quantum symmetric kac moody pairs the present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable kac moody algebras the resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras they give rise to triangular decompositions including a quantum analog of the iwasawa decomposition and they can be written explicitly in terms of generators and relations moreover their centers and their specializations are determined the constructions follow g letzter s theory of quantum symmetric pairs for semisimple lie algebras the main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable kac moody algebras due to kac and wang the resulting theory comprises various classes of examples which have previously appeared in the literature such as q onsager algebras and the twisted q yangians introduced by molev ragoucy and sorba

the asymptotic variance matrices of the sample correlation matrix in elliptical and normal situations and their proportionality abstract this note presents the asymptotic variance matrices of the sample correlation matrix in elliptical and normal situations and generalizes the scalar proportionality result to the multivariate case

ignorability in general incomplete data models summary rubin 1976 defined ignorability conditions for frequentist and bayes likelihood analyses of data subject to missing observations more recently heitjan rubin 1991 and heitjan 1993 generalised the rubin model to encompass other forms of incompleteness establishing ignorability conditions for bayes likelihood inferences only this paper extends the heitjan rubin model by explicitly defining the observed degree of coarseness as a data element this permits the development of a frequentist theory including a generalisation of missing completely at random the frequentist ignorability condition for missing data the model is applied in a number of incomplete data problems of general interest

automorphisms of borcherds superalgebras and fixed point subalgebras abstract let g be a borcherds superalgebra and let be a finite order automorphism of g satisfying certain conditions we prove that the fixed point subalgebra g is equal to a borcherds superalgebra we also compute the trace of on g from the twisted denominator identity of g and derive closed form formula for root multiplicities of g

classifying spaces of kac moody groups we study the structure of classifying spaces of kac moody groups from a homotopy theoretic point of view they behave in many respects as in the compact lie group case the mod p cohomology algebra is noetherian and lannes t functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p subgroups also spaces of maps from classifying spaces of finite p groups to classifying spaces of kac moody groups are described in terms of classifying spaces of centralizers while the classifying space of a kac moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p subgroups up to p completion we show that these properties are common to a larger class of groups also including parabolic subgroups of kac moody groups and centralizers of finite p subgroups

maps between classifying spaces of kac moody groups abstract we classify up to homotopy the self maps of the classifying space of any non affine kac moody group of rank two

a note on maximizing a special concave function subject to simultaneous loewner order constraints maximization of functions with multivariate arguments can be computationally difficult we show that for a special function which is proportional to the density of a wishart distribution reparametrization can lead to maximization of a concave function we use this fact to produce an algorithm which maximizes the function over a restricted parameter space of the form 0 l u where l u means that u l is a nonnegative definite matrix and l u means that u l is positive definite this restriction is often referred to as the loewner ordering

a sandwich type standard error estimator of sem models with multivariate time series structural equation models are increasingly used as a modeling tool for multivariate time series data in the social and behavioral sciences standard error estimators of sem models originally developed for independent data require modifications to accommodate the fact that time series data are inherently dependent in this article we extend a sandwich type standard error estimator of independent data to multivariate time series data one required element of this estimator is the asymptotic covariance matrix of concurrent and lagged correlations among manifest variables whose closed form expression has not been presented in the literature the performance of the adapted sandwich type standard error estimator is evaluated using a simulation study and further illustrated using an empirical example

twisted quantum affinizations and their vertex representations in this paper we generalize drinfeld s twisted quantum affine algebras to construct twisted quantum algebras for all simply laced generalized cartan matrices and present their vertex representation realizations

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

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

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

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

a classification of involutive automorphisms of an affine kac moody lie algebra abstract in this thesis we classify the conjugacy classes of involutions in aut g where g is an affine kac moody lie algebra we distinguish between two kinds of involutions those which preserve the conjugacy class of a borel subalgebra and those which don t we give a complete and non redundant list of representatives of involutions of the first kind and we compute their fixed points sets we prove that any involution of the first kind has a conjugate which leaves invariant the components of the gauss decomposition g n h n we also give a complete list of representatives of the conjugacy classes of involutions of the second kind

local model uncertainty and incomplete data bias with discussion problems of the analysis of data with incomplete observations are all too familiar in statistics they are doubly difficult if we are also uncertain about the choice of model we propose a general formulation for the discussion of such problems and develop approximations to the resulting bias of maximum likelihood estimates on the assumption that model departures are small loss of efficiency in parameter estimation due to incompleteness in the data has a dual interpretation the increase in variance when an assumed model is correct the bias in estimation when the model is incorrect examples include non ignorable missing data hidden confounders in observational studies and publication bias in meta analysis doubling variances before calculating confidence intervals or test statistics is suggested as a crude way of addressing the possibility of undetectably small departures from the model the problem of assessing the risk of lung cancer from passive smoking is used as a motivating example

discrete approximations for complex kac moody groups abstract we construct a map from the classifying space of a discrete kac moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological kac moody group and prove that it is a homology equivalence at primes q different from p this generalizes a classical result of quillen friedlander and mislin for lie groups as an application we construct unstable adams operations for general kac moody groups compatible with the frobenius homomorphism our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete kac moody groups

asymptotic biases in exploratory factor analysis and structural equation modeling formulas for the asymptotic biases of the parameter estimates in structural equation models are provided in the case of the wishart maximum likelihood estimation for normally and nonnormally distributed variables when multivariate normality is satisfied considerable simplification is obtained for the models of unstandardized variables formulas for the models of standardized variables are also provided numerical examples with monte carlo simulations in factor analysis show the accuracy of the formulas and suggest the asymptotic robustness of the asymptotic biases with normality assumption against nonnormal data some relationships between the asymptotic biases and other asymptotic values are discussed

dominant k theory and integrable highest weight representations of kac moody groups we give a topological interpretation of the highest weight representations of kac moody groups given the unitary form g of a kac moody group over c we define a version of equivariant k theory kg on the category of proper g cw complexes we then study kac moody groups of compact type in detail see section 2 for definitions in particular we show that the grothendieck group of integrable highest weight representations of a kac moody group g of compact type maps isomorphically onto k g eg where eg is the classifying space of proper g actions for the affine case this agrees very well with recent results of freed hopkins teleman we also explicitly compute kg eg for kac moody groups of extended compact type which includes the kac moody group e10

covering algebras ii isomorphism of loop algebras this paper studies the loop algebras that arise from pairs consisting of a symmetrizable kac moody lie algebra g and a finite order automorphism of g we obtain necessary and sufficient conditions for two such loop algebras to be isomorphic

turing patterns in three dimensions we investigate three dimensional turing patterns in two component reaction diffusion systems the fitzhugh nagumo equation the brusselator and the gray scott model are solved numerically in three dimensions several interconnected structures of domains as well as lamellar hexagonal and spherical domains are obtained as stable motionless equilibrium patterns the relative stability of these structures is studied analytically based on the reduction approximation the relation with the microphase separated structures in block copolymers is also discussed

on the performance of hadamard ratio detector based spectrum sensing for cognitive radios we consider the problem of multiantenna spectrum sensing ss in cognitive radios crs when the receivers are assumed to be uncalibrated across the antennas the performance of the hadamard ratio detector hrd is analyzed in such a scenario specifically we first derive the exact distribution of the hrd statistic under the null hypothesis which leads to an elaborate but closed form expression for the false alarm probability then we derive a simpler and tight closed form approximation for both the false alarm and detection probabilities by using a moment based approximation of the hrd statistical distribution under both hypotheses finally the accuracy of the obtained results is verified by simulations

coarsening at random in general sample spaces and random censoring in continuous time heitjan and rubin proposed a concept coarsening at random generalizing rubin s theory of missing at random their analysis was done in discrete sample spaces we propose a generalization to general sample spaces among heitjan and rubin s applications was right censoring in survival analysis we discuss the application of the generalized theory to various censoring patterns in continuous time and connect to the modern theory of random censoring

short note isotropic discrete laplacian operators from lattice hydrodynamics we show that discrete schemes developed for lattice hydrodynamics provide an elegant and physically transparent way of deriving laplacians with isotropic discretisation error isotropy is guaranteed whenever the laplacian weights follow from the discrete maxwell boltzmann equilibrium since these are by construction isotropic on the lattice we also point out that stencils using as few as 15 points in three dimensions generate isotropic laplacians these computationally efficient laplacians can be used in cell dynamical and hybrid lattice boltzmann simulations in favor of popular anisotropic laplacians which make use of larger stencils the method can be extended to provide discretisations of higher order and for other differential operators such the gradient divergence and curl

boltzmann cellular automata studies of the spinodal decomposition abstract a nonlinear galilean invariant 2 color boltzmann cellular automaton is used to study spinodal decomposition in 2 dimensional systems the initial conditions for the process correspond to a well mixed state of two fluids present in equal proportions we specify two types of such initial conditions corresponding to a correlated quench and to a deep quench in the former case a specific color is assigned randomly to each site in the latter individual states on each site are populated by the two colors about evenly with small fluctuations in the correlated quench situation the domain growth exponent is initially equal to 0 33 0 06 and for the late time stage it is equal to 0 66 0 06 in the deep quench situation on the other hand the effective exponent from initial value of 0 33 takes on the value of 0 82 0 05 for intermediate time scales and only at later times it crosses over to the exponent corresponding to the correlated quench spinodal decomposition taking place in a porous medium is slower and the effective exponents are nonuniversal inadequacies in the existing models of the surface tension in cellular automata are pointed out

ignorability and coarse data we present a general statistical model for data coarsening which includes as special cases rounded heaped censored partially categorized and missing data formally with coarse data observations are made not in the sample space of the random variable of interest but rather in its power set grouping is a special case in which the degree of coarsening is known and nonstochastic we establish simple conditions under which the possible stochastic nature of the coarsening mechanism can be ignored when drawing bayesian and likelihood inferences and thus the data can be validly treated as grouped data the conditions are that the data be coarsened at random a generalization of the condition missing at random and that the parameters of the data and the coarsening process be distinct applications of the general model and the ignorability condition are illustrated in a numerical example and described briefly in a variety of special cases

unitary structure in representations of infinite dimensional groups and a convexity theorem in this paper we show that a kac moody algebra fl a associated to a symmetrizable generalized cartan matrix a carries a contravariant hermitian form which is positive definite on all root spaces we deduce that every integrable highest weight g a module l a carries a contravariant positivedefinite hermitian form this allows us to define the moment map and prove a generalization of the schur horn kosta nt heckman atiya h pressley convexity theorem the proofs are based on an identity which also gives estimates for the action of 9 a on fl a and l a we hope that the main idea behind the paper is apparent it is to use the interplay between the coadjoint and the highest weight representations we are grateful to v guillemin for an introduction to the moment map

interconnected turing patterns in three dimensions we study numerically the turing pattern in three dimensions in a fitzhugh nagumo type reaction diffusion system we have found that interconnected periodic domain structures such as a gyroid fddd and perforated lamellar structures appear in three dimensions which never exist in lower dimensions the stability analysis of these structures is also performed by means of a mode expansion

a note on the estimator of the alpha coefficient for standardized variables under normality the asymptotic standard deviation sd of the alpha coefficient with standardized variables is derived under normality the research shows that the sd of the standardized alpha coefficient becomes smaller as the number of examinees and or items increase furthermore this research shows that the degree of the dependence of the sd on the number of items is a function of the average correlation coefficients when the average correlation approaches 1 the sd of the alpha coefficient decreases rapidly as the number of items increase with the order of p on the other hand when the items are only weakly correlated increasing the number of items decreases the sd of the alpha coefficient at a much slower rate

lamellar pattern formation in small world media abstract numerical and analytical techniques are used to investigate the effects of quenched disorder of small world networks on the phase ordering dynamics of lamellar patterns as modeled by the swift hohenberg equation morphologies for small and large values of the network randomness are quite different it is found that addition of shortcuts to an underlying regular lattice makes the growth of domains evolving from random initial conditions much slower at late times as the randomness increases the evolution is eventually frozen

tensor products and statistics abstract a dictionary between operator based and matrix based languages in multivariate statistical analysis is proposed then this formulary is applied to asymptotic factorial analyses especially for giving asymptotic covariance matrices and operators in an explicit form finally we present the mathematical foundations on which are based the functional tools i e tensor products of linear spaces of vectors and of operators

multiloop realization of extended affine lie algebras and lie tori an important theorem in the theory of infinite dimensional lie algebras states that any affine kac moody algebra can be realized that is to say constructed explicitly using loop algebras in this paper we consider the corresponding problem for a class of lie algebras called extended affine lie algebras ealas that generalize affine algebras ealas occur in families that are constructed from centreless lie tori so the realization problem for ealas reduces to the realization problem for centreless lie tori we show that all but one family of centreless lie tori can be realized using multiloop algebras in place of loop algebras we also obtain necessary and sufficient conditions for two centreless lie tori realized in this way to be isotopic a relation that corresponds to isomorphism of the corresponding families of ealas

representations of brauer category and categorification abstract we study representations of the locally unital and locally finite dimensional algebra b associated to the brauer category b 0 with defining parameter 0 over an algebraically closed field k with characteristic p 2 the grothendieck group k 0 b mod will be used to categorify the integrable highest weight sl k module v 0 1 2 with the fundamental weight 0 1 2 as its highest weight where b mod is a subcategory of b lfdmod in which each object has a finite flag and sl k is either sl or sl p depending on whether p 0 or 2 p as g modules c z k 0 b mod is isomorphic to v 0 1 2 where g is a lie subalgebra of sl k see definition 4 2 when p 0 standard b modules and projective covers of simple b modules correspond to monomial basis and so called quasi canonical basis of v 0 1 2 respectively

weighted chi squared tests for partial common principal component subspaces summary we consider tests of the null hypothesis that g covariance matrices have a partial common principal component subspace of dimension s our approach uses a dimensionality matrix which has its rank equal to s when the hypothesis holds the test can then be based on a statistic computed from the eigenvalues of an estimate of this dimensionality matrix the asymptotic distribution of this statistic is that of a linear combination of independent one degree of freedom chi squared random variables simulation results indicate that this test yields significance levels that come closer to the nominal level than do those of a previously proposed method the procedure is also extended to a test that g correlation matrices have a partial common principal component subspace

asymptotic properties of a correlation matrix under a two step monotone incomplete sample abstract this paper derives an asymptotic distribution for a correlation matrix in the context of a two step monotone incomplete sample drawn from n p q a multivariate normal population with mean and covariance matrix based on the result we perform hypothesis testing for the correlation matrix and investigate its accuracy using numerical simulations

isomorphisms of kac moody groups which preserve bounded subgroups a subgroup of a kac moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs in this paper we study the isomorphisms between kac moody groups over arbitrary fields of cardinality at least 4 which preserve the set of bounded subgroups we show that such an isomorphism between two such kac moody groups induces an isomorphism between the respective twin root data of these groups as a consequence we obtain the solution of the isomorphism problem for kac moody groups over finite fields of cardinality at least 4

correlations in two dimensional domain coarsening abstract we present single bit monte carlo simulations of the domain coarsening of a two dimensional binary system of size 1280 1280 from a quench at t to below the critical temperature t c with emphasis on cluster distributions and correlations we study both the conserved order parameter and the non conserved order parameter we measure the evolution of the mean cluster size and obtain both the distribution of cluster sizes and the correlations between clusters our results are compatible with recent experiments on ostwald ripening and numerical solutions of the cahn hilliard equation

spinodal decomposition in 3 space using a cell dynamical system cds to model the separation of phases we present the results of large scale simulations of spinodal decomposition in 3 space of a symmetric binary alloy and incompressible binary fluid at critical quench our main results are 1 the reliable determination of the asymptotic or the almost asymptotic form factors for these systems and 2 an understanding of the preasymptotic behavior of growth laws in terms of dispersion relations of interface fluctuations to achieve 1 it is indispensable to have i methods to analyze the data ii a scheme to simulate a system with fluid dynamic interactions and iii a demonstration of the self averaging nature of spherically averaged quantities item iii allows us to study sufficiently large systems with currently available computational resources a careful stability analysis of our cds model such as artificial pinning anisotropy etc is also given we must point out that the conventional data analysis coupled with artificial pinning and finite size effects may spuriously give the theoretically desired results especially in binary fluids the asymptotic form factors are estimated in collage form from our large scale late time simulation results and various theoretical asymptotic results such as tomita s sum rule we may conclude that for binary fluids the agreement between our form factor and ones obtained experimentally is excellent this also clearly demonstrates that apart from time and space scale changes binary polymer systems so far studied and low molecular weight systems are not distinct that is no polymer effect has been observed in the form factor in the case of binary alloys the agreement with our form factor and various experimental results is not as good as the binary fluid case this may be ascribed to various complications in solids anisotropy elastic effects etc

unbiased estimator for a covariance matrix in a three step monotone incomplete sample abstract in a monotone incomplete sample the maximum likelihood estimator mle for a mean vector is unbiased whereas the mle for a covariance matrix is not this study derives unbiased estimators ubes for a covariance matrix in a three step monotone incomplete sample we further investigate the properties of the estimators compare risks of the estimators and the mle and check accuracies through numerical simulation

lattice boltzmann langevin simulations of binary mixtures we report a hybrid numerical method for the solution of the model h fluctuating hydrodynamic equations for binary mixtures the momentum conservation equations with landau lifshitz stresses are solved using the fluctuating lattice boltzmann equation while the order parameter conservation equation with langevin fluxes is solved using stochastic method of lines two methods based on finite difference and finite volume are proposed for spatial discretization of the order parameter equation special care is taken to ensure that the fluctuation dissipation theorem is maintained at the lattice level in both cases the methods are benchmarked by comparing static and dynamic correlations and excellent agreement is found between analytical and numerical results the galilean invariance of the model is tested and found to be satisfactory thermally induced capillary fluctuations of the interface are captured accurately indicating that the model can be used to study nonlinear fluctuations

estimation of a multiplicative covariance structure in the large dimensional case abstract we propose a kronecker product model for correlation or covariance matrices in the large dimensional case the number of parameters of the model increases logarithmically with the dimension of the matrix we propose a minimum distance md estimator based on a log linear property of the model as well as a one step estimator which is a one step approximation to the quasi maximum likelihood estimator qmle we establish rates of convergence and central limit theorems clt for our estimators in the large dimensional case a specification test and tools for kronecker product model selection and inference are provided in a monte carlo study where a kronecker product model is correctly specified our estimators exhibit superior performance in an empirical application to portfolio choice for s p500 daily returns we demonstrate that certain kronecker product models are good approximations to the general covariance matrix

extensions of tr a 1 b 1 a b 0 for covariance matrices a b abstract let a b be p p positive definite symmetric matrices over the real field then tr a 1 b 1 a b 0 this inequality was used in obtaining the maximum likelihood estimates of certain covariance matrices several extensions of this inequality are obtained

estimating equations inference with missing data there is a large and growing body of literature on estimating equation ee as an estimation approach one basic property of ee that has been universally adopted in practice is that of unbiasedness and there are deep conceptual reasons why unbiasedness is a desirable ee characteristic this article deals with inference from ees when data are missing at random the investigation is motivated by the observation that direct imputation of missing data in ees generally leads to ees that are biased and thus violates a basic assumption of the ee approach the main contribution of this article is that it goes beyond existing imputation methods and proposes a procedure whereby one mitigates the effects of missing data through a reformulation of ees imputed through a kernel regression method these modified ees then constitute a basis for inference by the generalized method of moments gmm and empirical likelihood el asymptotic properties of the gmm and el estimators of the unknown parameters are derived a

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

missing at random likelihood ignorability and model completeness this paper provides further insight into the key concept of missing at random mar in incomplete data analysis following the usual selection modelling approach we envisage two models with separable parameters a model for the response of interest and a model for the missing data mechanism mdm if the response model is given by a complete density family then frequentist inference from the likelihood function ignoring the mdm is valid if and only if the mdm is mar this necessary and sufficient condition also holds more generally for models for coarse data such as censoring examples are given to show the necessity of the completeness of the underlying model for this equivalence to hold

principal component models for correlation matrices summary distributional theory regarding principal components is less well developed for correlation matrices than it is for covariance matrices the intent of this paper is to reduce this disparity methods are proposed that enable investigators to fit and to make inferences about flexible principal components models for correlation matrices the models allow arbitrary eigenvalue multiplicities and allow the distinct eigenvalues to be modelled parametrically or nonparametrically local parameterisations and implicit functions are used to construct full rank unconstrained parameterisations first order asymptotic distributions are obtained directly from the theory of estimating functions second order accurate distributions for making inferences under normality are obtained directly from likelihood theory simulation studies show that the bartlett correction is effective in controlling the size of the tests and that first order approximations to nonnull distributions are reasonably accurate the methods are illustrated on a dataset

a matrix derivation of the asymptotic covariance matrix of sample correlation coefficients abstract the asymptotic covariance matrix of the sample correlation matrix is derived in matrix form as an application of some new matrix theory in multivariate statistics

a class of pattern mixture models for normal incomplete data summary likelihood based methods are developed for analyzing a random sample on two continuous variables when values of one of the variables are missing normal maximum likelihood estimates when values are missing completely at random were derived by anderson 1957 they are also maximum likelihood providing the missing data mechanism is ignorable in rubin s 1976 sense that the mechanism depends only on observed data a new class of pattern mixture models little 1993 is described for the situation where missingness is assumed to depend on an arbitrary unspecified function of a linear combination of the two variables maximum likelihood for models in this class is straightforward and yields the estimates of anderson 1957 when missingness depends solely on the completely observed variable and the estimates of brown 1990 when missingness depends solely on the incompletely observed variable another choice of linear combination yields estimates from complete case analysis large sample and bayesian methods are described for this model the data do not supply information about the ratio of the coefficients of the linear combination that controls missingness if this ratio is not welldetermined based on prior knowledge a prior distribution can be specified and bayesian inference is then readily accomplished alternatively sensitivity of inferences can be displayed for a variety of choices of the ratio

ignorability and coarse data some biomedical examples heitjan and rubin 1991 annals of statistics 19 2244 2253 define data to be coarse when one observes not the exact value of the data but only some set a subset of the sample space that contains the exact value this definition covers a number of incomplete data problems arising in biomedicine including rounded heaped censored and missing data in analyzing coarse data it is common to proceed as though the degree of coarseness is fixed in advance in a word to ignore the randomness in the coarsening mechanism when coarsening is actually stochastic however inferences that ignore this randomness may be seriously misleading heitjan and rubin 1991 have proposed a general model of data coarsening and established conditions under which it is appropriate to ignore the stochastic nature of the coarsening the conditions are that the data be coarsened at random a generalization of missing at random rubin 1976 biometrika 63 581 592 and that the parameters of the data and the coarsening process be distinct this article presents detailed applications of the general model and the ignorability conditions to a variety of coarse data problems arising in biomedical statistics a reanalysis of the stanford heart transplant data crowley and hu 1977 journal of the american statistical association 72 27 36 reveals significant evidence that censoring of pretransplant survival times by transplantation was nonignorable suggesting a greater benefit from cardiac transplantation than previous analyses had found

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

isomorphisms of kac moody groups we give the solution of the isomorphism problem for kac moody groups over algebraically closed fields of any characteristic in particular we prove a conjecture of kac and peterson and compute the automorphism group of a kac moody group over an algebraically closed field of characteristic zero

elliptical multivariate analysis abstract in this paper we investigate the interpretation and the asymptotic properties of the well known statistics derived in multivariate analysis when the parent distribution of the observations is not necessary normal but belongs to the wider class of elliptical distributions including the normal our conclusion is that those statistics keep their geometric interpretation and consistency and that for most of those statistics the asymptotic distribution of t is n 0 1 k var n where varn is the covariance matrix under the normality assumption and where k is the joint kurtosis parameter for statistics of the x2 type used in asymptotic test criteria we find similar expressions it follows that all multivariate software may be utilized in the elliptical context under the provision that k is estimated in addition

diagnosing missing always at random in multivariate data models for analyzing multivariate data sets with missing values require strong often unassessable assumptions the most common of these is that the mechanism that created the missing data is ignorable a twofold assumption dependent on the mode of inference the first part which is the focus here under the bayesian and direct likelihood paradigms requires that the missing data are missing at random in contrast the frequentist likelihood paradigm demands that the missing data mechanism always produces missing at random data a condition known as missing always at random under certain regularity conditions assuming missing always at random leads to an assumption that can be tested using the observed data alone namely the missing data indicators only depend on fully observed variables here we propose three different diagnostic tests that not only indicate when this assumption is incorrect but also suggest which variables are the most likely culprits although missing always at random is not a necessary condition to ensure validity under the bayesian and direct likelihood paradigms it is sufficient and evidence for its violation should encourage the careful statistician to conduct targeted sensitivity analyses

hom lie structures on kac moody algebras abstract we describe hom lie structures on affine kac moody and related lie algebras and discuss the question when they form a jordan algebra

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

power analyses for correlations from clustered study designs power analysis constitutes an important component of modern clinical trials and research studies although a variety of methods and software packages are available almost all of them are focused on regression models with little attention paid to correlation analysis however the latter is arguably a simpler and more appropriate approach for modelling concurrent events especially in psychosocial research in this paper we discuss power and sample size estimation for correlation analysis arising from clustered study designs our approach is based on the asymptotic distribution of correlated pearson type estimates although this asymptotic distribution is easy to use in data analysis the presence of a large number of parameters creates a major problem for power analysis due to the lack of real data to estimate them by introducing a surrogacy type assumption we show that all nuisance parameters can be eliminated making it possible to perform power analysis based only on the parameters of interest simulation results suggest that power and sample size estimates obtained under the proposed approach are robust to this assumption copyright 2005 john wiley sons ltd

domain growth in weakly disordered magnets we present results from extensive numerical simulations of phase ordering dynamics in a mean field dynamical model of weakly disordered random magnets we apply a stringent test of dynamical scaling to confirm that the domain growth is superuniversal i e the scaling function for the disordered case is the same as for the pure case at least for the disorder amplitudes we consider however we are not able to ascertain a universal functional form for the domain growth law

ignorability sufficiency and ancillarity this paper explores the relationship between ignorability sufficiency and ancillarity in the coarse data model of heitjan and rubin bayes or likelihood ignorability has a natural relationship to sufficiency and frequentist ignorability an analogous relationship to ancillarity weaker conditions termed observed likelihood sufficiency observed specific sufficiency and observed ancillarity expand the concepts to models where the coarsening mechanism is sometimes but not always ignorable

orbits invariants and representations associated to involutions of reductive groups by an involution of an algebraic group or lie algebra we mean an automorphism 0 such that 0 2 is the identity automorphism let 0 be an involution of a complex reductive lie algebra q and let p denote the 1 eigenspace of 0 on q get g be the adjoint group of q and let k g glog go kostant and rallis 10 have made a detailed study of the action of k on p in particular they obtain very precise information concerning the orbits of k on p and the representation of k on c p the algebra of polynomial functions on p in this paper we consider a global analogue of the problem studied by kostant and rallis let g be a reductive affine algebraic group over an algebraically closed field k of characteristic 2 let 0 be an involution of g and let k gegi0 g g if g is semisimple and the base field is c then g k is the complexification of a riemannian symmetric space we study the action of k and of groups closely related to k by left translation on the coset space g k for technical reasons it is convenient to study a slightly different but equivalent problem let r g g be defined by r g g0 g i then z g p is a closed subvariety of g furthermore z is constant on left cosets of g modulo k and induces an isomorphism o g k p we have o k g k k o g k for k e k and geg thus p is stable under conjugation by k and a is a k isomorphism of affine k varieties we consider the action of k on p by conjugation let x e g have jordan decomposition x x x then we show that x e p if and only if both x and x are in p thus we have a jordan decomposition for elements of p this will play a key role in our results a torus s of g is o anisotropic if o s s 1 for every ses it is known that any two maximal 0 anisotropic tori of g are k let a be a maximal 0 anisotropic torus of g the following result characterizes semisimple elements of p

differentiating the pseudo determinant abstract a class of derivatives is defined for the pseudo determinant det a of a hermitian matrix a this class is shown to be non empty and to have a unique canonical member det a det a a where a is the moore penrose pseudo inverse the classic identity for the gradient of the determinant is thus reproduced examples are provided including the maximum likelihood problem for the rank deficient covariance matrix of the degenerate multivariate gaussian distribution