the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

state complexity of combined operations we study the state complexity of combined operations two particular combined operations are studied star of union and star of intersection it is shown that the state complexity of a combined operation is not necessarily similar to the combination of the individual state complexities of the participating operations

redescending m estimates of multivariate location and scatter a class of redescending m estimates of multivariate location and scatter are investigated sufficient conditions are given to ensure the existence and uniqueness of the estimates these results are applied to the multivariate t distribution with degrees of freedom nu geq 1

breakdown points of affine equivariant estimators of multivariate location and covariance matrices finite sample replacement breakdown points are derived for different types of estimators of multivariate location and covariance matrices the role of various equivariance properties is illustrated the breakdown point is related to a measure of performance based on large deviations probabilities finally we show that one step reweighting preserves the breakdown point

the masking breakdown point of multivariate outlier identification rules abstract in this article we consider simultaneous outlier identification rules for multivariate data generalizing the concept of so called outlier identifiers as presented by davies and gather for the case of univariate samples such multivariate outlier identifiers are based on estimators of location and covariance therefore it seems reasonable that characteristics of the estimators influence the behavior of outlier identifiers several authors mentioned that using estimators with low finite sample breakdown point is not recommended for identifying outliers to give a formal explanation we investigate how the finite sample breakdown points of estimators used in these identification rules influence the masking behavior of the rules

breakdown points of t type regression estimators summary to bound the influence of a leverage point generalised m estimators have been suggested however the usual generalised m estimator of regression has a breakdown point that is less than the inverse of its dimension this paper shows that dimension independent positive breakdown points can be attained by a class of well defined generalised mestimators with redescending scores the solution can be determined through optimisation of t type likelihood applied to properly weighted residuals the highest breakdown point of is attained by cauchy score these bounded influence and high breakdown estimators can be viewed as a fully iterated version of the one step generalised m estimates of simpson ruppert carroll 1992 with the two advantages of easier interpretability and avoidance of undesirable roots to estimating equations given the design dependent weights they can be computed via em algorithms empirical investigations show that they are highly competitive with other robust estimators of regression

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

on the stahel donoho estimator and depth weighted means of multivariate data the depth of multivariate data can be used to construct weighted means as robust estimators of location the use of projection depth leads to the stahel donoho estimator as a special case in contrast to maximal depth estimators the depth weighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature we also confirm through a finite sample study that the stahel donoho estimator achieves a desirable balance between robustness and efficiency at gaussian models

asymptotic behaviour of s estimates of multivariate location parameters and dispersion matrices it is shown under appropriate conditions that rousseeuw s minimum volume estimator and other s estimators of multivariate location and dispersion parameters are consistent under certain differentiability conditions the estimates are asymptotically normally distributed with a norming factor of n 1 2

breakdown and groups the concept of breakdown point was introduced by hampel ph d dissertation 1968 univ california berkeley ann math statist 42 1971 1887 1896 and developed further by among others huber robust statistics 1981 wiley new york and donoho and huber in a festschrift for erich l lehmann 1983 157 184 wadsworth belmont ca it has proved most successful in the context of location scale and regression problems attempts to extend the concept to other situations have not met with general acceptance in this paper we argue that this is connected to the fact that in the location scale and regression problems the translation and affine groups give rise to a definition of equivariance for statistical functionals comparisons in terms of breakdown points seem only useful when restricted to equivariant functionals and even here the connection between breakdown and equivariance is a tenuous one institute of mathematical statistics 2005

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

lieb thirring trace inequalities and a question of bourin in this paper we propose some conjectures in order to solve one of the questions posed by bourin regarding a special type of inequalities referred to as subadditivity inequalities in the case of the hilbert schmidt norm some of these conjectures are settled affirmatively for special cases in an algorithmic way by using some number theory tools

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

on some inequalities for the matrix power and karcher means abstract we establish a useful inequality for matrix power means of positive definite matrices it eventually provides not only a new characterization of the karcher mean cartan centroid riemannian geometric mean but also a simple proof of the convergence of power means to the karcher mean some new matrix mean inequalities on the karcher and power means are presented

a review on robust estimators applied to regression credibility the lack of robustness of regression credibility estimators leads to the development of this paper the appearance of outlier events including large claims catastrophic events can offset the result of the ordinary least squares regression technique and perturb the credibility premium estimation our proposal is to apply robust statistical procedures to the regression credibility estimation which are insensitive to the occurrence of outlier events in the data a review of robust estimators that appeared in the literature is provided including robust estimators that simultaneously attain maximum breakdown point and full asymptotic efficiency under normal errors these estimators are the following l1 estimators m estimators gm estimators lms estimators lts estimators s estimators mm estimators and robust rewls estimators applications of these estimators to regression credibility with two data sets are also presented herein

on the relation between s estimators and m estimators of multivariate location and covariance we discuss the relation between s estimators and m estimators of multivariate location and covariance as in the case of the estimation of a multiple regression parameter s estimators are shown to satisfy first order conditions of m estimators we show that the influence function if x s f of s functionals exists and is the same as that of corresponding m functionals also we show that s estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of m estimators finally we compare asymptotic variances and breakdown point of both types of estimators

projection based depth functions and associated medians order n uniform consistency depth regions and contours induced from projection based depth functions are investigated structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained affine equivariant multivariate medians induced from projection based depth functions are probed the limiting distributions as well as the strong and order n consistency of the sample projection medians are established the finite sample performance of projection medians is compared with that of a leading depth induced median the tukey halfspace median induced from the tukey halfspace depth function it turns out that with appropriate choices of univariate location and scale estimators the projection medians have a very high finite sample breakdown point and relative efficiency much higher than those of the halfspace median based on the results obtained it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators respectively

a cautionary note on robust covariance plug in methods many multivariate statistical methods rely heavily on the sample covariance matrix it is well known though that the sample covariance matrix is highly non robust one popular alternative approach for robustifying the multivariate method is to simply replace the role of the covariance matrix with some robust scatter matrix the aim of this paper is to point out that in some situations certain properties of the covariance matri x are needed for the corresponding robust plug in method to be a valid approach and that not all scatter matrices necessarily possess these important properties in particular the fol lowing three multivariate methods are discussed in this paper independent components analysis observational regression and graphical modeling for each case it is shown that using a symmetrized robust scatter matrix in place of the covariance matrix results in a proper robust multivariate method

diagnostics and robust estimation in multivariate data transformations abstract this article presents a method for detecting multivariate outliers that might be distorting the estimation of a transformation to normality it also proposes a robust estimator of the transformation parameter

a note on tyler s modification of the mad for the stahel donoho estimator in 1994 tyler proposed a modification of the mad in order to achieve the maximal possible finite sample breakdown point for the stahel donoho estimator based upon the median and the modified mad a proof is not given however and as far as we know has not been published yet moreover there seems to be an error concerning the exact definition of the modified mad which is needed to achieve this goal for the sake of theoretical completeness and in order to correct this error we calculate the finite sample breakdown point of modifications of the mad and point out which of these modifications lead to a stahel donoho estimator with maximal finite sample breakdown point

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

finite sample breakdown points of projection based multivariate location and scatter statistics finite sample breakdown points are obtained for two classes of projection based multivariate location and scatter statistics the stahel donoho statistics and the maronna yohai statistics the definition of these multivariate statistics are dependent on the value of location and scale statistics for all univariate projections of the data and consequently their properties depend on the nature of the corresponding univariate location and scale statistics used on the projected data the finite sample breakdown points of the multivariate statistics though are not directly related to those of the corresponding univariate location and scale statistics a uniform finite sample breakdown point concept is needed the median and the median absolute deviation about the median m a d are one possible choice for the univariate location and scale statistics respectively for sparse data sets in high dimensions though it is not recommended that the m a d be used as the univariate scale statistic for the projected data since its uniform finite sample breakdown point is shown to be much less than optimum a simple modification to the m a d however is shown to alleviate this problem for various reasons one may wish to consider univariate location and scale statistics other than the median and the m a d respectively a very broad and natural class of univariate location and scale statistics to consider for the projected data is the simultaneous m estimates of location and scale new results on their breakdown properties are given in this paper implicit formulas for the breakdown points of monotonic simultaneous m estimates of location and scale are known and they tend to imply rather low breakdown points for smooth choices of the defining weight functions it is shown here that this phenomenon does not occur for a large class of nonmonotonic simultaneous m estimates of location and scale furthermore explicit rather than implicit expressions for the uniform finite sample breakdown points are given for these nonmonotonic m estimates

robust and efficient one way manova tests we propose robust tests as alternatives to the classical wilks lambda test in one way manova the robust tests use highly robust and efficient multisample multivariate s estimators or mm estimators instead of the empirical covariances the properties of several robust test statistics are compared under the null hypothesis the distribution of the test statistics is proportional to a chi square distribution as an alternative to the asymptotic distribution we develop a fast robust bootstrap method to estimate the distribution under the null hypothesis we show when it is asymptotically correct to estimate the null distribution in this way and we use simulations to verify the performance of the bootstrap based tests in finite samples we also investigate the power of the new tests as well as their robustness against outliers finally we illustrate the use of these robust test statistics on a real data example some additional results are provided as supplemental material

the behavior of the stahel donoho robust multivariate estimator abstract the stahel donoho estimators t v of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w r where w is a weight function and r is a measure of outlyingness obtained by considering all univariate projections of the data it has a high breakdown point for all dimensions and order n consistency the asymptotic bias of v for point mass contamination for suitable weight functions is compared with that of rousseeuw s minimum volume ellipsoid mve estimator a simulation shows that for a suitable w t and v exhibit high efficiency for both normal and cauchy distributions and are better than their competitors for normal data with point mass contamination the performances of the estimators for detecting outliers are compared for both a real and a synthetic data set

computable robust estimation of multivariate location and shape in high dimension using compound estimators abstract estimation of multivariate shape and location in a fashion that is robust with respect to outliers and is affine equivariant represents a significant challenge the use of compound estimators that use a combinatorial estimator such as rousseeuw s minimum volume ellipsoid mve or minimum covariance determinant mcd to find good starting points for high efficiency robust estimators such as s estimators has been proposed in this article we indicate why this scheme will fail in high dimension due to combinatorial explosion in the space that must be searched for the mve or mcd we propose a meta algorithm based on partitioning the data that enables compound estimators to work in high dimension we show that even when the computational effort is restricted to a linear function of the number of data points the algorithm results in an estimator with good asymptotic properties extensive computational experiments are used to confirm that significant benefits accrue in finite samples as well we also g

asymptotics of reweighted estimators of multivariate location and scatter we investigate the asymptotic behavior ofa weighted sample mean and covariance where the weights are determined by the mahalanobis distances with respect to initial robust estimators we derive an explicit expansion for the weighted estimators from this expansion it can be seen that reweighting does not improve the rate ofconvergence ofthe initial estimators we also show that ifone uses smooth s estimators to determine the weights the weighted estimators are asymptotically normal finally we will compare the efficiency and local robustness of the reweighted s estimators with two other improvements of s estimators estimators and constrained m estimators

influence function and maximum bias of projection depth based estimators location estimators induced from depth functions increasingly have been pursued and studied in the literature among them are those induced from projection depth functions these projection depth based estimators have favorable properties among their competitors in particular they possess the best possible finite sample breakdown point robustness however robustness of estimators cannot be revealed by the finite sample breakdown point alone the influence function gross error sensitivity maximum bias and contamination sensitivity are also important aspects of robustness in this article we study these other robustness aspects of two types of projection depth based estimators projection medians and projection depth weighted means the latter includes the stahel donoho estimator as a special case exact maximum bias the influence function and contamination and gross error sensitivity are derived and studied for both types of estimators sharp upper bounds for the maximum bias and the influence functions are established comparisons based on these robustness criteria reveal that the projection depth based estimators enjoy desirable local as well as global robustness and are very competitive among their competitors

quotient complexity of ideal languages a language l over an alphabet s is a right left ideal if it satisfies email protected email protected l it is a two sided ideal if email protected email protected and an all sided ideal if email protected l the shuffle of s with l ideal languages are not only of interest from the theoretical point of view but also have applications to pattern matching we study the state complexity of common operations in the class of regular ideal languages but prefer to use the equivalent term quotient complexity which is the number of distinct left quotients of a language we find tight upper bounds on the complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of the minimal generator and also on the complexity of the minimal generator in terms of the complexity of the language moreover tight upper bounds on the complexity of union intersection set difference symmetric difference concatenation star and reversal of ideal languages are derived

robust multivariate regression we introduce a robust method for multivariate regression based on robust estimation of the joint location and scatter matrix of the explanatory and response variables as a robust estimator of location and scatter we use the minimum covariance determinant mcd estimator of rousseeuw based on simulations we investigate the finite sample performance and robustness of the estimator to increase the efficiency we propose a reweighted estimator selected from several possible reweighting schemes the resulting multivariate regression does not need much computation time and is applied to real datasets we show that the multivariate regression estimator has the appropriate equivariance properties has a bounded influence function and inherits the breakdown value of the mcd estimator these theoretical robustness properties confirm the good finite sample results obtained from the simulations

inequalities related to bourin and heinz means with a complex parameter abstract a conjecture posed by s hayajneh and f kittaneh claims that given a b positive matrices 0 t 1 and any unitarily invariant norm the following inequality holds a t b 1 t b t a 1 t a t b 1 t a 1 t b t recently r bhatia proved the inequality for the case of the frobenius norm and for t 1 4 3 4 in this paper using complex methods we extend this result to complex values of the parameter t z in the strip z c re z 1 4 3 4 we give an elementary proof of the fact that equality holds for some z in the strip if and only if a and b commute we also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm finally we give a counterexample for a related singular value inequality given by s j a t b 1 t b t a 1 t s j a b answering in the negative a question made by k audenaert and f kittaneh the methods of proof and examples can be adapted with no modifications to operator algebras infinite dimensional setting for instance it follows that the inequality above holds for hilbert schmidt operators with their banach algebra norm derived from the infinite trace of b h

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

tyler shape depth summaryin many problems from multivariate analysis the parameter of interest is a shape matrix a normalized version of the corresponding scatter or dispersion matrix in this article we propose a notion of depth for shape matrices that involves data points only through their directions from the centre of the distribution we refer to this concept as tyler shape depth since the resulting estimator of shape namely the deepest shape matrix is the median based counterpart of the m estimator of shape due to tyler 1987 besides estimation shape depth like its tyler antecedent also allows hypothesis testing on shape its main benefit however lies in the ranking of the shape matrices it provides the practical relevance of which is illustrated by applications to principal component analysis and shape based outlier detection we study the invariance quasi concavity and continuity properties of tyler shape depth the topological and boundedness properties of the corresponding depth regions and the existence of a deepest shape matrix and we prove fisher consistency in the elliptical case finally we derive a glivenko cantelli type result and establish almost sure consistency of the deepest shape matrix estimator

state complexity of pattern matching in regular languages abstract in a simple pattern matching problem one has a pattern w and a text t which are words over a finite alphabet one may ask whether w occurs in t and if so where more generally we may have a set p of patterns and a set t of texts where p and t are regular languages we are interested whether any word of t begins with a word of p ends with a word of p has a word of p as a factor or has a word of p as a subsequence thus we are interested in the languages p t p t p t and where is the shuffle operation the state complexity l of a regular language l is the number of states in the minimal deterministic finite automaton recognizing l we derive the following upper bounds on the state complexities of our pattern matching languages where p m and t n p t m n p t 2 m 1 n p t 2 m 2 1 n and we prove that these bounds are tight and that to meet them the alphabet must have at least two letters in the first three cases and at least m 1 letters in the last case

multidimensional trimming based on projection depth as estimators of location parameters univariate trimmed means are well known for their robustness and efficiency they can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy tailed models this paper introduces multidimensional trimmed means based on projection depth induced regions robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point the influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators it is found that the projection depth trimmed means are highly robust locally as well as globally asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions the strong consistency asymptotic representation and limiting distribution of the depth trimmed means are obtained relative to the mean and other leading competitors the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated simulation studies confirm the validity of the asymptotic efficiency results at finite samples

projection estimates of multivariate location in this paper we study the maximum asymptotic bias of the projection estimate for multivariate location based on univariate estimates of location and dispersion in particular we study the projection estimate that uses the median and median absolute deviation about the median mad as univariate location and dispersion estimates respectively this estimator may be considered a natural affine equivariant multivariate median for spherical distributions the maximum bias of this estimate depends only on the marginal distributions and not on the dimension and is approximately twice the maximum bias of the univariate median we also show that for multivariate normal distributions its maximum bias compares favorably with those of the donoho stahel minimum volume ellipsoid and minimum covariance determinant estimates in all these cases the maximum bias increases with the dimension p

some log majorizations and an extension of a determinantal inequality abstract an eigenvalue inequality involving a matrix connection and its dual is established and some log majorization type results are obtained in particular some eigenvalues inequalities considered by f hiai and m lin 9 an associated conjecture and a singular values inequality by l zou 20 are revisited a reformulation of the inequality det a u b det a b for positive semidefinite matrices a b with u a unitary matrix that appears in the polar decomposition of b a is also extended using some known norm inequalities associated to furuta inequality and araki cordes inequality

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 one step gm estimates and stability of inferences in linear regression abstract the folklore on one step estimation is that it inherits the breakdown point of the preliminary estimator and yet has the same large sample distribution as the fully iterated version as long as the preliminary estimate converges faster than n 1 4 where n is the sample size we investigate the extent to which this folklore is valid for one step gm estimators and their associated standard errors in linear regression we find that one step gm estimates based on newton raphson or scoring inherit the breakdown point of high breakdown point initial estimates such as least median of squares provided the usual weights that limit the influence of extreme points in the design space are based on location and scatter estimates with high breakdown points moreover these estimators have bounded influence functions and their standard errors can have high breakdown points the folklore concerning the large sample theory is correct assuming the regression errors are symmetrically distributed and homoscedastic

robust structural equation modeling with missing data and auxiliary variables the paper develops a two stage robust procedure for structural equation modeling sem and an r package rsem to facilitate the use of the procedure by applied researchers in the first stage m estimates of the saturated mean vector and covariance matrix of all variables are obtained those corresponding to the substantive variables are then fitted to the structural model in the second stage a sandwich type covariance matrix is used to obtain consistent standard errors se of the structural parameter estimates rescaled adjusted as well as corrected and f statistics are proposed for overall model evaluation using r and eqs the r package rsem combines the two stages and generates all the test statistics and consistent ses following the robust analysis multiple model fit indices and standardized solutions are provided in the corresponding output of eqs an example with open closed book examination data illustrates the proper use of the package the method is further applied to the analysis of a data set from the national longitudinal survey of youth 1997 cohort and results show that the developed procedure not only gives a better endorsement of the substantive models but also yields estimates with uniformly smaller standard errors than the normal distribution based maximum likelihood

principal components analysis based on multivariate mm estimators with fast and robust bootstrap we consider robust principal components analysis pca based on multivariate mm estimators we first study the robustness and efficiency of these estimators particularly in terms of eigenvalues and eigenvectors we then focus on inference procedures based on a fast and robust bootstrap for mm estimators this method is an alternative to the approach based on the asymptotic distribution of the estimators and can also be used to assess the stability of the principal components a formal consistency proof for the bootstrap method is given and its finite sample performance is investigated through simulations we illustrate the use of the robust pca and the bootstrap inference on a real dataset

projection estimators for generalized linear models we introduce a new class of robust estimators for generalized linear models which is an extension of the class of projection estimators for linear regression these projection estimators are defined using an initial robust estimator for a generalized linear model with only one unknown parameter we found a bound for the maximum asymptotic bias of the projection estimator caused by a fraction e of outlier contamination for small e this bias is approximately twice the maximum bias of the initial estimator independently of the number of regressors since these projection estimators are not asymptotically normal we define one step weighted m estimators starting at the projection estimators these estimators have the same asymptotic normal distribution as the m estimator and a degree of robustness close to the one of the projection estimator we perform a monte carlo simulation for the case of binomial and poisson regression with canonical links this study shows that the proposed estimators compare favorab

trace inequalities for products of positive definite matrices it is shown that for positive definite matrices a and b the inequality apbq bpaq 2 apbq aqbp 2 holds for all positive real number p and q for which p p q is in 1 4 3 4 this is a significant improvement on a recent result of hayajneh and kittaneh lieb thirring trace inequalities and a question of bourin j math phys 54 033504 2013

a unified approach to exploratory factor analysis with missing data nonnormal data and in the presence of outliers factor analysis is regularly used for analyzing survey data missing data data with outliers and consequently nonnormal data are very common for data obtained through questionnaires based on covariance matrix estimates for such nonstandard samples a unified approach for factor analysis is developed by generalizing the approach of maximum likelihood under constraints statistical properties of the estimates for factor loadings and error variances are obtained a rescaled bartlett corrected statistic is proposed for evaluating the number of factors equivariance and invariance of parameter estimates and their standard errors for canonical varimax and normalized varimax rotations are discussed numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures

nearest neighbor variance estimation nnve robust covariance estimation commonly proceeds by downweighting outliers in this article we measure the outlyingness of a data point by the standardized distance between the point and its kth nearest neighbor the appropriate weights for robust estimation are found by a model based mixture modeling approach that follows from considering the data cloud as a realization of a high dimensional point process to correct a potential bias when there are no outliers we introduce a boundary correction procedure that artificially adds in extra outlying points the resulting methodology is called nearest neighbor variance estimation nnve the strength of nnve is its robustness against a large proportion of noise points and against deviation from normality of the signal a consistency result for the method is established under some reasonable assumptions it is shown that the covariance estimate is bounded and that each point has only bounded influence on the final estimates nnve outperformed the popular mini

the percentage bend correlation coefficient a well known result is that the usual correlation coefficient is highly nonrobust very slight changes in only one of the marginal distributions can alter by a substantial amount there are a variety of methods for correcting this problem this paper identifies one particular method which is useful in psychometrics and provides a simple test for independence it is not recommended that the new test replace the usual test ofh 0 0 but the new test has important advantages over the usual test in terms of both type i errors and power

some inequalities for the matrix heron mean abstract in this paper we prove some norm inequalities for the matrix heron mean of two positive definite matrices we also prove a determinant inequality for the t power mean of two positive definite matrices as a consequence we obtain a determinant inequality for the matrix heron mean

on the existence and uniqueness of the maximum likelihood estimate of a vector valued parameter in fixed size samples the maximum likelihood estimate is shown to exist and to be unique if a twice continuously differentiable likelihood function is constant on the boundary of the parameter space and if the hessian matrix is negative definite whenever the gradient vector vanishes the condition of constancy on the boundary cannot be completely removed cf tarone and gruenhage 1975 the theory is illustrated with several examples

the asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions the asymptotic efficiency of the spatial sign covariance matrix relative to affine equivariant estimators of scatter is studied in particular the spatial sign covariance matrix is shown to be asymptotically inadmissible i e the asymptotic covariance matrix of the consistency corrected spatial sign covariance matrix is uniformly larger than that of its affine equivariant counterpart namely tyler s scatter matrix although the spatial sign covariance matrix has often been recommended when one is interested in principal components analysis its inefficiency is shown to be most severe in situations where principal components are of greatest interest simulation shows that the inefficiency of the spatial sign covariance matrix also holds for small sample sizes and that the asymptotic relative efficiency is a good approximation to the finite sample efficiency for relatively modest sample sizes

a lieb thirring inequality for singular values abstract let a and b be positive semidefinite matrices we investigate the conditions under which the lieb thirring inequality can be extended to singular values i e for which values of p the majorisation a p b p w ab p holds and for which values its reversed inequality ab p w a p b p

breakdown points for maximum likelihood estimators of location scale mixtures ml estimation based on mixtures of normal distributions is a widely used tool for cluster analysis however a single outlier can make the parameter estimation of at least one of the mixture components break down among others the estimation of mixtures of t distributions by mclachlan and peel finite mixture models 2000 wiley new york and the addition of a further mixture component accounting for noise by fraley and raftery the computer j 41 1998 578 588 were suggested as more robust alternatives in this paper the definition of an adequate robustness measure for cluster analysis is discussed and bounds for the breakdown points of the mentioned methods are given it turns out that the two alternatives while adding stability in the presence of outliers of moderate size do not possess a substantially better breakdown behavior than estimation based on normal mixtures if the number of clusters s is treated as fixed r additional points suffice for all three methods to let the parameters of r clusters explode only in the case of r s is this not possible for t mixtures the ability to estimate the number of mixture components for example by use of the bayesian information criterion of schwarz ann statist 6 1978 461 464 and to isolate gross outliers as clusters of one point is crucial for an improved breakdown behavior of all three techniques furthermore a mixture of normals with an improper uniform distribution is proposed to achieve more robustness in the case of a fixed number of components

the metrically trimmed mean as a robust estimator of location the metrically trimmed mean is defined as the average of observations remaining after a fixed number of outlying observations have been removed a metric the distance from the median is used to determine which points are outlying the influence curve and the asymptotic normality of the metrically trimmed mean are derived using von mises expansions the relative merits of the median the trimmed mean and the metrically trimmed mean are discussed in neighborhoods of nonparametric models with natural parameters it is observed that the metrically trimmed mean works well for the center of symmetry of a symmetric distribution function with asymmetric contamination a multivariate extension of the metrically trimmed mean is discussed