a characterization of markov equivalence classes for acyclic digraphs undirected graphs and acyclic digraphs adgs as well as their mutual extension to chain graphs are widely used to describe dependencies among variables in multivariate distributions in particular the likelihood functions of adg models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building bayesian networks for expert systems whereas the undirected graph associated with a dependence model is uniquely determined there may however be many adgs that determine the same dependence markov model thus the family of all adgs with a given set of vertices is naturally partitioned into markov equivalence classes each class being associated with a unique statistical model statistical procedures such as model selection or model averaging that fail to take into account these equivalence classes may incur substantial computational or other inefficiencies here it is shown that each markov equivalence class is uniquely determined by a single chain graph the essential graph that is itself simultaneously markov equivalent to all adgs in the equivalence class essential graphs are characterized a polynomial time algorithm for their construction is given and their applications to model selection and other statistical questions are described

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

global identifiability of linear structural equation models structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables we consider the classical case of linear relationships and additive gaussian noise terms we give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms global identifiability is understood to mean injectivity of the parametrization of the model and is fundamental in particular for applicability of standard statistical methodology

multiplicative lidskii s inequalities and optimal perturbations of frames abstract in this paper we study two design problems in frame theory on the one hand given a fixed finite frame f f j j i n for c d we compute those dual frames g of f that are optimal perturbations of the canonical dual frame for f under certain restrictions on the norms of the elements of g on the other hand we compute those v f v f j j i n for invertible operators v which are close to the identity that are optimal perturbations of f that is we compute the optimal perturbations of f among frames g g j j i n that have the same linear relations as f in both cases optimality is measured with respect to submajorization of the eigenvalues of the frame operators hence our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error the key tool for these results is a multiplicative analogue of lidskii s inequality in terms of log majorization and a characterization of the case of equality

causal inference for complex longitudinal data the continuous case we extend robins theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments in particular we establish versions of the key results of the discrete theory the g computation formula and a collection of powerful characterizations of the g null hypothesis of no treatment eect this is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past we also show that our assumptions concerning counterfactual variables place no restriction on the joint distribution of the observed variables thus in a precise sense these assumptions are for free or if you prefer harmless

graphical methods for efficient likelihood inference in gaussian covariance models in graphical modelling a bi directed graph encodes marginal independences among random variables that are identified with the vertices of the graph we show how to transform a bi directed graph into a maximal ancestral graph that i represents the same independence structure as the original bi directed graph and ii minimizes the number of arrowheads among all ancestral graphs satisfying i here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi directed edges in gaussian models this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts

wielandt and ky fan theorem for matrix pairs the generalization of wielandt and ky fan theorem is given for hermitian matrix pairs and some new eigenvalue perturbation estimates are obtained an application is made on a class of quadratic matrix pencils

estimation of a covariance matrix with zeros summary we consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero we first present an algorithm which we call iterative conditional fitting for computing the maximum likelihood estimate of the constrained covariance matrix under the assumption of multivariate normality in contrast to previous approaches this algorithm has guaranteed convergence properties dropping the assumption of multivariate normality we show how to estimate the covariance matrix in an empirical likelihood approach these approaches are then compared via simulation and on an example of gene expression

markov equivalence for ancestral graphs ancestral graphs can encode conditional independence relations that arise in directed acyclic graph dag models with latent and selection variables however for any ancestral graph there may be several other graphs to which it is markov equivalent we state and prove conditions under which two maximal ancestral graphs are markov equivalent to each other thereby extending analogous results for dags given by other authors these conditions lead to an algorithm for determining markov equivalence that runs in time that is polynomial in the number of vertices in the graph

multiplicative versions of klyachko s theorem in finite factors abstract let a and b be invertible positive elements in a ii 1 factor a and let s be the singular number on a we prove that exp k log s ab d s exp i log s a d s exp j log s b d s where i j k is an analogue of klyachko s list in this paper this family i j k must satisfy some hypotheses which are specific to operators a and b but we show that our family of inequalities includes the weak gelfand naimark inequality for all positive operators a and b

an incomplete data approach to the analysis of covariance structures in this paper linear structural equation models with latent variables are considered it is shown how many common models arise from incomplete observation of a relatively simple system subclasses of models with conditional independence interpretations are also discussed using an incomplete data point of view the relationships between the incomplete and complete data likelihoods assuming normality are highlighted for computing maximum likelihood estimates the em algorithm and alternatives are surveyed for the alternative algorithms simplified expressions for computing function values and derivatives are given likelihood ratio tests based on complete and incomplete data are related and an example on using their relationship to improve the fit of a model is given

a linear recognition algorithm for cographs cographs are the graphs formed from a single vertex under the closure of the operations of union and complement another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms furthermore it is known that cographs have a unique tree representation called a cotree using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general such problems include chromatic number clique determination clustering minimum weight domination isomorphism minimum fill in and hamiltonicity in this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation

covariance hypotheses which are linear in both the covariance and the inverse covariance both the covariance and the inverse covariance are products of models each of which consists of either i independent identically distributed random vectors which have a covariance with a real complex or quaternion structure or ii independent identically distributed random vectors with a parametrization of the covariance which is given by means of the clifford algebra the models i are well known for models ii we have found under the assumption that the distribution is nornal the exact distributions of the maximum likelihood estimates and the likelihood ratio test statistics 1 introduction the purpose of the present paper is to describe for the family of normal distributions the structure of those statistical hypotheses models which are linear in both the covariance and the inverse covariance in section 2 we show that under such a hypothesis the problem of maximum likelihood estimation of the covariance has an explicit solution and that these hypotheses are parametrized by jordan algebras quadratic subspaces based on the theory of jordan algebras it is then possible to give a canonical form for such hypotheses in section 3 we show that the jordan algebras are products of so called simple jordan algebras and in section 4 we prove that the hypotheses can be decomposed into products of hypotheses parametrized by simple jordan algebras the simple jordan algebras are classified according to their so called degree the only simple jordan algebras of degree 1 is the set of real numbers and corresponding to that we have the hypothesis under which the covariance is proportional to a known covariance in section 5 we give a canonical form for the hypotheses which are parametrized by simple jordan algebras of degree greater than 3 it tums out that these hypotheses are equivalent to those consisting of independent identically distributed random vectors which have a covariance with a real complex or quatemion structure and the distributions of the maximum likelihood estimates are given by wishart 1928 goodman 1963 and andersson 1975 respectively the statistical hypotheses which are parametrized by simple jordan algebras of degree 2 are new and in section 6 we give a complete solution to the problems of maximum likelihood inference it turns out that the distribution problems are not more complicated than those for the two dimensional normal distribution the mathematical structure of the hypotheses is closely related to the theory of spinors in quantum mechanics based on that theory we give a canonical form for such hypotheses in section 7

different asymptotic behavior of global solutions for a parabolic system with nonlinear gradient terms abstract in this paper we study the global existence and the different asymptotic behavior of mild solutions for the nonlinear parabolic system t u u a v p t v v b u q t 0 x r n where a b r n 1 1 p q 2 and p q q n 1 n 2 n 1 we prove in particular that if the initial values behave as u 0 x 1 x x x v 0 x 2 x x x as x 0 n 2 q q 2 p p and under suitable conditions on 1 2 then the resulting solutions are global furthermore although the scaling invariance properties of these initial values and the system are different we prove that some of the solutions are asymptotic to self similar solutions of appropriate asymptotic systems which depend on the values of and the asymptotic behavior estimates are given in the w 1 r n w 1 r n norm and are stable under some small perturbations the results of this paper complete those of al elaiw and tayachi 2010 1 known only for 2 q q and 2 p p

weyl type relative perturbation bounds for eigensystems of hermitian matrices we present a weyl type relative bound for eigenvalues of hermitian perturbations a e of not necessarily definite hermitian matrices a this bound given in function of the quantity a 1 2ea 1 2 2 that was already known in the definite case is shown to be valid as well in the indefinite case we also extend to the indefinite case relative eigenvector bounds which depend on the same quantity as a consequence new relative perturbation bounds for singular values and vectors are also obtained using matrix differential calculus techniques we obtain for eigenvalues a sharper first order bound involving the logarithm matrix function which is smaller than not only for small e as expected but for any perturbation

jensen and minkowski inequalities for operator means and anti norms abstract jensen inequalities for positive linear maps of choi and hansen pedersen type are established for a large class of operator matrix means such as some p means and some kubo ando means these results are also extensions of the minkowski determinantal inequality to this end we develop the study of anti norms a notion parallel to the symmetric norms in matrix analysis including functionals like schatten q norms for a parameter q 1 and the minkowski functional det 1 n a

relative perturbation theory iv sin 2θ theorems abstract the double angle theorems of davis and kahan bound the change in an invariant subspace when a hermitian matrix a is subject to an additive perturbation a a a a this paper supplies analogous results when a is subject to a congruential or multiplicative perturbation a a d ad the relative gaps that appear in the bounds involve the spectrum of only one matrix either a or a in contrast to the gaps that appear in the single angle bounds the double angle theorems do not directly bound the difference between the old invariant subspace s and the new one s but instead bound the difference between s and its reflection j s where the mirror is s and j reverses s the orthogonal complement of s the double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix d def d 1 jdj note that d is invariant under the transformation d d for 0 whereas the single angle theorems give bounds proportional to d s departure from the identity and from orthogonality the corresponding results for the singular value problem when a nonsquare matrix b is perturbed to b d 1 bd 2 are also presented

graphical gaussian modelling of multivariate time series with latent variable in time series analysis inference about causeeffect relationships among multiple times series is commonly based on the concept of granger causality which exploits temporal structure to achieve causal ordering of dependent variables one major problem in the application of granger causality for the identification of causal relationships is the possible presence of latent variables that affect the measured components and thus lead to so called spurious causalities in this paper we describe a new graphical approach for modelling the dependence structure of multivariate stationary time series that are affected by latent variables to this end we introduce dynamic maximal ancestral graphs dmags in which each time series is represented by a single vertex for gaussian processes this approach leads to vector autoregressive models with errors that are not independent but correlated according to the dashed edges in the graph we discuss identifiability of the parameters and show that these models can be viewed as graphical arma models that satisfy the granger causality restrictions encoded by the associated dynamic maximal ancestral graph

asymptotically self similar global solutions of a general semilinear heat equation we consider the general nonlinear heat equation partial t u delta u a u p 1 1 u g u u 0 varphi on 0 infty times i r n where a in i r p 1 1 2 n and g satisfies certain growth conditions we prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation we also prove that some of those global solutions are asymptotic for large time to self similar solutions of the single power nonlinear heat equation i e with g equiv 0

relative perturbation theory iii more bounds on eigenvalue variation abstract classically the relative error in ga 1 as an approximation to is measured by relative error in ga a the quantity log10 is usually used for the number of correct decimal digits in numerical results although this measure is clearly not a metric since it lacks symmetry between and ga in part i of this series two other kinds of relative distances which have much better mathematical properties have been introduced and employed to establish theories it is shown that these different measurements are topologically equivalent however the measure is more convenient to use in practice in this part we established relative perturbation bounds directly using the classical measure the new bounds for diagonalizable matrices are cleaner than the corresponding ones in part i and yield nice bounds for hermitian matrices too but when applied to nonnegative definite hermitian matrices the new bounds are weaker than those in part i

asymptotically self similar global solutions of a semilinear parabolic equation with a nonlinear gradient term we study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u 0 where a b q 1 p 1 for q 2p p 1 and 1 p 1 1 equivalently q n 2 n 1 we prove the existence of mild global solutions for small initial data with respect to some norm some of those solutions are asymptotically self similar

joint response graphs and separation induced by triangular systems summary we consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions the independences are captured by missing edges in a directed graph a matrix form of such a graph called the generating edge matrix is triangular so the distributions that are generated over such graphs are called triangular systems we study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models i e for alternative recursive factorizations of the given density using joint conditional distributions for this we introduce families of linear triangular equations which do not require assumptions of distributional form the strength of the associations that are implied by such linear families for chain graph models is derived the edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix it is shown how induced independences and dependences can be studied by graphs by edge matrix calculations and via the properties of densities some ways of using the results are illustrated

relative perturbation techniques for singular value problems a technique is presented for deriving bounds on the relative change in the singular values of a real matrix or the eigenvalues of a real symmetric matrix due to a perturbation as well as bounds on the angles between the unperturbed and perturbed singular vectors or eigenvectors the class of perturbations considered consists of all delta b for which b delta b d l bd r for some nonsingular matrices d l and d r this class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off diagonal block in a block triangular matrix many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations also some new relative perturbation and deflation results for the singular values and vectors of biacyclic triangular and shifted triangular matrices are presented

symmetry and lattice conditional independence in a multivariate normal distribution a class of multivariate normal models with symmetry restrictions given by a finite group and conditional independence restrictions given by a finite distributive lattice is defined and studied the statistical properties of these models including maximum likelihood inference invariance and hypothesis testing are discussed

stochastic independence algebraic independence and abstract connectedness abstract mutual stochastic independences among algebras and mutual algebraic independences among elements of semimodular lattices are observed to have a very similar behaviour we suggest abstract independence structures called i relations describing it presented examination of i relations resembles a theory of abstract connectedness a dual characterization of i relations by families of connected sets is found by means of a special galois connection representations of i relations in the matroid theory sense by algebras and by elements of lattices are discussed

computing maximum likelihood estimates in recursive linear models with correlated errors in recursive linear models the multivariate normal joint distribution of all variables exhibits a dependence structure induced by a recursive or acyclic system of linear structural equations these linear models have a long tradition and appear in seemingly unrelated regressions structural equation modelling and approaches to causal inference they are also related to gaussian graphical models via a classical representation known as a path diagram despite the models long history a number of problems remain open in this paper we address the problem of computing maximum likelihood estimates in the subclass of bow free recursive linear models the term bow free refers to the condition that the errors for variables i and j be uncorrelated if variable i occurs in the structural equation for variable j we introduce a new algorithm termed residual iterative conditional fitting ricf that can be implemented using only least squares computations in contrast to existing algorithms ricf has clear convergence properties and yields exact maximum likelihood estimates after the first iteration whenever the mle is available in closed form

global solutions and self similar solutions of semilinear wave equation we prove existence uniqueness and regularity results for the global solutions of the semilinear wave equations in particular we show existence of regular self similar solutions also we build some finite energy asymptotically self similar solutions

wishart distributions for decomposable graphs when considering a graphical gaussian model n g markov with respect to a decomposable graph g the parameter space of interest for the precision parameter is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g the parameter space for the scale parameter of n g is the cone q g dual to p g of incomplete matrices with submatrices corresponding to the cliques of g being positive definite in this paper we construct on the cones q g and p g two families of wishart distributions namely the type i and type ii wisharts they can be viewed as generalizations of the hyper wishart and the inverse of the hyper inverse wishart as defined by dawid and lauritzen ann statist 21 1993 1272 1317 we show that the type i and ii wisharts have properties similar to those of the hyper and hyper inverse wishart indeed the inverse of the type ii wishart forms a conjugate family of priors for the covariance parameter of the graphical gaussian model and is strong directed hyper markov for every direction given to the graph by a perfect order of its cliques while the type i wishart is weak hyper markov moreover the inverse type ii wishart as a conjugate family presents the advantage of having a multidimensional shape parameter thus offering flexibility for the choice of a prior both type i and ii wishart distributions depend on multivariate shape parameters a shape parameter is acceptable if and only if it satisfies a certain eigenvalue property we show that the sets of acceptable shape parameters for a noncomplete g have dimension equal to at least one plus the number of cliques in g these families as conjugate families are richer than the traditional diaconis ylvisaker conjugate families which all have a shape parameter set of dimension one a decomposable graph which does not contain a three link chain as an induced subgraph is said to be homogeneous in this case our wisharts are particular cases of the wisharts on homogeneous cones as defined by andersson and wojnar j theoret probab 17 2004 781 818 and the dimension of the shape parameter set is even larger than in the nonhomogeneous case it is indeed equal to the number of cliques plus the number of distinct minimal separators using the model where g is a three link chain we show by computing a 7 tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one

maximum likelihood estimation of the parameters of a multivariate normal distribution abstract this paper provides an exposition of alternative approaches for obtaining maximum likelihood estimators mle for the parameters of a multivariate normal distribution under different assumptions about the parameters a central focus is on two general techniques namely matrix differentiation and matrix transformations these are systematically applied to derive the mle of the means under a rank constraint and of the covariances when there are missing observations derivations using induction and inequalities are also included to illustrate alternative methods other examples such as a connection with an econometric model are included although the paper is primarily expository some of the proofs are new

absolute and relative weyl theorems for generalized eigenvalue problems abstract weyl type eigenvalue perturbation theories are derived for hermitian definite pencils a b in which b is positive definite the results provide a one to one correspondence between the original and perturbed eigenvalues and give a uniform perturbation bound we give both absolute and relative perturbation results defined in the standard euclidean metric instead of the chordal metric that is often used

ancestral graph markov models this paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all dag independence models this class of graphs called maximal ancestral graphs has two attractive features there is at most one edge between each pair of vertices every missing edge corresponds to an independence relation these features lead to a simple parameterization of the corresponding set of distributions in the gaussian case

multimodality of the likelihood in the bivariate seemingly unrelated regressions model summary we analyse the simplest two equation seemingly unrelated regressions model and demonstrate that its likelihood may have up to five stationary points and thus there may be up to three local modes consequently the estimates obtained via iterative estimation methods may depend on starting values we further show that the probability of multi modality vanishes asymptotically monte carlo simulations suggest that multimodality rarely occurs if the seemingly unrelated regressions model is true but can become more frequent if the model is misspecified the existence of multimodality in the likelihood for seemingly unrelated regressions models contradicts several claims in the literature

what can we learn from the path equations identifiability constraints equivalence procedures are given for determining identified parameters finding constraints on the covariances and checking equivalence in acyclic recursive linear path models with correlated error terms disturbances by inspection of the path equations aided by simple recursions this provides a useful and general alternative to the employment of directed acyclic graph theory for such purposes

the lidskii mirsky wielandt theorem additive and multiplicative versions we use a simple matrix splitting technique to give an elementary new proof of the lidskii mirsky wielandt theorem and to obtain a multiplicative analog of the lidskii mirsky wielandt theorem which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of hermitian matrices and singular values of general matrices we apply our bound to obtain numerous bounds on the matching distance between the eigenvalues and singular values of matrices our results strengthen and generalize those in the literature

on the causal interpretation of acyclic mixed graphs under multivariate normality abstract in multivariate statistics acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden i e latent or unobserved variables indeed under multivariate normality every mixed graph corresponds to a set of covariance matrices that contains as a full dimensional subset the covariance matrices associated with a causally interpretable acyclic digraph this digraph generally has some of its nodes corresponding to hidden variables we seek to clarify for which mixed graphs there exists an acyclic digraph whose hidden variable model coincides with the mixed graph model restricting to the tractable setting of chain graphs and multivariate normality we show that decomposability of the bidirected part of the chain graph is necessary and sufficient for equality between the mixed graph model and some hidden variable model given by an acyclic digraph

likelihood ratio tests and singularities many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi algebraic subsets of the parameter space we consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions we show that the assumptions of chernoff s theorem hold for semi algebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point at boundary points or singularities the tangent cone need not be a linear space and limiting distributions other than chi square distributions may arise while boundary points often lead to mixtures of chi square distributions singularities give rise to nonstandard limits we demonstrate that minima of chi square random variables are important for locally identifiable models and in a study of the factor analysis model with one factor we reveal connections to eigenvalues of wishart matrices

an extension of the results of asmussen and edwards on collapsibility in contingency tables summary asmussen edwards 1983 defined necessary and sufficient conditions for collapsibility of a hierarchical log linear model for a multidimensional contingency table we have shown that for decomposable log linear models these conditions can be combined with various graph theoretic algorithms to provide useful classes of sub tables which are collapsible onto in particular the sahr algorithm finds the minimal set onto which the model can be collapsed and which contains a sub table of interest in the context of expert systems by reducing a probabilistic influence network onto only the relevant nodes the algorithms reduce the required computation and simplify interpretation

maximum likelihood analysis of linear mediation models with treatment mediator interaction this research concerns a mediation model where the mediator model is linear and the outcome model is also linear but with a treatment mediator interaction term and a residual correlated with the residual of the mediator model assuming the treatment is randomly assigned parameters in this mediation model are shown to be partially identifiable under the normality assumption on the residual of the mediator and the residual of the outcome explicit full information maximum likelihood estimates of model parameters are introduced given the correlation between the residual for the mediator and the residual for the outcome a consistent variance matrix of these estimates is derived currently the coefficients of this mediation model are estimated using the iterative feasible generalized least squares ifgls method that is originally developed for seemingly unrelated regressions surs we argue that this mediation model is not a system of surs while the ifgls estimates are consistent their variance matrix is not theoretical comparisons of the fimle variance matrix and the ifgls variance matrix are conducted our results are demonstrated by simulation studies and an empirical study the fimle method has been implemented in a freely available r package imediate

the cauchy problem for a coupled semilinear parabolic system abstract we consider a semilinear parabolic coupled system u t u f u v v t v g u v in r n r where the maximum principle and the minimum principle fail for the solution itself and also for its first derivatives with respect to x under the assumption that f and g present a subquadratic growth near the origin we prove the global existence and uniqueness of the solutions to the cauchy problem and the existence of nonnegative solutions the maximum principle and the minimum principle are replaced by the existence of some invariant regions in r 2n 2 for u v u x1 v x1 u xn v xn