herman s theory revisited we prove that a c 2 smooth orientation preserving circle diffeomorphism with rotation number in diophantine class d 0 1 is c 1 smoothly conjugate to a rigid rotation we also derive the most precise version of denjoy s inequality for such diffeomorphisms

completely continuous composition operators a composition operator tbf f o b is completely continuous on h1 if and only if lbl 1 a e if the adjoint operator tb is completely continuous on vmoa then tb is completely continuous on h1 examples are given to show that the converse fails in general two results are given concerning the relationship between the complete continuity of an operator and of its adjoint in the presence of certain separability conditions on the underlying banach space

bias due to missing exposure data using complete case analysis in the proportional hazards regression model we studied bias due to missing exposure data in the proportional hazards regression model when using complete case analysis cca eleven missing data scenarios were considered one with missing completely at random mcar four missing at random mar and six non ignorable missingness scenarios with a variety of hazard ratios censoring fractions missingness fractions and sample sizes when missingness was mcar or dependent only on the exposure there was negligible bias 2 3 per cent that was similar to the difference between the estimate in the full data set with no missing data and the true parameter in contrast substantial bias occurred when missingness was dependent on outcome or both outcome and exposure for models with hazard ratio of 3 5 a sample size of 400 20 per cent censoring and 40 per cent missing data the relative bias for the hazard ratio ranged between 7 per cent and 64 per cent we observed important differences in the direction and magnitude of biases under the various missing data mechanisms for example in scenarios where missingness was associated with longer or shorter follow up the biases were notably different although both mechanisms are mar the hazard ratio was underestimated with larger bias when missingness was associated with longer follow up and overestimated with smaller bias when associated with shorter follow up if it is known that missingness is associated with a less frequently observed outcome or with both the outcome and exposure cca may result in an invalid inference and other methods for handling missing data should be considered

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

a nonparametric method for dealing with mismeasured covariate data abstract mismeasurement of covariate data is a frequent problem in statistical data analysis however when true and mismeasured data are obtained for a subsample of the observations it is possible to estimate the parameters relating the outcome to the covariate of interest maximum likelihood methods that rely on parametric models for the mismeasurement have not met with much success realistic models for the mismeasurement process are difficult to construct the form of the likelihood is often intractable and more important such methods are not robust to model misspecification we propose an easily implemented method that is nonparametric with respect to the mismeasurement process and that is applicable when mismeasurement is due to the problem of missing data errors in variables or use of imperfect surrogate covariates specifically denote the outcome variable by y the covariate data subject to mismeasurement by x and the remaining covariates including perhaps surrogates or mismeasured values

proportional hazards regression with missing covariates abstract nonparametric maximum likelihood npml is used to estimate regression parameters in a proportional hazards regression model with missing covariates the npml estimator is shown to be consistent and asymptotically normally distributed under some conditions em type algorithms are applied to solve the maximization problem variance estimates of the regression parameters are obtained by a profile likelihood approach that uses em aided numerical differentiation simulation results indicate that the npml estimates of the regression parameters are more efficient than the approximate partial likelihood estimates and estimates from complete case analysis when missing covariates are missing completely at random and that the proposed method corrects for bias when the missing covariates are missing at random

adjoints of composition operators on hardy spaces of the half plane abstract building on techniques used in the case of the disc we use a variety of methods to develop formulae for the adjoints of composition operators on hardy spaces of the upper half plane in doing so we prove a slight extension of a known necessary condition for the boundedness of such operators and use it to provide a complete classification of the bounded composition operators with rational symbol we then consider some specific examples comparing our formulae with each other and with other easily deduced formulae for simple cases

likelihood based methods for missing covariates in the cox proportional hazards model problems associated with missing covariate data are well known but often ignored we present a method for estimating the parameters in the cox proportional hazards model when the missing data are missing at random mar and censoring is noninformative due to the computational burden of this method we introduce an approximation that allows us to use a weighted expectation maximization em algorithm to estimate the parameters more easily when the missing covariates are continuous rather than categorical we implement a monte carlo version of the em algorithm along with the gibbs sampler to obtain parameter estimates we also give the asymptotic distribution of these estimates the primary advantage of this method over complete case analysis is that it produces more efficient parameter estimates and corrects for bias in the mar setting to motivate the methodology we present an analysis of a phase iii melanoma clinical trial conducted by the eastern cooperative oncology group

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

hypercyclic operators that commute with the bergman backward shift the backward shift b on the bergman space of the unit disc is known to be hypercyclic meaning it has a dense orbit here we ask which operators that commute with b inherit its hypercyclicity we show that the problem reduces to the study of operators of the form 9p b where so is a holomorphic self map of the unit disc that multiplies the dirichlet space into itself and that the question of hypercyclicity for such an operator depends on how freely 9p z is allowed to approach the unit circle as lz s 1

regression with missing x s a review abstract the literature of regression analysis with missing values of the independent variables is reviewed six classes of procedures are distinguished complete case analysis available case methods least squares on imputed data maximum likelihood bayesian methods and multiple imputation methods are compared and illustrated when missing data are confined to one independent variable and extensions to more general patterns are indicated attention is paid to the performance of methods when the missing data are not missing completely at random least squares methods that fill in missing x s using only data on the x s are contrasted with likelihood based methods that use data on the x s and y the latter approach is preferred and provides methods for elaboration of the basic normal linear regression model it is suggested that more widely distributed software is needed that advances beyond complete case analysis available case analysis and naive imputation methods bayesian simulation methods and mu

an application of atomic decomposition in bergman spaces to the study of differences of composition operators abstract in this paper we characterize bounded differences of composition operators from standard weighted bergman spaces a p into lebesgue spaces l q when 1 and p q in the process we also find an interesting connection between the difference operator and certain weighted composition operators answering a question raised in saukko 2011 18 by combining the results of this paper with those of saukko 2011 18 we obtain a complete characterization of bounded and compact differences between standard weighted bergman spaces

a new proof of m herman s theorem a new proof of the m herman theorem on the smooth conjugacy of a circle map is presented here it is based on the thermodynamic representation of dynamical systems and the study of the ergodic properties for the corresponding radom variables

carleson embeddings and some classes of operators on weighted bergman spaces we prove carleson type embedding theorems for weighted bergman spaces with bekolle weights we use this to study properties of toeplitz type operators integration operators and composition operators acting on such spaces in particular we investigate the membership of these operators to schatten class ideals

a remark on denjoy s inequality for pl circle homeomorphisms with two break points abstract it is well known that for a p homeomorphism f of the circle s 1 r z with irrational rotation number f the denjoy s inequality log d f q n v holds where v is the total variation of log d f and q n n 1 are the first return times of f let h be a piecewise linear pl circle homeomorphism with two break points a 0 c 0 irrational rotation number h and total jump ratio h 1 denote by b n h the partition determined by the break points of h q n and by h the unique h invariant probability measure it is shown that the derivative d h q n is constant on every element of b n h and takes either two or three values furthermore we prove that log d h q n can be expressed in terms of h measures of some intervals of the partition b n h multiplied by the logarithm of the jump ratio h a 0 of h at the break point a 0

difference of composition operators between different hardy spaces abstract some estimates for the norm and essential norm of the difference of two composition operators between different hardy spaces are given in this paper

fitting cox s proportional hazards models from survey data summary the proportional hazards model makes a number of assumptions which may not always be completely satisfied however fitting such models can have both descriptive and analytical value we discuss the implications of the survey design for large scale studies we propose a weighted estimator and derive a design based estimator for the variance of the estimated parameters the derivation requires using methods not previously used in the context of analysis of survey data we conduct a small simulation study comparing the proposed method with others in the literature

subnormality and composition operators on h2 if is a non constant analytic function defined on the unit disk d such that d d the composition operator c is the operator on the hardy space h2 of the unit disk defined by c f f in this paper we investigate the relationship between properties of the symbol and the subnormality of the operators c and c the main theorem shows under a regularity hypothesis that c is subnormal if and only if is in a restricted class of linear fractional transformations the two previously known special cases the affine transformations associated with the cesaro operator and the automorphisms are included as the extreme examples the proof of the main theorem consists of verifying a moment condition and yields in addition the construction of a measure that exhibits cgj as multiplication by an analytic function on p2 and the identification of the minimal normal extension as a weighted sum of shifts

differentiability at the tip of arnold tongues for diophantine rotations numerical studies and renormalization group explanations we study numerically the regularity of arnold tongues corresponding to diophantine rotation numbers of circle maps at the edge of validity of kam theorem this serves as a good test for the numerical stability of two different algorithms we find empirically that arnold tongues are only finitely differentiable at the tip we also find several scaling properties of the sobolev norms of the conjugacy near the breakdown we also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the sobolev regularity we also uncover empirically some other patterns which require explanation

composition operators belonging to operator ideals abstract with each composition operator c f f acting between classical bergman spaces ap and aq 1 p q a if and only if the diagonal operator d lp lq given by ak ak k belongs to a conversely we show that for any max 2 q 2 p 0 and r there exists a composition operator c ap aq whose associated sequence is equivalent to k log k k 2 this provides an effective tool to separate quasi banach ideals by means of composition operators the results can be generalized to standard weighted bergman spaces and beyond

estimating equations with incomplete categorical covariates in the cox model incomplete covariate data is a common occurrence in many studies in which the outcome is survival time when a full likelihood is specified a useful technique for obtaining parameter estimates is the em algorithm we propose a set of estimating equations to estimate the parameters of cox s proportional hazards model when some covariate values are missing these estimating equations can be solved by an algorithm similar to the em algorithm because of the computational burden of finding a solution to these estimating equations we propose obtaining parameter estimates via monte carlo methods asymptotic variances of the parameter estimates are also derived we present a clinical trials example with three covariates two of which have some missing values

adjoints of composition operators with rational symbol abstract building on techniques developed by cowen and gallardo gutierrez we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the hardy space h 2 we consider some specific examples comparing our formula with several results that were previously known

on conjugacies between asymmetric bernoulli shifts abstract this paper investigates conjugacies between asymmetric bernoulli shifts it is shown that there exist a unique increasing conjugacy and a unique decreasing conjugacy we respectively construct a sequence of functions to approximate these two conjugacies and give an estimation for the error of the approximation we also present explicit formulae of these two conjugacies it is shown that these two conjugacies are singular holder continuous and not differentiable

essential norm of composition operators between weighted hardy spaces abstract an asymptotic formula for the essential norm of composition operators acting between two weighted hardy spaces h w 1 and h w 2 where w1 and w2 are two admissible weight functions is given the boundedness of the operators is also characterized

on using the cox proportional hazards model with missing covariates summary we propose two methods for handling missing covariates in using the cox proportional hazards model the maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information also it is biased when the probability of missingness depends on the failure or censoring time our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information the proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates also the proposed estimator is more efficient than the estimator suggested previously by lin ying 1993 when data are missing completely at random

trace ideal criteria for toeplitz operators for a complex measure on the open unit disk u define an operator t on a hilbert space h of analytic functions with reproducing kernel k z w by u for a certain scale of hilbert spaces h 1 which includes the hardy space h2 and weighted bergman spaces a2 conditions are obtained which imply t belongs to a schatten ideal sp if is a positive measure then these conditions are necessary and sufficient application to composition operators and restriction operators are indicated

mean growth of koenigs eigenfunctions in 1884 g koenigs solved schroeder s functional equation f o a af in the following context is a given holomorphic function mapping the open unit disk u into itself and fixing a point a e u f is holomorphic on u and a is a complex scalar koenigs showed that if 0 equation for p has a unique holomorphic solution cr satisfying a co p a a and o 0 1 moreover he showed that the only other solutions are the obvious ones given by constant multiples of powers of a we call a the koenigs eigenfunction of motivated by fundamental issues in operator theory and function theory we seek to understand the growth of integral means of koenigs eigenfunctions for 0 of to belong to the hardy space hp and show that the condition is necessary when is analytic on the closed disk for many mappings the condition may be expressed as a relationship between p a and derivatives of at points on ou that are fixed by some iterate of our work depends upon a formula we establish for the essential spectral radius of any composition operator on the hardy space hp department of mathematics washington and lee university lexington virginia 24450 e mail address pbourdonfwlu edu department of mathematics michigan state university east lansing michigan 48824 e mail address shapirofmath msu edu this content downloaded from 157 55 39 211 on sat 25 jun 2016 06 33 26 utc all use subject to http about jstor org terms

a structure theorem in one dimensional dynamics on considere la classe 7b a des applications c f 0 1 0 1 telles que f 0 f 1 0 et f a un unique point critique c 0 1 si le point critique de f 7b a est non plat alors f n a pas d intervalle errant

boundary convergence of vector valued pseudocontinuable functions abstract in the first part of the paper we discuss a multi dimensional analogue of the well known construction by d clark that allows one to study families of spectral measures of perturbations of the model contraction in the second part we present extensions of the relevant results on the boundary behavior of pseudocontinuable functions we show that although the most direct analogue of the scalar theorem on the existence of boundary values for pseudocontinuable functions with respect to clark measures fails in the non scalar situation suitable vector valued versions of such results can be found

on shapiroʼs compactness criterion for composition operators abstract we give an elementary and direct proof of the identity lim sup w 1 n w 1 w lim sup a 1 1 a 2 1 1 a h 2 2 for any analytic self map of z z 1 where n denotes the nevanlinna counting function of we further show that one can find analytic self maps of z z 1 where the composition operator c on the hardy space h 2 is compact such that n h 2 tends to zero at an arbitrarily slow rate as n even in the case that is univalent among these are new examples where c is compact on h 2 but not in any of the schatten classes

a new class of operators and a description of adjoints of composition operators abstract starting with a general formula precise but difficult to use for the adjoint of a composition operator on a functional hilbert space we compute an explicit formula on the classical hardy hilbert space for the adjoint of a composition operator with rational symbol to provide a foundation for this formula we study an extension to the definitions of composition weighted composition and toeplitz operators to include symbols that are multiple valued functions these definitions can be made on any banach space of analytic functions on a plane domain but in this work our attention is focused on the basic properties needed for the application to operators on the standard hardy and bergman hilbert spaces on the unit disk

log linear analysis of censored survival data with partially observed covariates abstract log linear models provide a flexible means of extending life table techniques for the analysis of censored survival data with categorical covariates as discussed by holford 1980 and laird and olivier 1981 we extend this methodology to incorporate cases in which one or more of the categorical covariates are sometimes missing maximum likelihood estimates of the parameters are calculated using data from all cases this can result in large gains in efficiency over standard methods that require the exclusion of cases with incomplete data with this approach we assume that the hazard function conditional on the covariates is a stepwise function over disjoint intervals of time the model has two parts a log linear model describing the hazard parameters and a multinomial model describing the probabilities in the contingency table defined by the covariates the main interest is in the model for the hazard parameters we show how to calculate maximum likelihood estimates of parameters of the mo

adjoints of rationally induced composition operators abstract we give an elementary proof of a formula recently obtained by hammond moorhouse and robbins for the adjoint of a rationally induced composition operator on the hardy space h 2 christopher hammond jennifer moorhouse marian e robbins adjoints of composition operators with rational symbol j math anal appl 341 2008 626 639 we discuss some variants and implications of this formula and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator

incomplete data in generalized linear models abstract this article examines incomplete data for the class of generalized linear models in which incompleteness is due to partially missing covariates on some observations under the assumption that the missing data are missing at random it is shown that the e step of the em algorithm for any generalized linear model can be expressed as a weighted complete data log likelihood when the unobserved covariates are assumed to come from a discrete distribution with finite range expressing the e step in this manner allows for a straightforward maximization in the m step thus leading to maximum likelihood estimates mle s for the parameters asymptotic variances of the mle s are also derived and results are illustrated with two examples

indicator and stratification methods for missing explanatory variables in multiple linear regression abstract the statistical literature and folklore contain many methods for handling missing explanatory variable data in multiple linear regression one such approach is to incorporate into the regression model an indicator variable for whether an explanatory variable is observed another approach is to stratify the model based on the range of values for an explanatory variable with a separate stratum for those individuals in which the explanatory variable is missing for a least squares regression analysis using either of these two missing data approaches the exact biases of the estimators for the regression coefficients and the residual variance are derived and reported the complete case analysis in which individuals with any missing data are omitted is also investigated theoretically and is found to be free of bias in many situations though often wasteful of information a numerical evaluation of the bias of two missing indicator methods and the complete case analysis is reported the missing indi

toeplitz operators on harmonically weighted bergman spaces and applications to composition operators on dirichlet spaces abstract in this paper we consider toeplitz operators acting on bergman spaces associated with harmonic weights we give a complete description of those operators which belong to p schatten classes we also obtain an upper estimate for the reproducing kernel of these spaces applications to composition operators on dirichlet spaces are given

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

rigidity of some abelian by cyclic solvable group actions on mathbb t n in this paper we study a natural class of groups that act as affine transformations of mathbb t n we investigate whether these solvable abelian by cyclic groups can act smoothly and nonaffinely on mathbb t n while remaining homotopic to the affine actions in the affine actions elliptic and hyperbolic dynamics coexist forcing a priori complicated dynamics in nonaffine perturbations we first show using the kam method that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action provided certain diophantine conditions on the action are met in dimension two under natural dynamical hypotheses we get a complete classification of such actions namely any such group action by c r diffeomorphisms can be conjugated to the affine action by c r varepsilon conjugacy next we show that in any dimension c 1 small perturbations can be conjugated to an affine action via c 1 varepsilon conjugacy the method is a generalization of the herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics

cox regression with incomplete covariate measurements abstract this article provides a general solution to the problem of missing covariate data under the cox regression model the estimating function for the vector of regression parameters is an approximation to the partial likelihood score function with full covariate measurements and reduces to the pseudolikelihood score function of self and prentice in the special setting of case cohort designs the resulting parameter estimator is consistent and asymptotically normal with a covariance matrix for which a simple and consistent estimator is provided extensive simulation studies show that the large sample approximations are adequate for practical use the proposed approach tends to be more efficient than the complete case analysis especially for large cohorts with infrequent failures for case cohort designs the new methodology offers a variance covariance estimator that is much easier to calculate than the existing ones and allows multiple subcohort augmentations to improve efficiency real data taken f

combination laws for scaling exponents and relation to the geometry of renormalization operators the principle of approximate combination of scaling exponents renormalization group has become a standard tool for describing universal properties of different routes to chaos period doubling in unimodal maps quasiperiodic transitions in circle maps dynamics on the boundaries of siegel disks destruction of invariant circles of area preserving twist maps and others the universal scaling exponents for each route are related to the properties of the corresponding renormalization operators