assessing the order of dependence for partially exchangeable binary data abstract the problem we consider is how to assess the order of serial dependence within partially exchangeable binary sequences we obtain exact conditional tests comparing any two orders by finding the conditional distribution of data given certain transition counts these tests are facilitated with a new monte carlo scheme asymptotic tests are also discussed in particular we show that the likelihood ratio tests have an asymptotic chi squared distribution thus generalizing the results of billingsley for the particular case of markov chains we apply these methods to several datasets and perform a simulation to study their properties

randomized polynuclear growth with a columnar defect we study a variant of poly nuclear growth where the level boundaries perform continuous time discrete space random walks and study how its asymptotic behavior is affected by the presence of a columnar defect on the line we prove that there is a non trivial phase transition in the strength of the perturbation above which the law of large numbers for the height function is modified

object tracking using deformable templates we propose a method for object tracking using prototype based deformable template models to track an object in an image sequence we use a criterion which combines two terms the frame to frame deviations of the object shape and the fidelity of the modeled shape to the input image the deformable template model utilizes the prior shape information which is extracted from the previous frames along with a systematic shape deformation scheme to model the object shape in a new frame the following image information is used in the tracking process 1 edge and gradient information the object boundary consists of pixels with large image gradient 2 region consistency the same object region possesses consistent color and texture throughout the sequence and 3 interframe motion the boundary of a moving object is characterized by large interframe motion the tracking proceeds by optimizing an objective function which combines both the shape deformation and the fidelity of the modeled shape to the current image in terms of gradient texture and interframe motion the inherent structure in the deformable template together with region motion and image gradient cues makes the proposed algorithm relatively insensitive to the adverse effects of weak image features and moderate amounts of occlusion

nonnegative idempotent matrices and the minus partial order abstract we describe the structure of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus order

the star partial order and the eigenprojection at 0 on ep matrices abstract the space of n n complex matrices with the star partial order is considered in the first part of this paper the class of ep matrices is analyzed and several properties related to this order are given in addition some information about predecessors and successors of a given ep matrix is obtained the second part is dedicated to the study of some properties that relate the eigenprojection at 0 with the star and sharp partial orders

a bayesian population model with hierarchical mixture priors applied to blood count data abstract population pharmacokinetic and pharmacodynamic studies require analyzing nonlinear growth curves fit to multiple measurements from study subjects we propose a class of nonlinear population models with nonparametric second stage priors for analyzing such data the proposed models apply a flexible class of mixtures to implement the nonparametric second stage the discussion is based on a pharmacodynamic study involving longitudinal data consisting of hematologic profiles i e blood counts measured over time of cancer patients undergoing chemotherapy we describe a full posterior analysis in a bayesian framework this includes prediction of future observations profiles and end points for new patients estimation of the mean response function for observed individuals and inference on population characteristics the mixture model is specified and given a hyperprior distribution by means of a dirichlet processes prior on the mixing measure estimation is implemented by a combination of various m

infimum of a pair of matrices abstract in this paper answers are provided to the following questions given a matrix e1 of order k m does there exist a matrix e2 of order m k m such that the matrix e e 1 e 2 is square idempotent and of the same rank as e1 further when is the choice of e2 unique given a subspace t virtually disjoint with the row span of e1 when does there exist a matrix g g 1 g 2 such that g is square idempotent of rank equal to rank g1 rank e1 and g1 is congruent to e1 mod t these results are then used to determine if a pair of matrices a and b has an infimum in the minus partial order further the existence of the sharp order infimum is studied

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

a schur complement approach to a general extrapolation algorithm this paper is devoted to a schur complement approach to the e transformation which is the most general scalar sequence transformation known so far for accelerating the convergence a new derivation of known results on schur complements is given in the first part of the paper then schur complements and their properties are used to obtain various interpretations of the e transformation the recursive rules of the e algorithm for its implementation are also recovered new results on its kernel are derived and issues on its convergence are discussed this approach can be extended to the vector case thus leading to new vector sequence transformations

star sharp core and dual core partial order in rings with involution the star sharp core and dual core partial orders are already known and investigated in the set of complex matrices in this paper these orders are extensively studied in the context of arbitrary ring with involution it is shown that the condition a b where is one of the considered partial orders defines appropriate idempotents and thus the suitable matrix representations of a and b are obtained using these representations we proved that a b if and only if g b g a where g a is appropriate subset of the set of all generalized inverses of a if is the minus partial order the conditions under which a b implies a b are determined also several well known results are generalized and some new results are proved

directed polymers on trees a martingale approach the authors use martingale methods and simple convexity arguments to compute rigorously the limiting free energy in the problem of directed polymers on a tree the limit is a degenerate random variable and convergence holds almost surely the only assumption on the common distribution of the random potentials attached to the bonds of the tree is that its laplace transform exists everywhere in 0 infinity

shorted matrices an extended concept and some applications the shorting of an operator hitherto considered by krein 11 and by anderson and trapp 3 only for positive operators is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite some applications of the shorted matrix in mathematical statistics are discussed

a semi parametric bayesian approach to generalized linear mixed models summary the linear mixed e ects model with normal errors is a popular model for the analysis of repeated measures and longitudinal data the generalized linear model is useful for data that have non normal errors but where the errors are uncorrelated a descendant of these two models generates a model for correlated data with non normal errors called the generalized linear mixed model glmm frequentist attempts to fit these models generally rely on approximate results and inference relies on asymptotic assumptions recent advances in computing technology have made bayesian approaches to this class of models computationally feasible markov chain monte carlo methods can be used to obtain exact inference for these models as demonstrated by zeger and karim 6 in the linear or generalized linear mixed model the random e ects are typically taken to have a fully parametric distribution such as the normal distribution in this paper we extend the glmm by allowing the random e ects to have a non parametric prior distribution we do this using a dirichlet process prior for the general distribution of the random e ects the approach easily extends to more general population models we perform computations for the models using the gibbs sampler 1998 john wiley sons ltd

semiparametric regression for clustered data we consider estimation in a semiparametric partially generalised linear model for clustered data using estimating equations a marginal model is assumed where the mean of the outcome variable depends on some covariates parametrically and a cluster level covariate nonparametrically a profile kernel method allowing for working correlation matrices is developed we show that the nonparametric part of the model can be estimated using standard nonparametric methods including smoothing parameter estimation and the parametric part of the model can be estimated in a profile fashion the asymptotic distributions of the parameter estimators are derived and the optimal estimators of both the nonparametric and parametric parts are shown to be obtained when the working correlation matrix equals the actual correlation matrix the asymptotic covariance matrix of the parameter estimator is consistently estimated by the sandwich estimator we show that the semiparametric efficient score takes on a simple form and our profile kernel method is semiparametric efficient the results for the case where the nonparametric part of the model is an observation level covariate are noted to be dramatically different

more on maximal and minimal ranks of schur complements with applications the maximal and minimal ranks of the schur complement d cgb are determined with respect to generalized inverse g of a matrix a where g is taken respectively as least squares g inverse a 1 3 minimum norm g inverse a 1 4 as well as a 1 2 3 a 1 2 4 and a 1 3 4 various consequences are also presented

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

unpredictable paths and percolation we construct a nearest neighbor process s n on z that is less predictable than simple random walk in the sense that given the process until time n the conditional probability that s n k x is uniformly bounded by ck for some 1 2 from this process we obtain a probability measure on oriented paths in z 3 such that the number of intersections of two paths chosen independently according to has an exponential tail for d 4 the uniform measure on oriented paths from the origin in z d has this property we show that on any graph where such a measure on paths exists oriented percolation clusters are transient if the retention parameter p is close enough to 1 this yields an extension of a theorem of grimmett kesten and zhang who proved that supercritical percolation clusters in z d are transient for all d 3

searching for a trail of evidence in a maze consider a graph with a set of vertices and oriented edges connecting pairs of vertices each vertex is associated with a random variable and these are assumed to be independent in this setting suppose we wish to solve the following hypothesis testing problem under the null the random variables have common distribution n 0 1 while under the alternative there is an unknown path along which random variables have distribution n 1 0 and distribution n 0 1 away from it for which values of the mean shiftcan one reliably detect and for which values is this impossible this paper develops detection thresholds for two types of common graphs which exhibit a different behavior the first is the usual regular lattice with vertices of the form i j 0 i i j i and j has the parity of i and oriented edges i j i 1 j s where s 1 we show that for paths of length m start ing at the origin the hypotheses become distinguishable in a minimax sense ifm log m while they are not ifm log m we derive equivalent results in a bayesian setting where one assumes that all paths are equally likely there the asymptotic threshold ism m 1 4 we obtain corresponding results for trees where the threshold is of order 1 and independent of the size of the tree for distributions other than the gaussian and for other graphs the concept of predictability profile first introduced by benjamini pemantle and peres plays a crucial role in our analysis

energy landscape for large average submatrix detection problems in gaussian random matrices the problem of finding large average submatrices of a real valued matrix arises in the exploratory analysis of data from a variety of disciplines ranging from genomics to social sciences in this paper we provide a detailed asymptotic analysis of large average submatrices of an n times n gaussian random matrix the first part of the paper addresses global maxima for fixed k we identify the average and the joint distribution of the k times k submatrix having largest average value as a dual result we establish that the size of the largest square sub matrix with average bigger than a fixed positive constant is with high probability equal to one of two consecutive integers that depend on the threshold and the matrix dimension n the second part of the paper addresses local maxima specifically we consider submatrices with dominant row and column sums that arise as the local optima of iterative search procedures for large average submatrices for fixed k we identify the limiting average value and joint distribution of a k times k submatrix conditioned to be a local maxima in order to understand the density of such local optima and explain the quick convergence of such iterative procedures we analyze the number l n k of local maxima beginning with exact asymptotic expressions for the mean and fluctuation behavior of l n k for fixed k the mean of l n k is theta n k log n k 1 2 while the standard deviation is theta n 2k 2 k 1 log n k 2 k 1 our principal result is a gaussian central limit theorem for l n k that is based on a new variant of stein s method

unbiased and minimum variance unbiased estimation of estimable functions for fixed linear models with arbitrary covariance structure consider a general linear model for a column vector y of data having e y xa and var y ur2h where a is a vector of unknown parameters and x and h are given matrices that are possibly deficient in rank let b ty where t is any matrix of maximum rank such that th 4 the estimation of a linear function of a by functions of the form c a y where c and a are permitted to depend on b is investigated allowing c gnd a to depend on b expands the class of unbiased estimators in a nontrivial way however it does not add to the class of linear functions of a that are estimable any minimumvariance unbiased estimator is identically for y in the column space of x h equal to the estimator that has minimum variance among strictly linear unbiased estimators

the minus order and range additivity abstract we study the minus order on the algebra of bounded linear operators on a hilbert space by giving a characterization in terms of range additivity we show that the intrinsic nature of the minus order is algebraic applications to generalized inverses of the sum of two operators to systems of operator equations and to optimization problems are also presented

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

a semiparametric bayesian model for randomised block designs summary a model is proposed for a bayesian semiparametric analysis of randomised block experiments the model is a hierarchical model in which a dirichlet process is inserted at the middle stage for the distribution of the block effects this model allows an arbitrary distribution of block effects and it results in effective estimates of treatment contrasts block effects and the distribution of block effects an effective computational strategy is presented for describing the posterior distribution

a note on the rank subtractivity ordering abstract some well known matrix partial orderings can be created by supplementing the so called rank subtractivity partial ordering by an appropriate preordering in this note a new partial ordering with the same property is introduced this new ordering is seen to be weaker than the other orderings but still preserving some inheritance properties

distinguishing infections on different graph topologies the history of infections and epidemics holds famous examples where understanding containing and ultimately treating an outbreak began with understanding its mode of spread influenza hiv and most computer viruses spread person to person device to device and through contact networks cholera cancer and seasonal allergies on the other hand do not in this paper we study two fundamental questions of detection first given a snapshot view of a perhaps vanishingly small fraction of those infected under what conditions is an epidemic spreading via contact e g influenza distinguishable from a random illness operating independently of any contact network e g seasonal allergies second if we do have an epidemic under what conditions is it possible to determine which network of interactions is the main cause of the spread the causative network without any knowledge of the epidemic other than the identity of a minuscule subsample of infected nodes the core therefore of this paper is to obtain an understanding of the diagnostic power of network information we derive sufficient conditions that networks must satisfy for these problems to be identifiable and produce efficient highly scalable algorithms that solve these problems we show that the identifiability condition we give is fairly mild and in particular is satisfied by two common graph topologies the d dimensional grid and the erdos renyi graphs

a method for combining inference across related nonparametric bayesian models summary we consider the problem of combining inference in related nonparametric bayes models analogous to parametric hierarchical models the hierarchical extension formalizes borrowing strength across the related submodels in the nonparametric context modelling is complicated by the fact that the random quantities over which we define the hierarchy are infinite dimensional we discuss a formal definition of such a hierarchical model the approach includes a regression at the level of the nonparametric model for the special case of dirichlet process mixtures we develop a markov chain monte carlo scheme to allow efficient implementation of full posterior inference in the given model

an anova model for dependent random measures we consider dependent nonparametric models for related random probability distributions for example the random distributions might be indexed by a categorical covariate indicating the treatment levels in a clinical trial and might represent random effects distributions under the respective treatment combinations we propose a model that describes dependence across random distributions in an analysis of variance anova type fashion we define a probability model in such a way that marginally each random measure follows a dirichlet process dp and use the dependent dirichlet process to define the desired dependence across the related random measures the resulting probability model can alternatively be described as a mixture of anova models with a dp prior on the unknown mixing measure the main features of the proposed approach are ease of interpretation and computational simplicity because the model follows the standard anovastructure interpretation and inference parallels conventions for anova mode

semiparametric regression for clustered data using generalized estimating equations we consider estimation in a semiparametric generalized linear model for clustered data using estimating equations our results apply to the case where the number of observations per cluster is finite whereas the number of clusters is large the mean of the outcome variable is of the form g xt t where g is a link function x and t are covariates is an unknown parameter vector and t is an unknown smooth function kernel estimating equations proposed previously in the literature are used to estimate the infinite dimensional nonparametric function t and a profile based estimating equation is used to estimate the finite dimensional parameter vector we show that for clustered data this conventional profile kernel method often fails to yield a n consistent estimator of along with appropriate inference unless working independence is assumed or t is artificially undersmoothed in which case asymptotic inference is possible to gain insight into these results we derive the se

a bayesian semiparametric model for bivariate sparse longitudinal data mixed effects models have recently become popular for analyzing sparse longitudinal data that arise naturally in biological agricultural and biomedical studies traditional approaches assume independent residuals over time and explain the longitudinal dependence by random effects however when bivariate or multivariate traits are measured longitudinally this fundamental assumption is likely to be violated because of intertrait dependence over time we provide a more general framework where the dependence of the observations from the same subject over time is not assumed to be explained completely by the random effects of the model we propose a novel mixed model based approach and estimate the error covariance structure nonparametrically under a generalized linear model framework we use penalized splines to model the general effect of time and we consider a dirichlet process mixture of normal prior for the random effects distribution we analyze blood pressure data from the framingham heart study where body mass index gender and time are treated as covariates we compare our method with traditional methods including parametric modeling of the random effects and independent residual errors over time we conduct extensive simulation studies to investigate the practical usefulness of the proposed method the current approach is very helpful in analyzing bivariate irregular longitudinal traits copyright 2013 john wiley sons ltd

perturbation bounds for constrained and weighted least squares problems we derive perturbation bounds for the constrained and weighted linear least squares ls problems both the full rank and rank deficient cases are considered the analysis generalizes some results of earlier works

clustering based on pairwise distances when the data is of mixed dimensions in the context of clustering we consider a generative model in a euclidean ambient space with clusters of different shapes dimensions sizes and densities in an asymptotic setting where the number of points becomes large we obtain theoretical guaranties for some emblematic methods based on pairwise distances a simple algorithm based on the extraction of connected components in a neighborhood graph hierarchical clustering with single linkage and the spectral clustering method of ng jordan and weiss the methods are shown to enjoy some near optimal properties in terms of separation between clusters and robustness to outliers the local scaling method of zelnik manor and perona is shown to lead to a near optimal choice for the scale in the first and third methods we also provide a lower bound on the spectral gap to consistently choose the correct number of clusters in the spectral method

partial orders based on outer inverses classes of outer inverses are used to create new partial orders and unify the classification of some of the old partial orders these classes are then compared with those obtained by using g inverses inner inverses

bilateral shorted operators and parallel sums in this paper we study shorted operators relative to two different subspaces for bounded operators on infinite dimensional hilbert spaces we define two notions of complementability in the sense of ando for operators and study the properties of the shorted operators when they can be defined we use these facts in order to define and study the notions of parallel sum and substraction in this hilbertian context

on parallel summability of matrices abstract it is shown that the generalized inverses characterize the parallel sum the almost positive definite a p d matrices introduced by duffin and morley 2 are of two types whose intersection is the class of quasi positive definite matrices mitra and puri 7 the a p d matrices of any one type form a saturated subclass of pairwise parallel summable a p d matrices

linear preservers for matrix inequalities and partial orderings we characterize the linear preservers of minus partial order on matrix algebras the developed approach allows us to classify linear transformations that live fixed several matrix relations arising as extremal cases in some classical matrix inequalities including rank additivity relation and related properties applications to the determinant preservers are considered

detection of an anomalous cluster in a network we consider the problem of detecting whether or not in a given sensor network there is a cluster of sensors which exhibit an unusual behavior formally suppose we are given a set of nodes and attach a random variable to each node we observe a realization of this process and want to decide between the following two hypotheses under the null the variables are i i d standard normal under the alternative there is a cluster of variables that are i i d normal with positive mean and unit variance while the rest are i i d standard normal we also address surveillance settings where each sensor in the network collects information over time the resulting model is similar now with a time series attached to each node we again observe the process over time and want to decide between the null where all the variables are i i d standard normal and the alternative where there is an emerging cluster of i i d normal variables with positive mean and unit variance the growth models used to represent the emerging cluster are quite general and in particular include cellular automata used in modelling epidemics in both settings we consider classes of clusters that are quite general for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic by far the most popular method in practice achieves that same rate within a logarithmic factor our results are not limited to the normal location model but generalize to any one parameter exponential family when the anoma lous clusters are large enough

lattice properties of the order for complex matrices abstract it is shown that the set c m n of complex m n matrices forms a lower semilattice under the partial ordering a b defined by a a a b aa ba where a denotes the conjugate transpose of a as a special case of a result for division rings it is further shown that over any field f for m n 2 and any proper involution of f2 2 the corresponding intersections a b all exist

on the limitation of spectral methods from the gaussian hidden clique problem to rank one perturbations of gaussian tensors we consider the following detection problem given a realization of a symmetric matrix x of dimension n distinguish between the hypothesis that all upper triangular variables are independent and identically distributed i i d gaussians variables with mean 0 and variance 1 and the hypothesis where x is the sum of such matrix and an independent rank one perturbation this setup applies to the situation where under the alternative there is a planted principal submatrix b of size l for which all upper triangular variables are i i d gaussians with mean 1 and variance 1 whereas all other upper triangular elements of x not in b are i i d gaussians variables with mean 0 and variance 1 we refer to this as the gaussian hidden clique problem when l 1 epsilon sqrt n epsilon 0 it is possible to solve this detection problem with probability 1 o n 1 by computing the spectrum of x and considering the largest eigenvalue of x we prove that this condition is tight in the following sense when l no algorithm that examines only the eigenvalues of x can detect the existence of a hidden gaussian clique with error probability vanishing as n to infty we prove this result as an immediate consequence of a more general result on rank one perturbations of k dimensional gaussian tensors in this context we establish a lower bound on the critical signal to noise ratio below which a rank one signal cannot be detected

dirichlet process mixed generalized linear models abstract although generalized linear models glm s are an attractive and widely used class of models they are limited in the range of density shapes that they can provide for instance they are unimodal exponential families of densities in the response variable with tail behavior determined by the implicit mean variance relationship dirichlet process dp mixing adds considerable flexibility to these models using such mixing we develop models that we call dpmglm s which still retain the glm character with regard to the mean overdispersed glm s oglm s provide an alternative class of models to cope with extra variability in samples we show that how oglm s may be dp mixed leading to what we call dpmoglm s these models are extremely rich moreover recent computational advances enable them to be fitted straightforwardly we illustrate this with both simulated and real datasets we also address the question of choosing between the glm oglm dpmglm and dpmoglm finally we consider extensions by

independent and identically distributed monte carlo algorithms for semiparametric linear mixed models hybrid versions of independent and identically distributed weighted chinese restaurant wcr algorithms are developed for inference in semiparametric linear mixed models under minimal assumptions for the random effects distributions the wcr method of working with the posterior partition structure leads to rao blackwell estimates for higher order moments of random effects such as skewness and kurtosis and can be used to estimate densities for random effects a key feature of our approach is the manner in which we incorporate external estimates into our algorithms the use of such information leads to simplified computational procedures reduces the amount of user input required for specifying models and results in numerical stability and accuracy the resulting procedures are automated and can be readily used in standard statistical software our methods are tested by simulation and illustrated by application to a longitudinal study involving chronic renal disease

some more properties of core partial order in this paper the core partial order introduced by baksalary and trenkler has been studied further new characterizations of the core partial order have been derived relationship between the core partial order and some known partial orders has been also investigated

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

characterizations of the best linear unbiased estimator in the general gauss markov model with the use of matrix partial orderings abstract under the general gauss markov model y x 2 v two new characterizations of blue x are derived involving the lowner and rank subtractivity partial orderings between the dispersion matrix of blue x and the dispersion matrix of y as particular cases of these characterizations three new criteria for the equality between olse x and blue x are given

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

extending some results and proofs for the singular linear model abstract this note gives some new forms and shortened proofs for results on the linear model with singular dispersion matrix

automorphisms of mn partially ordered by rank subtractivity ordering let mn be the space of all n n matrices with coefficients in r or c where n 3 the rank subtractivity or minus partial order on mn is defined by a rsb iff rankb ranka rank b a we characterize bijective mappings on mn such that a rsb iff a rs b a b mn

reverse order laws for generalized inverses of multiple matrix products abstract in this article we study reverse order laws for generalized inverses and reflexive generalized inverses of the products of multiple matrices a 1 a n and the products of generalized inverses and reflexive generalized inverses of a n a 1 by applying the multiple product singular value decomposition we obtain necessary and sufficient conditions for one side inclusion relation of reverse order law for generalized inverses and necessary and sufficient conditions of reverse order law for reflexive generalized inverses

rank equalities for idempotent and involutary matrices abstract we establish several rank equalities for idempotent and involutary matrices in particular we obtain new formulas for the rank of the difference the sum the product and the commutator of idempotent or involutary matrices extensions to scalar potent matrices are also included our matrices are complex and are not necessarily hermitian

on some matrix equalities for generalized inverses with applications abstract necessary and sufficient conditions are derived for the matrix equality a pn q to hold where a and n are generalized inverses of matrices some consequences and applications are also given in particular necessary and sufficient conditions are derived for the additive decompositions c a b and a a 1 a k to hold

on combinatorial testing problems we study a class of hypothesis testing problems in which upon observing the realization of an n dimensional gaussian vector one has to decide whether the vector was drawn from a standard normal distribution or alternatively whether there is a subset of the components belonging to a certain given class of sets whose elements have been contaminated that is have a mean different from zero we establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk the combinatorial and geometric structure of the class of sets is shown to play a crucial role the bounds are illustrated on various examples

detecting anomalous activity on networks with the graph fourier scan statistic we consider the problem of deciding based on a single noisy measurement at each vertex of a given graph whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation this problem is relevant to several applications such as surveillance disease outbreak detection biomedical imaging environmental monitoring etc since the activations in these problems often tend to be localized to small groups of vertices in the graphs we model such activity by a class of signals that are elevated over a possibly disconnected cluster with low cut size relative to its size we analyze the corresponding generalized likelihood ratio glr statistics and relate it to the problem of finding a sparsest cut in the graph we develop a convex relaxation of the glr statistic based on spectral graph theory which we call the graph fourier scan statistic gfss in our main theoretical result we show that the performance of the gfss depends explicitly on the spectral properties of the graph to assess the optimality of the gfss we prove an information theoretic lower bound for the detection of anomalous activity on graphs because the gfss requires the specification of a tuning parameter we develop an adaptive version of the gfss using these results we are able to characterize in a very explicit form the performance of the gfss on a few notable graph topologies we demonstrate that the gfss can efficiently detect a simulated arsenic contamination in groundwater

simultaneous diagonalization of rectangular matrices abstract a matrix d is said to be diagonal if its i j th element is null whenever i and j are unequal for a set a of matrices a of the same order the paper gives necessary and sufficient conditions for nonsingular matrices s and t to exist such that sa t d is diagonal for each matrix a in the set

nonnegative group monotone matrices and the minus partial order bapat et al previously described a class of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus partial order in this paper we improve upon that description by first presenting a more general result that gives the precise structure of nonnegative matrices dominated by a group monotone matrix under the minus partial order as a special case we derive the complete class of nonnegative matrices dominated by a nonnegative idempotent matrix that includes the class obtained by bapat et al

the least squares g inverse for the sum of multiple matrices in this paper we give the necessary and sufficient conditions for the set identities a 1 a 2 a k 1 3 a 1 1 3 a 2 1 3 a k 1 3 and a 1 a 2 a k 1 4 a 1 1 4 a 2 1 4 a k 1 4 which concern the least squares g inverse 1 3 inverse and the minimum norm g inverse 1 4 inverse of sum of k matrices

detection of correlations with adaptive sensing the problem of detecting correlations from samples of a high dimensional gaussian vector has recently received a lot of attention in most existing work detection procedures are provided with a full sample however following common wisdom in experimental design the experimenter may have the capacity to make targeted measurements in an on line and adaptive manner in this paper we investigate such adaptive sensing procedures for detecting positive correlations it is shown that using the same number of measurements adaptive procedures are able to detect significantly weaker correlations than their nonadaptive counterparts we also establish minimax lower bounds that show the limitations of any procedure keywords sequential testing adaptive sensing high dimensional detection highdimensional detection sequential testing sparse covariance matrices sparse principal component analysis

nearest neighbor walks with low predictability profile and percolation in 2 epsilon dimensions a few years ago grimmett kesten and zhang proved that for supercritical bond percolation on z 3 simple random walk on the infinite cluster is a s transient we generalize this result to a class of wedges in z 3 including for any e 0 1 the wedge w e x y z z 3 x 0 z x e which can be thought of as representing a 2 e dimensional lattice our proof builds on recent work of benjamini pemantle and peres and involves the construction of finite energy flows using nearest neighbor walks on z with low predictability profile along the way we obtain some new results on attainable decay rates for predictability profiles of nearest neighbor walks

nonparametric detection of geometric structures over networks nonparametric detection of the possible existence of an anomalous structure over a network is investigated nodes corresponding to the anomalous structure if one exists receive samples generated by a distribution q which is different from a distribution p generating samples for other nodes if an anomalous structure does not exist all nodes receive samples generated by p it is assumed that the distributions p and q are arbitrary and unknown the goal is to design statistically consistent tests with probability of errors converging to zero as the network size becomes asymptotically large kernel based tests are proposed based on maximum mean discrepancy which measures the distance between mean embeddings of distributions into a reproducing kernel hilbert space detection of an anomalous interval over a line network is first studied sufficient conditions on minimum and maximum sizes of candidate anomalous intervals are characterized in order to guarantee that the proposed test is consistent it is also shown that certain necessary conditions must hold in order to guarantee that any test is universally consistent comparison of sufficient and necessary conditions yields that the proposed test is order level optimal and nearly optimal respectively in terms of minimum and maximum sizes of candidate anomalous intervals generalization of the results to other networks is further developed numerical results are provided to demonstrate the performance of the proposed tests

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

some further remarks on the singular linear model abstract inspired by a recent article of searle 1994 we consider some specific features of estimation of x in the general linear model y x v where the model matrix x need not have full column rank and the dispersion matrix v can be singular particular attention is paid to the problems related to the invariance of some matrix expressions with respect to the choice of generalized inverses these invariance properties yield interesting matrix algebra while characterizing the best linear unbiased estimator of x

unpredictable nearest neighbor processes benjamini pemantle and peres constructed nearest neighbor processes which have predictability profiles that decay faster than that of the simple random walk haggstrom and mossel found processes with even faster decaying predictability profiles we prove that the rate of decay achieved by haggstrom and mossel is optimal

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

optimal detection of sparse mixtures against a given null distribution detection of sparse signals arises in a wide range of modern scientific studies the focus so far has been mainly on gaussian mixture models in this paper we consider the detection problem under a general sparse mixture model and obtain explicit expressions for the detection boundary under mild regularity conditions in addition for gaussian null hypothesis we establish the adaptive optimality of the higher criticism procedure for all sparse mixtures satisfying the same conditions in particular the general results obtained in this paper recover and extend in a unified manner the previously known results on sparse detection far beyond the conventional gaussian model and other exponential families

reverse order laws for the generalized inverses of multiple matrix products by using the ranks of matrices this article gives necessary and sufficient conditions for the reverse product a n a 2a 1of the moore penrose inverses of a1 a2 an to a 1 inverse inner inverse 1 2 inverse reflexive inner inverse 1 3 inverse 1 4 inverse 1 2 3 inverse 1 2 4 inverse or moore penrose inverse of matrix product a a1a2 an

statistical data analysis and inference prologue c radhakrishna rao s contributions to statistics y dodge nonlinear estimation in linear models f hampel on the maximum number of factors in two construction methods for orthogonal arrays a s hedayat j stufken some geometrical aspects of data analysis and statistics j m oller inferences and test of hypotheses on bayes and empirical bayes two stage allocation procedures for selection problems s s gupta t c liang multi sample functional statistical data analysis e parzen discussing truth or falsity by computing a q value w schaafsma j tolboom b van der meulen linear pivotals and the bayes non bayes compromise g a barnard bayes modified minimax experimental design r v canfield a simple class of tests locally better than the score test t k chandra comparison of tests in the presence of a nuisance parameter r mukerjee tests for redundancy of some variables in multivariate analysis y fujikoshi model choice in the context of simultaneous inference t havranek o soudsky first zeros of empirical characteristic functions and extreme values of gaussian processes j husler parameter and variance estimation comparison of experiments with weighted distributions m j bayarri m h de groot optimal estimation for weighted distributions semi parametric models v p godambe m b rajarshi on efficiency for quasi likelihood and composite quasi likelihood methods c c heyde improved estimators of dispersion of an inverse gaussian distribution n pal b k sinha applications of edgeworth expansions to bootstrap a review g j babu dual poincare type inequalities via the cramer rao and the cauchy schwarz inequalities and related characterizations th cacoullos some optimality results on stein s two stage sampling j k ghosh r mukerjee the cramer frechet rao inequality for sequential estimation in non regular case z govindarajulu i vincze whither delete k jackknifing for smooth statistical functionals p k sen distributions weighted distributions and characterization three useful expressions for expectations involving a wishart matrix and its inverse g p h styan characterizations of distributions via moments of order statistics a survey and comparison of methods g d lin constancy of regression of a polynomial of sample average of positive random variables on their ratios characterizes gamma distribution a kagan probing encountered data meta analysis and weighted distribution methods g p patil c taillie further results on identification when the parameters are partially unknown s el khattabi f streit linear models and matrices weighted least squares estimation in the general gauss markov model j k baksalary s puntanen evaluating pre test predictors of success in linear regression models a cohen h b sackrowitz optimum invariant tests in mixed linear models with two variance components th mathew

semiparametric bayesian inference for multilevel repeated measurement data we discuss inference for data with repeated measurements at multiple levels the motivating example is data with blood counts from cancer patients undergoing multiple cycles of chemotherapy with days nested within cycles some inference questions relate to repeated measurements over days within cycle while other questions are concerned with the dependence across cycles when the desired inference relates to both levels of repetition it becomes important to reflect the data structure in the model we develop a semiparametric bayesian modeling approach restricting attention to two levels of repeated measurements for the top level longitudinal sampling model we use random effects to introduce the desired dependence across repeated measurements we use a nonparametric prior for the random effects distribution inference about dependence across second level repetition is implemented by the clustering implied in the nonparametric random effects model practical use of the model requires that the posterior distribution on the latent random effects be reasonably precise

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses