statistical estimation in varying coefficient models varying coefficient models are a useful extension of classical linear models they arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age the appeal of these models is that the coef cient functions can easily be estimated via a simple local regression this yields a simple one step estimation procedure we show that such a one step method cannot be optimal when different coefficient functions admit different degrees of smoothness this drawback can be repaired by using our proposed two step estimation procedure the asymptotic mean squared error for the two step procedure is obtained and is shown to achieve the optimal rate of convergence a few simulation studies show that the gain by the two step procedure can be quite substantial the methodology is illustrated by an application to an environmental data set

divergence and quasimorphisms of right angled artin groups we give a group theoretic characterization of geodesics with superlinear divergence in the cayley graph of a right angled artin group a with connected defining graph we use this to prove that the divergence of a is linear if is a join and quadratic otherwise as an application we give a complete description of the cut points in any asymptotic cone of a we also show that every non abelian subgroup of a has an infinite dimensional space of non trivial quasimorphisms

the geometry of purely loxodromic subgroups of right angled artin groups we prove that finitely generated purely loxodromic subgroups of a right angled artin group a fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups mod s in particular such subgroups are quasiconvex in a in addition we identify a milder condition for a finitely generated subgroup of a that guarantees it is free undistorted and retains finite generation when intersected with a for subgraphs of these results have applications to both the study of convex cocompactness in mod s and the way in which certain groups can embed in right angled artin groups

automorphisms of graph groups abstract given a graph v e the graph group on is the group generated by the vertex set v with the defining relations being that two adjacent vertices commute in the following the centralizer problem for graph groups is solved and it is shown that for some graph groups aut f is finitely generated by generators which are natural analogues to the nielsen automorphisms for a free group

cofinality spectrum theorems in model theory set theory and general topology we connect and solve two longstanding open problems in quite different areas the modeltheoretic question of whether sop2 is maximal in keisler s order and the question from general topology set theory of whether p t the oldest problem on cardinal invariants of the continuum we do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely using model theoretic methods by a cofinality spectrum problem s we essentially mean the data of a pair of models m m1 which code sufficient set theory possibly in an expanded language along with a distinguished set of formulas s which define linear orders in m1 let ts the treetops of s be the smallest regular cardinal such that one of a set of derived trees in m1 has a strictly increasing sequence with no upper bound let c s ts be the set of pairs of regular cardinals 1 2 such that 1 2 or what is equivalent such that d contains no cuts for cf is good we obtain several consequences notably existence of a minimum keisler class among the non simple theories

semiparametric analysis of the additive risk model in contrast to the proportional hazards model the additive risk model specifies that the hazard function associated with a set of possibly time varying covariates is the sum of rather than the product of the baseline hazard function and the regression function of covariates this formulation describes a different aspect of the association between covariates and the failure time than the proportional hazards model and is more plausible than the latter for many applications in the present paper simple procedures with high efficiencies are developed for making inference about the regression parameters under the additive risk model with an unspecified baseline hazard function

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography

modeling partly conditional means with longitudinal data abstract we propose a general modeling approach to longitudinal data that is a hybrid of the marginal regression models of zeger and liang and of the classical transition models such as used in time series analyses rather than conditioning at time t only on covariate values as is typical with the marginal approach or on the entire history of the process up to t as is typical with the transition model approach we suggest models that condition on a subset of the process history estimation proceeds using generalized estimating equation methodology but with the restriction that the working covariance matrix is diagonal the proposed regression models share common features with cox regression models for failure time data in that they are composed of a nuisance baseline function of time and a simple parametric function of the covariates two illustrative examples are presented

general topology meets model theory on p and t cantor proved in 1874 cantor g 1874 j reine angew math 77 258 262 that the continuum is uncountable and hilbert s first problem asks whether it is the smallest uncountable cardinal a program arose to study cardinal invariants of the continuum which measure the size of the continuum in various ways by godel godel k 1939 proc natl acad sci usa 25 4 220 224 and cohen cohen p 1963 proc natl acad sci usa 50 6 1143 1148 hilbert s first problem is independent of zfc zermelo fraenkel set theory with the axiom of choice much work both before and since has been done on inequalities between these cardinal invariants but some basic questions have remained open despite cohen s introduction of forcing the oldest and perhaps most famous of these is whether which was proved in a special case by rothberger rothberger f 1948 fund math 35 29 46 building on hausdorff hausdorff 1936 fund math 26 241 255 in this paper we explain how our work on the structure of keisler s order a large scale classification problem in model theory led to the solution of this problem in zfc as well as of an a priori unrelated open question in model theory

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

time series modelling with semiparametric factor dynamics high dimensional regression problems which reveal dynamic behavior are typically analyzed by time propagation of a few number of factors the inference on the whole system is then based on the low dimensional time series analysis such high dimensional problems occur frequently in many different fields of science in this article we address the problem of inference when the factors and factor loadings are estimated by semiparametric methods this more flexible modeling approach poses an important question is it justified from an inferential point of view to base statistical inference on the estimated times series factors we show that the difference of the inference based on the estimated time series and true unobserved time series is asymptotically negligible our results justify fitting vector autoregressive processes to the estimated factors which allows one to study the dynamics of the whole high dimensional system with a low dimensional representation we illustrate the theory with a simulati

superrigidity for irreducible lattices and geometric splitting we prove general superrigidity results for actions of irreducible lattices on cat 0 spaces first in terms of the ideal boundary and then for the intrinsic geometry including for infinite dimensional spaces in particular one obtains a new and self contained proof of margulis superrigidity theorem for uniform irreducible lattices in non simple groups the proofs rely on simple geometric arguments including a splitting theorem which can be viewed as an infinite dimensional and singular generalization of the lawson yau gromoll wolf theorem appendix a gives a very elementary proof of commensurator superrigidity appendix b proves that all our results also hold for certain non uniform lattices

dynamical correlation for multivariate longitudinal data nonparametric methodology for longitudinal data analysis is becoming increasingly popular the analysis of multivariate longitudinal data where data on several time courses are recorded for each subject has received considerably less attention despite its importance for practical data analysis in particular there is a need for measures and estimates to capture dependency between the components of vector valued longitudinal data we propose and analyze a simple and effective nonparametric method to quantify the covariation of components of multivariate longitudinal observations which are viewed as realizations of a random process this includes the notion of a correlation between derivatives and time shifted versions the concept of dynamical correlation is based on a scalar product obtained from pairs of standardized smoothed curves the proposed method can be used when observation times are irregular and not matching between subjects or between responses within a subject for higher dimensional dat

asymptotic cones of finitely presented groups let g be a connected semisimple lie group with at least one absolutely simple factor s such that r rank s 2 and let be a uniform lattice in g a if ch holds then has a unique asymptotic cone up to homeomorphism b if ch fails then has 22 asymptotic cones up to homeomorphism

constructing regular ultrafilters from a model theoretic point of view this paper contributes to the set theoretic side of understanding keisler s order we consider properties of ultralters which aect saturation of unstable theories the lower conality lcf 0 d of 0 modulo d saturation of the minimum unstable theory the random graph exi bility goodness goodness for equality and realization of symmetric cuts we work in zfc except when noted as several constructions appeal to complete ultralters thus assume a measurable car dinal the main results are as follows first we investigate the strength of exibility known to be detected by non low theories assuming 0 is measurable we construct a regular ultralter on 2 which is exible but not good and which moreover has large lcf 0 but does not even saturate models of the random graph this implies a that exibility alone cannot characterize saturation of any theory however b by separating exibility from goodness we remove a main obstacle to proving non low does not imply maximal since exible is precisely ok this also shows that c from a set theoretic point of view consistently ok need not imply good addressing a prob lem from dow 1985 second under no additional assumptions we prove that there is a loss of saturation in regular ultrapowers of unstable theories and also give a new proof that there is a loss of saturation in ultrapowers of non simple theories more precisely for d regular onand m a model of an unstable theory m d is not 2 saturated and for m a model of a non simple theory and m d is not saturated in the third part of the paper we investigate realization and omission of symmetric cuts signicant both because of the maximality of the strict order property in keisler s order and by recent work of the authors on sop2 we prove that if d is a complete ultralter on any ultrapower of a suciently saturated model of linear or der will have no cuts and that ifd is also normal it will have a cut we apply this to prove that for any n assuming the existence of n measurable cardinals below there is a regular ultralter d onsuch that any d ultrapower of a model of linear order will haven alternations of cuts as dened below moreover d will saturate all stable theories but will not 2 saturate any unstable theory whereis the smallest measurable cardinal used in the construction

the solution to a conjecture of tits on the subgroup generated by the squares of the generators of an artin group let a be an artin group with standard generating set s s s tits conjectured that the only relations in a amongst the squares of the generators are consequences of the obvious ones namely that s 2 and t 2 commute whenever s and t commute for s t s in this paper we prove tits conjecture for all artin groups in fact given a number m s 2 for each s s we show that the elements t s s ms s s generate a subgroup that has a finite presentation in which the only defining relations are that t s and t t commute if s and t commute

nonparametric smoothing estimates of time varying coefficient models with longitudinal data this paper considers nonparametric estimation in a varying coefficient model with repeated measurements y ij x ij t ij for i 1 n and j 1 n i where x ij x ij x ijk t and y ij x ij t ij denote the jth outcome covariate and time design points respectively of the ith subject the model considered here is y ij x ij t t ij e i t ij where t 0 t k t t for k 0 are smooth nonparametric functions of interest and e i t is a zero mean stochastic process the measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject two nonparametric estimators of t namely a smoothing spline and a locally weighted polynomial are derived for such repeatedly measured data a crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters asymptotic properties such as consistency rates of convergence and asymptotic mean squared errors are established for kernel estimators a special case of the local polynomials these asymptotic results give useful insights into the reliability of our general estimation methods an example of predicting the growth of children born to hiv infected mothers based on gender hiv status and maternal vitamin a levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies

selective covering properties of product spaces ii spaces we study productive properties of spaces and their relation to other classic and modern selective covering properties among other things we prove the following results 1 solving a problem of f jordan we show that for every unbounded tower set x r of cardinality 1 the space cp x is productively frechet urysohn in particular the set x is productively 2 solving problems of scheepers and weiss and proving a conjecture of babinkostova scheepers we prove that assuming the continuum hypothesis there are spaces whose product is not even menger 3 solving a problem of scheepers tall we show that the properties and gerlits nagy are preserved by cohen forcing moreover every hurewicz space that remains hurewicz in a cohen extension must be rothberger and thus we apply our results to solve a large number of additional problems and use arhangel skiuo duality to obtain results concerning local properties of function spaces and countable topo logical groups

contracting elements and random walks we define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups pseudo anosovs in mapping class groups rank one isometries in groups acting properly on proper cat 0 spaces elements acting hyperbolically on the bass serre tree in graph manifold groups we also define a related notion of weakly contracting element and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in out fn n 3 we prove that any simple random walk in a non elementary finitely generated subgroup containing a weakly contracting element ends up in a non weakly contracting element with exponentially decaying probability also we show that each weakly contracting element is contained in a hyperbolically embedded elementary subgroup

divergence in right angled coxeter groups let w be a 2 dimensional right angled coxeter group we characterise such w with linear and quadratic divergence and construct right angled coxeter groups with divergence poly nomial of arbitrary degree our proofs use the structure of walls in the davis complex

generalised structured models summary we present a general class of nonlinear regression and time series models that we call generalised structured models the class is a natural generalisation of generalised additive models and it includes generalised interaction models structured volatility models visual garch generalised autoregressive conditional heteroscedasticity models and varying coefficient models we discuss estimation principles including smoothing splines and a generalisation of the projection approach of mammen et al 1999 we finish the paper with some theoretical considerations about the asymptotic performance of the estimator for the general class of generalised structured models

thick metric spaces relative hyperbolicity and quasi isometric rigidity we study the geometry of non relatively hyperbolic groups generalizing a result of schwartz any quasi isometric image of a non relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup this implies that a group being relatively hyperbolic with non relatively hyperbolic peripheral subgroups is a quasi isometry invariant as an application artin groups are relatively hyperbolic if and only if freely decomposable we also introduce a new quasi isometry invariant of metric spaces called metrically thick which is sufficient for a metric space to be non hyperbolic relative to any non trivial collection of subsets thick finitely generated groups include mapping class groups of most surfaces outer automorphism groups of most free groups certain artin groups and others non uniform lattices in higher rank semisimple lie groups are thick and hence non relatively hyperbolic in contrast with rank one which provided the motivating examples of relatively hyperbolic groups mapping class groups are the first examples of non relatively hyperbolic groups having cut points in any asymptotic cone resolving several questions of drutu and sapir about the structure of relatively hyperbolic groups outside of group theory teichmuller spaces for surfaces of sufficiently large complexity are thick with respect to the weil peterson metric in contrast with brock farb s hyperbolicity result in low complexity

projected partial likelihood and its application to longitudinal data summary an estimating equation which we call the projected partial score is introduced for longitudinal data analysis the estimating equation is obtained by projecting the partial likelihood score function onto the vector space spanned by a class of conditionally linear estimating equations we demonstrate that removing certain terms from the projection of the full likelihood score does not alter important inferential properties of the estimating equation and doing so is advantageous in handling missing data and time varying covariates within a prequential frame of reference it is shown that the estimating equation is optimal among the largest collection of estimating equations determined by the conditional moments furthermore the method possesses similar properties to generalised estimating equations in particular the correct conditional variance specification is necessary for efficiency but not for asymptotic consistency and distribution theory

a dividing line within simple unstable theories abstract we give the first zfc dividing line in keisler s order among the unstable theories specifically among the simple unstable theories that is for any infinite cardinal for which there is 2 we construct a regular ultrafilter d on so that i for any model m of a stable theory or of the random graph m d is saturated but ii if th n is not simple or not low then n d is not saturated the non saturation result relies on the notion of flexible ultrafilters to prove the saturation result we develop a property of a class of simple theories called qr 1 generalizing the fact that whenever b is a set of parameters in some sufficiently saturated model of the random graph b and 2 then there is a set a with a so that any nonalgebraic p s b is finitely realized in a in addition to giving information about simple unstable theories our proof reframes the problem of saturation of ultrapowers in several key ways we give a new characterization of good filters in terms of excellence a measure of the accuracy of the quotient boolean algebra we introduce and develop the notion of moral ultrafilters on boolean algebras we prove a so called separation of variables result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more set theoretic stage building an excellent filter followed by a more model theoretic stage building so called moral ultrafilters on the quotient boolean algebra a process which highlights the complexity of certain patterns arising from first order formulas in certain boolean algebras

age dependent u shaped risk functions and aalen s additive risk model epidemiologic studies of time to some failure often show a quadratic relation between the risk of failure and covariate s we study the nadir for a given covariate i e the value of the covariate associated with the lowest risk supposing a u shape within aalen s additive risk model this model was applied since the effect of the covariate s is allowed to vary over time and as a consequence a given nadir can vary over time we propose a test for the null hypothesis that the nadir is time independent and if this is the case an estimate of the nadir large sample properties of the test statistic and estimator are derived the methods are illustrated with data where time to death is related to body mass index

divergence in lattices in semisimple lie groups and graphs of groups divergence functions of a metric space estimate the length of a path connecting two points a b at distance 3 s is a finite set of valuations of a number field k including all infinite valuations and o s is the corresponding ring of s integers

a nonparametric dynamic additive regression model for longitudinal data in this work we study additive dynamic regression models for longitudinal data these models provide a flexible and nonparametric method for investigating the time dynamics of longitudinal data the methodology is aimed at data where measurements are recorded at random time points we model the conditional mean of responses given the full internal history and possibly time varying covariates we derive the asymptotic distribution for a new nonparametric least squares estimator of the cumulative time varying regression functions based on the asymptotic results confidence bands may be computed and inference about time varying coefficients may be drawn we propose two estimators of the cumulative regression function one estimator that involves smoothing and one that does not the latter however has twice the variance as the smoothing based estimator goodness of fit of the model is considered using martingale residuals finally we also discuss how partly conditional mean models in which the mean of the response is regressed onto selected time varying covariates may be analysed in the same framework we apply the methods to longitudinal data on height development for cystic fibrosis patients

a linear regression model for the analysis of life times a linear model is suggested for the influence of covariates on the intensity function this approach is less vulnerable than the cox model to problems of inconsistency when covariates are deleted or the precision of covariate measurements is changed a method of non parametric estimation of regression functions is presented this results in plots that may give information on the change over time in the influence of covariates a test method and two goodness of fit plots are also given the approach is illustrated by simulation as well as by data from a clinical trial of treatment of carcinoma of the oropharynx

existence of optimal ultrafilters and the fundamental complexity of simple theories abstract in the first edition of classification theory the second author characterized the stable theories in terms of saturation of ultrapowers prior to this theorem stability had already been defined in terms of counting types and the unstable formula theorem was known a contribution of the ultrapower characterization was that it involved sorting out the global theory and introducing nonforking seminal for the development of stability theory prior to the present paper there had been no such ultrapower characterization of an unstable class in the present paper we first establish the existence of so called optimal ultrafilters on suitable boolean algebras which are to simple theories as keisler s good ultrafilters 15 are to all first order theories then assuming a supercompact cardinal we characterize the simple theories in terms of saturation of ultrapowers to do so we lay the groundwork for analyzing the global structure of simple theories in zfc via complexity of certain amalgamation patterns this brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability

further results on the non parametric linear regression model in survival analysis this paper gives further developments of a non parametric linear regression model in survival analysis three subjects are studied first martingale residuals originally developed for the cox model are introduced for our linear model their theory is developed and they are shown to be useful for judging goodness of fit the second focus of the paper is on the use of bootstrap replications to judge which features of the cumulative regression plots are likely to reflect real phenomena and not merely random variation in particular this is applied to judging whether the effect of a covariate disappears over time a problem for which no formal test exists the third subject is density type or kernel estimation of the regression functions themselves this might give more direct information than the cumulative plots the approaches are illustrated by data from a clinical trial of carcinoma of the oropharynx and by survival times of grafts in renal patients

a partly parametric additive risk model summary aalen s additive risk model allows the influence of each covariate to vary separately over time although allowing greater flexibility of temporal structure than a cox model aalen s model is more limited in the number of covariates it can handle we introduce a partly parametric version of aalen s model in which the influence of only a few covariates varies nonparametrically over time and that of the remaining covariates is constant efficient procedures for fitting this new model are developed and studied the approach is applied to data from the medical research council s myelomatosis trials it is the first step of a taylor series expansion of a general hazard function about the zero of the covariate vector however in estimating the unknown functions in such a general model there is a variance bias trade off that may be critical in small and medium samples also after fitting the model one does not have parameters or formulae that are easily reported we propose a model that takes the additive structure of aalen s model and imposes parametric constraints to obtain a semiparametric submodel which may be more appropriate in some applications the model will be illustrated with data from clinical trials on myelomatosis covariates include treatment sex and four age strata which will be treated parametrically together with serum levels of haemoglobin and f32 microglobulin whose effects will be investigated nonparametrically the additive form can be interpreted loosely in terms of unobserved competing risks since the hazard function for the minimum of independent random vari ables is the sum of the hazard functions for the individual variables microglobulin levels are related to kidney function and tumour mass whereas haemoglobin is unaffected by kidney function hence one might anticipate that the hazard function associated with each covariate represents a different cause of death

algorithmic problems in right angled artin groups complexity and applications abstract in this paper we consider several classical and novel algorithmic problems for right angled artin groups some of which are closely related to graph theoretic problems and study their computational complexity we study these problems with a view towards applications to cryptography

weighted least squares estimation for aalen s additive risk model abstract cox s proportional hazards model has so far been the most popular model for the regression analysis of censored survival data however the additive risk model of aalen can provide a useful and biologically more plausible alternative aalen s model stipulates that the conditional hazard function for a subject whose covariates are y y 1 yp has the form h t y y t where 1 p is an unknown vector of hazard functions this article discusses inference for based on a weighted least squares wls estimator of the vector of cumulative hazard functions the asymptotic distribution of the wls estimator is derived and used to obtain confidence intervals and bands for the cumulative hazard functions both a grouped data and a continuous data version of the estimator are examined an extensive simulation study is carried out the method is applied to grouped data on the incidence of cancer mortality among japanese atomic bomb survivors

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures