relationships between partial orders of matrices and their powers abstract baksalary and pukelsheim linear algebra appl 151 1991 135 considered the problem of how an order between two hermitian nonnegative definite matrices a and b is related to the corresponding order between the squares a 2 and b 2 in the sense of the star partial ordering the minus partial ordering and the lowner partial ordering in the present paper possibilities of generalizing and strengthening their results are studied from two points of view by widening the class of matrices considered and by replacing the squares by arbitrary powers

invertibility of a linear combination of two matrices and partial orderings abstract in this paper we present some results relating different matrix partial orderings and the invertibility of a linear combination of ep matrices

inverse problem for the acoustic wave equation a probabilistic approach to approximations and uniqueness by using path integral representation of the laplace transform of the solutions of the acoustic wave equation in a heterogeneous medium we examine questions related to the inverse problem consisting of finding the velocity of propagation of sound in a domain starting from measurements of the wave field on the surface of the domain

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

on the existence of positive solutions for semilinear elliptic equations with neumann boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded lipschitz domainsd d d 3 having compact boundary with nonlinear neumann boundary conditions on the boundary ofd for this we use an implicit probabilistic representation schauder s fixed point theorem and a recently proved sobolev inequality forw 1 2 d special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

perturbation of the drazin inverse for matrices with equal eigenprojections at zero abstract let a denote the eigenprojection of a matrix a corresponding to the eigenvalue 0 we characterize matrices b such that b a and derive from the results 1 error bounds for the drazin inverse of a perturbation 2 improvement of error bounds given recently by wei 10 wei and wang 11 and castro gonzalez et al 4 and 3 a new characterization of ep matrices

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

causal diagrams for empirical research summary the primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject matter information in particular the paper develops a principled nonparametric framework for causal inference in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data if so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions otherwise the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained

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

the probabilistic solution of the third boundary value problem for second order elliptic equations using standard reflected brownian motion srbm and martingales we define in the spirit of stroock and varadhan see s v the probabilistic solution of the boundary value problem begin gathered frac 1 2 delta u qu 0 in d hfill frac partial u partial n cu f on partial d hfill end gathered whered is a bounded domain withc3 boundary andn is the inward unit oormal vector on d the assumptions forq c andf are quite general

ignorability for general longitudinal data likelihood factors that can be disregarded for inference are termed ignorable we demonstrate that close ties exist between ignorability and identification of causal effects by covariate adjustment a graphical condition stability plays a role analogous to that of missingness at random but is applicable to general longitudinal data our formulation of ignorability does not depend on any notion of missing data so is appealing in situations where missing data may not actually exist several examples illustrate how stability may be assessed

inter and extrapolatory properties of matrix partial orderings in this paper we study some matrix partial orderings with respect to the preserving the relation a b for original a or and b and the linear combinations of a and b the property that between above respectively a and below respectively b one can locate convex affine combinations of original matrices will be called interpolatory extrapolatory respectively besides classical orderings we introduce and discuss in the mentioned aspect the notion of f orderings

further relationships between certain partial orders of matrices and their squares abstract theorem 3 of baksalary and pukelsheim linear algebra appl 151 1991 135 asserts that if both a and b are hermitian nonnegative definite matrices then the star order a b between them and the star order a 2 b 2 between their squares are equivalent and they imply the commutativity property ab ba in this paper relationships between the three conditions mentioned above are reinvestigated in situations where the assumptions on a and b are completely or partially relaxed some results concerning the star order are obtained as corollaries to corresponding results referring to the left star and right star orders introduced by baksalary and mitra linear algebra appl 149 1991 73

free fermions and the classical compact groups there is a close connection between the ground state of non interacting fermions in a box with classical absorbing reflecting and periodic boundary conditions and the eigenvalue statistics of the classical compact groups the associated determinantal point processes can be extended in two natural directions i we consider the full family of admissible quantum boundary conditions i e self adjoint extensions for the laplacian on a bounded interval and the corresponding projection correlation kernels ii we construct the grand canonical extensions at finite temperature of the projection kernels interpolating from poisson to random matrix eigenvalue statistics the scaling limits in the bulk and at the edges are studied in a unified framework and the question of universality is addressed whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is a priori not obvious we complete the picture by constructing a finite temperature extension of the haar measure on the classical compact groups the eigenvalue statistics of the resulting grand canonical matrix models of random size corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions

every missingness not at random model has a missingness at random counterpart with equal fit over the last decade a variety of models to analyse incomplete multivariate and longitudinal data have been proposed many of which allowing for the missingness to be not at random in the sense that the unobserved measurements influence the process governing missingness in addition to influences coming from observed measurements and or covariates the fundamental problems that are implied by such models to which we refer as sensitivity to unverifiable modelling assumptions has in turn sparked off various strands of research in what is now termed sensitivity analysis the nature of sensitivity originates from the fact that a missingness not at random mnar model is not fully verifiable from the data rendering the empirical distinction between mnar and missingness at random mar where only covariates and observed outcomes influence missingness difficult or even impossible unless we are willing to accept the posited mnar model in an unquestioning way we show that the empirical distinction between mar and mnar is not possible in the sense that each mnar model fit to a set of observed data can be reproduced exactly by an mar counterpart of course such a pair of models will produce different predictions of the unobserved outcomes given the observed outcomes theoretical considerations are supplemented with an illustration that is based on the slovenian public opinion survey which has been analysed before in the context of sensitivity analysis

markov processes on the lipschitz boundary for the neumann and robin problems abstract we investigate the markov process on the boundary of a bounded lipschitz domain associated to the neumann and robin boundary value problems we first construct l p semigroups of sub markovian contractions on the boundary generated by the boundary conditions and we show that they are induced by the transition functions of the forthcoming processes as in the smooth boundary case the process on the boundary is obtained by the time change with the inverse of a continuous additive functional of the reflected brownian motion the robin problem is treated with a kato type l p perturbation method using the revuz correspondence an exceptional polar set occurs on the boundary we make the link with the dirichlet forms approach

characterizations of ep matrices and weighted ep matrices abstract a square complex matrix a is said to be ep if a and its conjugate transpose a have the same range in this paper we first collect a group of known characterizations of ep matrix and give some new characterizations of ep matrices then we define weighted ep matrix and present a wealth of characterizations for weighted ep matrix through various rank formulas for matrices and their generalized inverses

remarks on the sharp partial order and the ordering of squares of matrices abstract in this note we revisit the sharp partial order introduced by mitra s k mitra on group inverses and the sharp order linear algebra appl 92 1987 17 37 we recall some already known facts from certain matrix decompositions and derive new statements relating our discussion to recent results in the literature concerned with partial orders between matrices and their squares

joint analysis of longitudinal data with informative observation times and a dependent terminal event in many longitudinal studies repeated measures are often correlated with observation times also there may exist a dependent terminal event such as death that stops the follow up in this article we propose a new joint model for the analysis of longitudinal data in the presence of both informative observation times and a dependent terminal event via latent variables estimating equation approaches are developed for parameter estimation and the resulting estimators are shown to be consistent and asymptotically normal in addition some graphical and numerical procedures are presented for model checking simulation studies demonstrate that the proposed method performs well for practical settings an application to a medical cost study of chronic heart failure patients from the university of virginia health system is provided

armitage lecture 2010 understanding treatment effects the value of integrating longitudinal data and survival analysis there is a single minded focus on events in survival analysis and we often ignore longitudinal data that are collected together with the event data this is due to a lack of methodology but also a result of the artificial distinction between survival and longitudinal data analyses understanding the dynamics of such processes is important but has been hampered by a lack of appreciation of the difference between confirmatory and exploratory causal inferences the latter represents an attempt at elucidating mechanisms by applying mediation analysis to statistical data and will usually be of a more tentative character than a confirmatory analysis the concept of local independence and the associated graphs are useful this is related to granger causality an important method from econometrics that is generally undervalued by statisticians this causality concept is different from the counterfactual one since it lacks lacks the intervention aspect the notion that one can intervene at will in naturally occurring processes which seems to underly much of modern causal inference is problematic when studying mediation and mechanisms it is natural to assume a stochastic process point of view when analyzing dynamic relationships we present some examples to illustrate this it is not clear how survival analysis must be developed to handle the complex life history data that are increasingly being collected today we give some suggestions copyright 2012 john wiley sons ltd

operators with equal projections related to their generalized inverses in this article we characterize operators on banach spaces which have the same projections related to their outer or inner generalized inverses as corollaries we obtain well known results for the drazin inverse of bounded operators

on the robin problem in fractal domains research partially supported by national science foundation nsf grants dms 0244737 and dms 0303310

conditional transformation of drift formula and potential theory for 1 2 b using conditional brownian motion and the transformation of drift formula of cameron martin girsarov maruyama we give integral conditions on a vector fieldb which imply the harmonic measures and green functions for 1 2 and 1 2 b on a bounded lipschitz domaind are equivalent by equivalent we mean there exist two sided inequalities with constants depending only onb andd this enables one to conclude the potential theory for 1 2 b ond and 1 2 ond are the same

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

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

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

partial orders in rings based on generalized inverses unified theory abstract the unified theory for matrix partial orders based on generalized inverses has already been done by mitra we consider a special kind of ring r which generalizes the ring of linear operators on finite dimensional vector space and extend mitra s approach to it thus we find necessary and sufficient condition for a g based relation to be a partial order on r also some known results are generalized and some new results are proved

stochastic differential equations for multi dimensional domain with reflecting boundary in this paper we prove that there exists a unique solution of the skorohod equation for a domain inr d with a reflecting boundary condition we remove the admissibility condition of the domain which is assumed in the work 4 of lions and sznitman we first consider a deterministic case and then discuss a stochastic case

analysis of semiparametric regression models for repeated outcomes in the presence of missing data abstract we propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data the proposed estimators do not require full specification of the likelihood they can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random these estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable the likelihood based parametric g computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance but because of possible model misspecification the parametric g computation algorithm estimator in contrast with the proposed w

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

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

probabilistic approach to the neumann problem the basic problem considered in this paper is to solve the following neumann boundary value problem probabilistically where we assume that q is in a certain functional class to be specified below and is a bounded measurable function on the boundary we give a martingale formulation of the neumann problem and show that this formulation is essentially equivalent to the classical formulation the paper culminates in an explicit formula for the solution of this problem in terms of reflecting brownian motion and its boundary local time

the kpz limit of asep with boundary it was recently proved in corwin and shen cpam cs16 that under weakly asymmetric scaling the height functions for asep with sources and sinks converges to the hopf cole solution of the kpz equation with inhomogeneous neumann boundary conditions in their assumptions cs16 chose positive values for the neumann boundary condition and they assumed initial data which is close to stationarity by developing more extensive heat kernel estimates we clarify and extend their results to negative values of the neumann boundary parameters and we also show how to generalize their results to empty initial data which is very far from stationarity combining our result with barraquand et al duke math j bbcw17 we obtain the laplace transform of the one point distribution for half line kpz and use this to confirm t1 3 scale goe tracy widom long time fluctuations at the origin

clarifying missing at random and related definitions and implications when coupled with exchangeability we clarify the key concept of missingness at random in incomplete data analysis we first distinguish between data being missing at random and the missingness mechanism being a missing at random one which we call missing always at random and which is more restrictive we further discuss how in general neither of these conditions is a statement about conditional independence we then consider the implication of the more restrictive missing always at random assumption when coupled with full unit exchangeability for the matrix of the variables of interest and the missingness indicators the conditional distribution of the missingness indicators for any variable that can have a missing value can depend only on variables that are always fully observed we discuss implications of this for modelling missingness mechanisms

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

a simple proof of the product theorem for ep matrices abstract we give a simple proof of a recent theorem of hartwig and katz on the product of ep matrices

ordering of nonnegative definite matrices with application to comparison of linear models abstract for any real nonnegative definite matrices m 1 and m 2 several necessary and sufficient conditions for m 1 m 2 are presented the conditions are used for the comparison of linear models

characterizations of matrices which eigenprojections at zero are equal to a fixed perturbation in this paper we characterize matrices b such that their eigenprojection b p corresponding to the eigenvalue 0 is a fixed perturbation of the eigenprojection of the matrix a i e b p a p s from this result we derive upper bounds for b email protected and for b d a email protected in terms of email protected and a d b a the error estimate is applied to perturbed linear systems