approximation of solutions of hammerstein equations with bounded strongly accretive nonlinear operators abstract suppose x is a real q uniformly smooth banach space and f k x x are bounded strongly accretive maps with d k f x x let u denote the unique solution of the hammerstein equation u k f u 0 a new explicit coupled iteration process is shown to converge strongly to u no invertibility assumption is imposed on k and the operators k and f need not be defined on compact subsets of x furthermore our new technique of proof is of independent interest finally some interesting open questions are included

a general class of pattern mixture models for nonignorable dropout with many possible dropout times in this article we consider the problem of fitting pattern mixture models to longitudinal data when there are many unique dropout times we propose a marginally specified latent class pattern mixture model the marginal mean is assumed to follow a generalized linear model whereas the mean conditional on the latent class and random effects is specified separately because the dimension of the parameter vector of interest the marginal regression coefficients does not depend on the assumed number of latent classes we propose to treat the number of latent classes as a random variable we specify a prior distribution for the number of classes and calculate approximate posterior model probabilities in order to avoid the complications with implementing a fully bayesian model we propose a simple approximation to these posterior probabilities the ideas are illustrated using data from a longitudinal study of depression in hiv infected women

an alternative parameterization of the general linear mixture model for longitudinal data with non ignorable drop outs this paper considers the mixture model methodology for handling non ignorable drop outs in longitudinal studies with continuous outcomes recently hogan and laird have developed a mixture model for non ignorable drop outs which is a standard linear mixed effects model except that the parameters which characterize change over time depend also upon time of drop out that is the mean response is linear in time other covariates and drop out time and their interactions one of the key attractions of the mixture modelling approach to drop outs is that it is relatively easy to explore the sensitivity of results to model specification however the main drawback of mixture models is that the parameters that are ordinarily of interest are not immediately available but require marginalization of the distribution of outcome over drop out times furthermore although a linear model is assumed for the conditional mean of the outcome vector given time of drop out after marginalization the unconditional mean of the outcome vector is not in general linear in the regression parameters as a result it is not possible to parsimoniously describe the effects of covariates on the marginal distribution of the outcome in terms of regression coefficients the need to explicitly average over the distribution of the drop out times and the absence of regression coefficients that describe the effects of covariates on the outcome are two unappealing features of the mixture modelling approach in this paper we describe a particular parameterization of the general linear mixture model that circumvents both of these problems

fixed point iterations for certain nonlinear mappings abstract in a recent paper schu established the convergence of the mann iterates of a completely continuous asymptotically nonexpansive mapping on a hilbert space in this paper we extend the result of schu to uniformly convex spaces we also obtain the same generalization for the ishikawa iteration process

a generalized steepest descent approximation for the zeros of m accretive operators a necessary and sufficient condition is proved for a generalized steepest descent approximation to converge to the zeros of m accretive operators related results deal with the convergence of the scheme to fixed points of pseudocontractive maps

matching conditional and marginal shapes in binary random intercept models using a bridge distribution function summary random effects logistic regression models are often used to model clustered binary response data regression parameters in these models have a conditional subject specific interpretation in that they quantify regression effects for each cluster very often the logistic functional shape conditional on the random effects does not carry over to the marginal scale thus parameters in these models usually do not have an explicit marginal population averaged interpretation we study a bridge distribution function for the random effect in the random intercept logistic regression model under this distributional assumption the marginal functional shape is still of logistic form and thus regression parameters have an explicit marginal interpretation the main advantage of this approach is that likelihood inference can be obtained for either marginal or conditional regression inference within a single model framework the generality of the results and some properties of the bridge distribution functions are discussed an example is used for illustration

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

approximation methods for nonlinearm accretive operator equations abstract suppose e is a real reflexive banach space and t d t e e is a locally lipschitzian m accretive operator where the domain of t d t is a proper subset of e for any f e approximation methods are constructed which converge strongly to the solution of the equation x tx f explicit error estimates are given and convergence is at least as fast a geometric progression

hamiltonian square roots of skew hamiltonian matrices abstract we present a constructive existence proof that every real skew hamiltonian matrix w has a real hamiltonian square root the key step in this construction shows how one may bring any such w into a real quasi jordan canonical form via symplectic similarity we show further that every w has infinitely many real hamiltonian square roots and give a lower bound on the dimension of the set of all such square roots some extensions to complex matrices are also presented

iterative solution of nonlinear equations with strongly accretive operators abstract let e be a real banach space with a uniformly convex dual suppose t e e is a strongly accretive map with bounded range such that for each e the equation tx has a solution in e it is proved that each of the two well known fixed point iteration methods the mann and ishikawa iteration methods under suitable conditions converges strongly to a solution of the equation tx furthermore our method shows that such a solution is necessarily unique explicit error estimates are given our results resolve in the affirmative an open problem j math anal appl 151 no 2 1990 460 and generalize important known results

iterative approximation of solutions of nonlinear equations of hammerstein type abstract suppose x is a real q uniformly smooth banach space and f k x x are lipschitz strongly accretive maps with d k f x x let u denote the unique solution of the hammerstein equation u k f u 0 an iteration process recently introduced by chidume and zegeye is shown to converge strongly to u no invertibility assumption is imposed on k and the operators k and f need not be defined on compact subsets of x furthermore our new technique of proof is of independent interest finally some interesting open questions are included

marginally specified logistic normal models for longitudinal binary data summary likelihood based inference for longitudinal binary data can be obtained using a generalized linear mixed model breslow n and clayton d g 1993 journal of the american statistical association88 9 25 wolfinger r and o connell m 1993 journal of statistical computation and simulation48 233 243 given the recent improvements in computational approaches alternatively fitzmaurice and laird 1993 biometrika80 141 151 molenberghs and lesaffre 1994 journal of the american statistical association89 633 644 and heagerty and zeger 1996 journal of the american statistical association91 1024 1036 have developed a likelihood based inference that adopts a marginal mean regression parameter and completes full specification of the joint multivariate distribution through either canonical and or marginal higher moment assumptions each of these marginal approaches is computationally intense and currently limited to small cluster sizes in this manuscript an alternative parameterization of the logistic normal random effects model is adopted and both likelihood and estimating equation approaches to parameter estimation are studied a key feature of the proposed approach is that marginal regression parameters are adopted that still permit individual level predictions or contrasts an example is presented where scientific interest is in both the mean response and the covariance among repeated measurements

smoothing spline based score tests for proportional hazards models we propose score type tests for the proportional hazards assumption and for covariate effects in the cox model using the natural smoothing spline representation of the corresponding nonparametric functions of time or covariate the tests are based on the penalized partial likelihood and are derived by viewing the inverse of the smoothing parameter as a variance component and testing an equivalent null hypothesis that the variance component is zero we show that the tests have a size close to the nominal level and good power against general alternatives and we apply them to data from a cancer clinical trial

steepest descent method for equilibrium points of nonlinear systems with accretive operators let e be a real normed linear space and let a e 2e be a bounded uniformly continuous strongly accretive multi valued map with nonempty closed values such that the inclusion 0 ax has a solution x e it is proved that both the ishikawa and the mann iteration processes are bounded and converge strongly to x related results deal with the convergence of the iteration processes to fixed points of strongly pseudocontractive maps and the iterative solution of the equation 0 x fx for an accretive map f furthermore the ishikawa and mann iteration methods with errors are discussed and proofs are sketched on how our theorems extend to these methods our method of proof is of independent interest

on chidume s open questions and approximate solutions of multivalued strongly accretive mapping equations in banach spaces abstract some new convergence theorems of the ishikawa iteration process for multi valued pseudo contractive mappings and multi valued accretive mappings are obtained which affirmatively answer the open questions put forth by 4

global iterative schemes for accretive operators abstract let e be a real q uniformly smooth banach space and a e 2e be an m accretive operator which satisfies a linear growth condition of the form ax c 1 x for some constant c 0 and for all x e it is proved that if two real sequences n and n satisfy appropriate conditions the sequence xn generated from arbitrary x0 e by xn 1 xn n un n xn z un axn n 0 converges strongly to some x a 1 0 furthermore if e is a reflexive banach space with a uniformly g teaux differentiable norm and if every weakly compact convex subset of e has the fixed point property for nonexpansive mappings and a d a e 2e is m accretive then for arbitrary z x0 e the sequence xn defined by xn 1 n un 1 n xn 1 z xn en for un axn where en e is such that en

a necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations abstract a necessary and sufficient condition is established which ensures the strong convergence of the steepest descent approximation to a solution of equations involving quasi accretive operators defined on a uniformly smooth banach space

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

iterative solution of nonlinear equations of the φ strongly accretive type abstract letq 1 and letebe a realq uniformly smooth banach space lett e ebe a lipschitz strongly accretive operator and suppose the equationtx f f e has a solution it is proved that under suitable conditions on the real sequences n n 0 and n n 0 the iteration process formula converges strongly to the unique solution of the equationtx f a related result deals with the approximation of fixed points of hemicontractive operators a class of operators which is much more general than the important class of strongly pseudocontractive operators

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

marginalized transition models and likelihood inference for longitudinal categorical data summary marginal generalized linear models are now frequently used for the analysis of longitudinal data semiparametric inference for marginal models was introduced by liang and zeger 1986 biometrics73 13 22 this article develops a general parametric class of serial dependence models that permits likelihood based marginal regression analysis of binary response data the methods naturally extend the first order markov models of azzalini 1994 biometrika81 767 775 and prove computationally feasible for long series

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

note on the factorization of a square matrix into two hermitian or symmetric matrices this paper presents elementary proofs of the factorization of a square matrix into two hermitian or symmetric matrices

factorizations of a matrix into symmetric matrices with prescribed nullities abstract necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k 2 symmetric matrices with prescribed nullities a generalization of the case k 2 to hermitian matrices is also presented

modeling longitudinal data with nonignorable dropouts using a latent dropout class model summary in longitudinal studies with dropout pattern mixture models form an attractive modeling framework to account for nonignorable missing data however pattern mixture models assume that the components of the mixture distribution are entirely determined by the dropout times that is two subjects with the same dropout time have the same distribution for their response with probability one as that is unlikely to be the case this assumption made lead to classification error in addition if there are certain dropout patterns with very few subjects which often occurs when the number of observation times is relatively large pattern specific parameters may be weakly identified or require identifying restrictions we propose an alternative approach which is a latent class model the dropout time is assumed to be related to the unobserved latent class membership where the number of classes is less than the number of observed patterns a regression model for the response is specified conditional on the latent variable this is a type of shared parameter model where the shared parameter is discrete parameter estimates are obtained using the method of maximum likelihood averaging the estimates of the conditional parameters over the distribution of the latent variable yields estimates of the marginal regression parameters the methodology is illustrated using longitudinal data on depression from a study of hiv in women