matrix chebyshev polynomials and continued fractions in the first part we expose the notion of continued fractions in the matrix case in this paper we are interested in their connection with matrix orthogonal polynomials in the second part matrix continued fractions are used to develop the notion of matrix chebyshev polynomials in the case of hermitian coefficients in the recurrence formula we give the explicit formula for the stieltjes transform the support of the orthogonality measure and its density as a corollary we get the extension of the matrix version of the blumenthal theorem proved in j approx theory 84 1 1996 96 the third part contains examples of matrix orthogonal polynomials

perfect cocycles through stochastic differential equations we prove that if is a random dynamical system cocycle for whicht t x is a semimartingale then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment helix and conversely this relation is succinctly expressed as semimartingale cocycle exp semimartingale helix to implement it we lift stochastic calculus from the traditional one sided time to two sided timet and make this consistent with ergodic theory we also prove a general theorem on the perfection of a crude cocycle thus solving a problem which was open for more than ten years

the generalized logit linear item response model for binary designed items this paper introduces the generalized logit linear item response model gllirm which represents the item solving process as a series of dichotomous operations or steps the gllirm assumes that the probability function of the item response is a logistic function of a linear composite of basic parameters which describe the operations and the coefficients depend on three design matrices x y and z the gllirm provides a tool for testing hypotheses on the item solving process and generalizes existing models an empirical application is included in which the model is applied to evaluate sources of difficulty and pairwise item interactions in a logical analysis test

analysis of residuals for the multinomial item response model using the item response model as developed on the multinomial distribution asymptotic variances are obtained for residuals associated with response patterns and first and second order marginal frequencies of manifest variables when the model does not fit well an examination of these residuals may reveal the source of the poor fit finally a limited information test of fit for the model is developed by using residuals defined for the first and second order marginals model evaluation based on residuals for these marginals is particularly useful when the response pattern frequencies are sparse

high dimensional maximum marginal likelihood item factor analysis by adaptive quadrature although the bock aitkin likelihood based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations a serious limitation of the method is its reliance on fixed point gauss hermite g h quadrature in the solution of the likelihood equations and likelihood ratio tests when the number of latent dimensions is large computational considerations require that the number of quadrature points per dimension be few but with large numbers of items the dispersion of the likelihood given the response pattern becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space in this paper we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data in particular we show that adaptive g h quadrature combined with mean and covariance adjustments at each iteration of an em algorithm produces an accurate fast converging solution with as few as two points per dimension evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions unlike an earlier application of adaptive gibbs sampling to this problem by meng and schilling the simulations also confirm the validity of the present method in calculating likelihood ratio chi square statistics for determining the number of factors required in the model finally we apply the method to a sample of real data from a test of teacher qualifications

on a class of matrix orthogonal polynomials on the real line abstract it is shown how to construct matrix orthogonal polynomials on the real line provided we have a sequence of scalar orthogonal polynomials over an algebraic harmonic curve we consider the moment matrix associated with a distribution on and related to a basis of the linear space of polynomials obtained through the polynomial which defines the curve this particular moment matrix can be regarded as a hankel block matrix and thus matrix orthogonal polynomials appear in a natural way

a survey on orthogonal matrix polynomials satisfying second order differential equations the subject of orthogonal polynomials cuts across a large piece of mathematics and its applications two notable examples are mathematical physics in the 19th and 20th centuries as well as the theory of spherical functions for symmetric spaces it is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of scalar valued orthogonal polynomials matrix orthogonality on the real line has been sporadically studied during the last half century since krein devoted some papers to the subject in 1949 see ams translations series 2 vol 97 providence rhode island 1971 pp 75 143 dokl akad nauk sssr 69 2 1949 125 in the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case the most recent of these results is the discovery of important examples of orthogonal matrix polynomials many families of orthogonal matrix polynomials have been found that as the classical families of hermite laguerre and jacobi in the scalar case satisfy second order differential equations with coefficients independent of n the aim of this paper is to give an overview of the techniques that have led to these examples a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples

transformations of matricial α stieltjes non negative definite sequences abstract we investigate the inner structure of power moment sequences of matrix measures on the right semiaxis where is a given real number to a given matrix sequence we associate in a bijective way a new sequence of matrices which we call the right stieltjes parametrization thereby one to one correspondences between power moment sequences on the right semiaxis with additional properties and particular sequences of non negative hermitian matrices are established we consider distinguished transformations of matrix sequences the study of which was suggested by considering some natural transformations of matrix measures on an interval a main theme is to describe the right stieltjes parametrization of the transformed sequence in terms of the right stieltjes parametrization of the original sequence

a rasch model for partial credit scoring a unidimensional latent trait model for responses scored in two or more ordered categories is developed this partial credit model is a member of the family of latent trait models which share the property of parameter separability and so permit specifically objective comparisons of persons and items the model can be viewed as an extension of andrich s rating scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item the difference between the parameters in this model and the category boundaries in samejima s graded response model is demonstrated an unconditional maximum likelihood procedure for estimating the model parameters is developed

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

psychometric latent response models in this paper some psychometric models will be presented that belong to the larger class oflatent response models lrms first lrms are introduced by means of an application in the field ofcomponential item response theory embretson 1980 1984 second a general definition of lrms not specific for the psychometric subclass is given third some more psychometric lrms and examples of how they can be applied are presented fourth a method for obtaining maximum likelihood ml and some maximum a posteriori map estimates of the parameters of lrms is presented this method is then applied to theconjunctive rasch model fifth and last an application of the conjunctive rasch model is presented this model was applied to responses to typical verbal ability items open synonym items

matrix valued orthogonal polynomials on the real line some extensions of the classical theory in the work presented below the classical subject of orthogonal polynomials on the real line is discussed in the matrix setting an analogue of the determinant definition of orthogonal polynomials is presented the classical properties such as the recurrence relation the kernel polynomials and the christoffel darboux formula are discussed a function for the system of matrix valued orthogonal polynomials on the real line is presented some properties of the functions are investigated

a convex optimization model for finding non negative polynomials this paper presents a convex optimization model for the problem of finding some polynomials for which certain linear combinations are non negative polynomials this model is then applied to solve several filter design problems we first reformulate some low pass filter design problems with finite or infinite impulse response as optimization problems over non negative real or complex polynomials whose feasibility problems can be solved by applying our model the whole optimization problems are then solved by using a combination of a bisection search procedure on an appropriate parameter and our convex optimization model to solve the feasibility problems some numerical examples illustrate the method

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

the statistical analysis of general processing tree models with the em algorithm multinomial processing tree models assume that an observed behavior category can arise from one or more processing sequences represented as branches in a tree these models form a subclass of parametric multinomial models and they provide a substantively motivated alternative to loglinear models we consider the usual case where branch probabilities are products of nonnegative integer powers in the parameters 0 s 1 and their complements 1 s a version of the em algorithm is constructed that has very strong properties first the e step and the m step are both analytic and computationally easy therefore a fast pc program can be constructed for obtaining mles for large numbers of parameters second a closed form expression for the observed fisher information matrix is obtained for the entire class third it is proved that the algorithm necessarily converges to a local maximum and this is a stronger result than for the exponential family as a whole fourth we show how the algorithm can handle quite general hypothesis tests concerning restrictions on the model parameters fifth we extend the algorithm to handle the read and cressie power divergence family of goodness of fit statistics the paper includes an example to illustrate some of these results

marginal maximum likelihood estimation of item parameters application of an em algorithm maximum likelihood estimation of item parameters in the marginal distribution integrating over the distribution of ability becomes practical when computing procedures based on an em algorithm are used by characterizing the ability distribution empirically arbitrary assumptions about its form are avoided the em procedure is shown to apply to general item response models lacking simple sufficient statistics for ability this includes models with more than one latent dimension

a real nullstellensatz for free modules abstract let a be the algebra of all n n matrices with entries from r x 1 x d and let g 1 g m f a we will show that f a v 0 for every a r d and v r n such that g i a v 0 for all i if and only if f belongs to the smallest real left ideal of a which contains g 1 g m here a left ideal j of a is real if for every h 1 h k a such that h 1 t h 1 h k t h k j j t we have that h 1 h k j we call this result the one sided real nullstellensatz for matrix polynomials we first prove by induction on n that it holds when g 1 g m f have zeros everywhere except in the first row this auxiliary result can be formulated as a real nullstellensatz for the free module r x 1 x d n

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

equivalent linear logistic test models this paper is about the linear logistic test model lltm we demonstrate that there are infinitely many equivalent ways to specify a model an implication is that there may well be many ways to change the specification of a given lltm and achieve the same improvement in model fit to illustrate this phenomenon we analyze a real data set using a lagrange multiplier test for the specification of the model this lagrange multiplier test is similar to the modification index used in structural equation modeling

the nevanlinna pick interpolation problems and power moment problems for matrix valued functions abstract this paper is concerned with the study of several types of matrix interpolation problems including the matrix nevanlinna pick interpolation problem in the nevanlinna class n p the matrix nevanlinna pick interpolation problem in the stieltjes class l p the truncated matrix power moment problem on the real axis the stieltjes matrix power moment problem we use the so called block hankel vector approach for the treatment of these problems and make explicit the correspondence between solutions to the nevanlinna pick interpolation problem and solutions to an associated power moment problem with the block hankel vector under consideration this enables one not only to find existence criteria for the solutions of the nevanlinna pick interpolation problem but also to describe the solutions based on the results of power moment problem and vice versa

orthogonal matrix polynomials and applications orthogonal matrix polynomials on the real line or on the unit circle have properties which are natural generalizations of properties of scalar orthogonal polynomials appropriately modified for the matrix calculus we show that orthogonal matrix polynomials both on the real line and on the unit circle appear at various places and we describe some of them the spectral theory of doubly infinite jacobi matrices can be described using orthogonal 2 x 2 matrix polynomials on the real line scalar orthogonal polynomials with a sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line orthogonal matrix polynomials on the unit circle are related to unitary block hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing finally we show how orthogonal matrix polynomials can be used for gaussian quadrature of matrix valued functions parallel algorithms for this purpose have been implemented using pvm and some examples are worked out

on dyukarev s resolvent matrix for a truncated stieltjes matrix moment problem under the view of orthogonal matrix polynomials abstract the main result of this paper is a new representation of yu m dyukarev s resolvent matrix for the non degenerate truncated matricial version of the classical stieltjes moment tmcsm problem this new representation is based on the use of a quadruple of q q matrix polynomials which are characterized by certain orthogonality properties in this way useful new interrelations between the moment problems of stieltjes and hamburger which are associated with a stieltjes positive definite sequence s j j 0 2 n of complex q q matrices are found this includes an explicit formula connecting the resolvent matrices used by yu m dyukarev and i v kovalishina respectively furthermore the coefficients in the recurrence formulas for the orthogonal polynomials with respect to a stieltjes positive definite sequence are expressed in terms of the dyukarev stieltjes parameters additionally we obtain two different representations for each extremal solution of the tmcsm problem via matrix continued fraction expansions

a bayesian random effects model for testlets standard item response theory irt models fit to dichotomous examination responses ignore the fact that sets of items testlets often come from a single common stimuli e g a reading comprehension passage in this setting all items given to an examinee are unlikely to be conditionally independent given examinee proficiency models that assume conditional independence will overestimate the precision with which examinee proficiency is measured overstatement of precision may lead to inaccurate inferences such as prematurely ending an examination in which the stopping rule is based on the estimated standard error of examinee proficiency e g an adaptive test to model examinations that may be a mixture of independent items and testlets we modified one standard irt model to include an additional random effect for items nested within the same testlet we use a bayesian framework to facilitate posterior inference via a data augmented gibbs sampler dags tanner wong 1987 the modified and standard irt models are both applied to a data set from a disclosed form of the sat we also provide simulation results that indicates that the degree of precision bias is a function of the variability of the testlet effects as well as the testlet design

synchronization by noise for order preserving random dynamical systems we provide sufficient conditions for weak synchronization stabilization by noise for order preserving random dynamical systems on polish spaces that is under these conditions we prove the existence of a weak point attractor consisting of a single random point this generalizes previous results in two directions first we do not restrict to banach spaces and second we do not require the partial order to be admissible nor normal as a second main result and application we prove weak synchronization by noise for stochastic porous media equations with additive noise

remarks on equivalent linear logistic test models by bechger verstralen and verhelst 2002 this paper discusses a new form of specifying and normalizing a linear logistic test model lltm as suggested by bechger verstralen and verhelst psychometrika 2002 it is shown that there are infinitely many ways to specify the same normalization moreover the relationship between some of their results and equivalent previous results in the literature is clarified and it is shown that the goals of estimating and testing a single element of the weight matrix for which they propose new methods can be reached by means of simple well known tools already implemented in published lltm software

n extremal matrices of measures for an indeterminate matrix moment problem in this paper we study the n extremal matrices of measures associated to a completely indeterminate matrix moment problem i e those matrices of measure w solutions of a completely indeterminate matrix moment problem for which the linear space of matrix polynomials is dense in the corresponding l2 w

on the truncated matricial stieltjes moment problem m α sj j 0m abstract this paper gives via stieltjes transform a complete description of the solution set of a matricial truncated stieltjes type power moment problem in the non degenerate and degenerate cases the approach is based on the schur type algorithm which was worked out in the papers 9 10 furthermore the subset of parameters is determined which corresponds to another truncated matricial stieltjes type moment problem

estimating difficulty from polytomous categorical data a comprehensive analysis of difficulty for multiple choice items requires information at different levels the test the items and the alternatives this paper introduces a new parameterization of the nominal categories model ncm for analyzing difficulty at these three levels the new parameterization is referred to as the ne ncm and is statistically equivalent to the ncm the ne ncm is applied to a sample of responses from a logical analysis test the results suggest that the individuals execute a self terminated response process that is mostly determined by working memory load

modification indices for the 2 pl and the nominal response model in this paper it is shown that various violations of the 2 pl model and the nominal response model can be evaluated using the lagrange multiplier test or the equivalent efficient score test the tests presented here focus on violation of local stochastic independence and insufficient capture of the form of the item characteristic curves primarily the tests are item oriented diagnostic tools but taken together they also serve the purpose of evaluation of global model fit a useful feature of lagrange multiplier statistics is that they are evaluated using maximum likelihood estimates of the null model only that is the parameters of alternative models need not be estimated as numerical examples an application to real data and some power studies are presented

polynomial interpolation and gaussian quadrature for matrix valued functions abstract the techniques for polynomial interpolation and gaussian quadrature are generalized to matrix valued functions it is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

a unified treatment for the matrix stieltjes moment problem in both nondegenerate and degenerate cases abstract this paper presents a unified method for the solution of the matrix stieltjes moment problem in both the nondegenerate and degenerate cases based on the use of our recent investigation of the matrix hamburger moment problem as a by product a complete solution of the matrix nevanlinna pick interpolation problem in the class of the matrix valued stieltjes functions is derived as well

on matrix hurwitz type polynomials and their interrelations to stieltjes positive definite sequences and orthogonal matrix polynomials abstract this paper is a direct continuation of the author s recent investigations 4 on the non degenerate truncated matricial stieltjes problem inspired by a characterization of scalar hurwitz polynomials via continued fractions see e g gantmacher 15 a class of matrix hurwitz type polynomials is introduced it is shown that this class is intimately related with stieltjes positive definite sequences of matrices and can be expressed in terms of the associated stieltjes quadruple of sequences of left orthogonal matrix polynomials it is shown that a monic matrix polynomial is a matrix hurwitz type polynomial if and only if the sequence of its markov parameters is a stieltjes positive definite sequence

minimal random attractors abstract it is well known that random attractors of a random dynamical system are generally not unique we show that for general pullback attractors and weak attractors there is always a minimal in the sense of smallest random attractor which attracts a given family of possibly random sets we provide an example which shows that this property need not hold for forward attractors we point out that our concept of a random attractor is very general the family of sets which are attracted is allowed to be completely arbitrary

conjunctive item response theory kernels conjunctive item response models are introduced such that a sufficient statistics for latent traits are not necessarily additive in item scores b items are not necessarily locally independent and c existing compensatory additive item response models including the binomial rasch logistic and general locally independent model are special cases simple estimates and hypothesis tests for conjunctive models are introduced and evaluated as well conjunctive models are also identified with cognitive models that assume the existence of several individually necessary component processes for a global ability it is concluded that conjunctive models and methods may show promise for constructing improved tests and uncovering conjunctive cognitive structure it is also concluded that conjunctive item response theory may help to clarify the relationships between local dependence multidimensionality and item response function form

relative asymptotics for orthogonal matrix polynomials in this paper we study sequences of matrix polynomials that satisfy a non symmetric recurrence relation to study this kind of sequences we use a vector interpretation of the matrix orthogonality in the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class we study the generalized matrix chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples the concept of a dirac delta functional is introduced we show how the vector model that includes a dirac delta functional is a representation of a discrete sobolev inner product it also allows to reinterpret such perturbations in the usual matrix nevai class finally the relative asymptotics between a polynomial in the generalized matrix nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a dirac delta functional is deduced

identifiability and equivalence of gllirm models the generalized logit linear item response model gllirm is a linearly constrained nominal categories model ncm that computes the scale and intercept parameters for categories as a weighted sum of basic parameters this paper addresses the problems of the identifiability of the basic parameters and the equivalence between different gllirm models it is shown that the identifiability of the basic parameters depends on the size and rank of the coefficient matrix of the linear functions moreover two models are observationally equivalent if the product of the respective coefficient matrices has full column rank finally the paper also explores the relations between the parameters of nested models

a multidimensional item response model constrained latent class analysis using the gibbs sampler and posterior predictive checks in this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models the parameters of this latent class model will be estimated using an application of the gibbs sampler it will be illustrated that the gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences model fit will be investigated using posterior predictive checks checks for manifest monotonicity the agreement between the observed and expected conditional association structure marginal local homogeneity and the number of latent classes will be presented

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

the truncated hamburger matrix moment problems in the nondegenerate and degenerate cases and matrix continued fractions abstract the present paper deals simultaneously with the nondegenerate and degenerate truncated hamburger matrix moment problems in a unified way based on the use of the schur algorithm involving matrix continued fractions a full analysis of them together with a relative matrix moment problem on the real axis is given with the help of the correspondence between the moment problem on the real axis and the nevanlinna pick np interpolation the solutions of the nontangential np interpolation in the nevanlinna class are derived as an application

principal representations and canonical moment sequences for distributions on an interval abstract an initial expository introduction to moment space geometry and canonical moment sequences for distributions on the unit interval leads to the exposure of previously unknown but striking symmetries in stieltjes transforms for these distributions when the transforms are viewed as moment functions the n th and the n 1 st canonical moments corresponding to a moment point interior to the space of the first n moments are shown to depend in a very simple way on the canonical intervals determined by the two principal representations of the moment point for example the n th canonical moment is just the sum of the lengths of the upper canonical intervals

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples