions an additional rescaled statistic can be constructed that also asymptotically follows a chi square distribution our results provide a rationale for applying these rescaled statistics to general non normal distributions and also provide insight into issues related to level 1 and level 2 sample sizes

on the asymptotics of maronna s robust pca the eigenvalue decomposition evd parameters of the second order statistics are ubiquitous in statistical analysis and signal processing notably the evd of the m estimators of the scatter matrix is a popular choice to perform robust probabilistic pca or other dimension reduction related applications towards the goal of characterizing this process this paper derives new asymptotics for the evd parameters i e eigenvalues eigenvectors and principal subspace of m estimators in the context of complex elliptically symmetric distributions first their gaussian asymptotic distribution is obtained by extending standard results on the sample covariance matrix in the gaussian context second their convergence towards the evd parameters of a gaussian core wishart equivalent is derived this second result represents the main contribution in the sense that it quantifies when it is acceptable to directly rely on well established results on the evd of wishart distributed matrix for characterizing the evd of m estimators finally some examples intrinsic bias analysis rank estimation and low rank adaptive filtering illustrate where the obtained results can be leveraged

optimal shrinkage covariance matrix estimation under random sampling from elliptical distributions this paper considers the problem of estimating a high dimensional covariance matrix in a low sample support situation where the sample size is smaller or not much larger than the dimensionality of the data which could potentially be very large we develop a regularized sample covariance matrix rscm estimator which can be applied in commonly occurring high dimensional data problems the proposed rscm estimator is based on estimators of the unknown optimal oracle shrinkage parameters that yield the minimum mean squared error between the rscm and the true covariance matrix when the data is sampled from an unspecified elliptically symmetric distribution we propose two variants of the rscm estimator which differ in the approach in which they estimate the underlying sphericity parameter involved in the theoretical optimal shrinkage parameter the performance of the proposed rscm estimators are evaluated with numerical simulation studies in particular when the sample sizes are low the proposed rscm estimators often show a significant improvement over the conventional rscm estimator by ledoit and wolf 2004 we further evaluate the performance of the proposed estimators in a portfolio optimization problem with real data wherein the proposed methods are able to outperform the benchmark methods

on normal theory based inference for multilevel models with distributional violations data in social and behavioral sciences are often hierarchically organized though seldom normal yet normal theory based inference procedures are routinely used for analyzing multilevel models based on this observation simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions for characterizing the distribution of parameter estimates sandwich type covariance matrices are derived standard errors based on these covariance matrices remain consistent under distributional violations implications of various covariance estimators are also discussed for evaluating the quality of a multilevel model a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model the rescaled statistic improving the likelihood ratio statistic by estimating one extra parameter approaches the same mean as its reference distribution a simulation study with a 2 level factor model implies that the rescaled statistic is preferable

structural equation modeling with heavy tailed distributions data in social and behavioral sciences typically possess heavy tails structural equation modeling is commonly used in analyzing interrelations among variables of such data classical methods for structural equation modeling fit a proposed model to the sample covariance matrix which can lead to very inefficient parameter estimates by fitting a structural model to a robust covariance matrix for data with heavy tails one generally gets more efficient parameter estimates because many robust procedures are available we propose using the empirical efficiency of a set of invariant parameter estimates in identifying an optimal robust procedure within the class of elliptical distributions analytical results show that the robust procedure leading to the most efficient parameter estimates also yields a most powerful test statistic examples illustrate the merit of the proposed procedure the relevance of this procedure to data analysis in a broader context is noted

semiparametrically efficient rank based inference for shape i optimal rank based tests for sphericity we propose a class of rank based procedures for testing that the shape matrix v of an elliptical distribution with unspecified center of symmetry scale and radial density has some fixed value v 0 this includes for v 0 i k the problem of testing for sphericity as an important particular case the proposed tests are invariant under translations monotone radial transformations rotations and reflections with respect to the estimated center of symmetry they are valid without any moment assumption for adequately chosen scores they are locally asymptotically maximin in the le cam sense at given radial densities they are strictly distribution free when the center of symmetry is specified and asymptotically so when it must be estimated the multivariate ranks used throughout are those of the distances in the metric associated with the null value v 0 of the shape matrix between the observations and the estimated center of the distribution local powers against elliptical alternatives and asymptotic relative efficiencies ares are derived with respect to the adjusted mauchly test a modified version of the gaussian likelihood ratio procedure proposed by muirhead and waternaux biometrika 67 1980 31 43 or equivalently with respect to an extension of the test for sphericity introduced by john biometrika 59 1972 169 173 for gaussian scores these ares are uniformly larger than one irrespective of the actual radial density necessary and or sufficient conditions for consistency under nonlocal possibly nonelliptical alternatives are given finite sample performance is investigated via a monte carlo study

some tests for common principal component subspaces in several groups summary an approximate test based on sample eigenprojections is obtained for testing the hypothesis that the subspaces spanned by the first m principal components of several different covariance matrices are identical this procedure can fairly easily be extended to an analogous test of hypothesis concerning correlation matrices extensions to a robust principal components analysis are also considered in recent years principal components analysis has been generalized from the typical single population setting to that of several populations this paper focuses on an extension that may be useful in those situations in which one needs a variance preserving simultaneous reduction of dimensionality in several groups this sort of reduction will be possible when the subspace spanned by the most important principal component vectors of a particular group is the same as that of any other group formally suppose the same p variables are being measured on objects in g different groups with the covariance